Applied Numerical Mathematics 61 (January 2011)

Editors-in-Chief Robert Beauwens Facultés des Sciences Appliquées Service de Métrologie Nucléaire Université Libre de Br...

Author: Robert Beauwens | Martin Berzins

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Editors-in-Chief Robert Beauwens Facultés des Sciences Appliquées Service de Métrologie Nucléaire Université Libre de Bruxelles Ave. F.D. Roosevelt 50 1050 Brussels, Belgium Martin Berzins School of Computing University of Utah 50 S. Central Campus Dr., Rm 3190 Salt Lake City, UT 84112-9205, USA

Founding Editor and Editor Emeritus R. Vichnevetsky

Editor Emeritus Joseph E. Flaherty

Senior Editors M. Ainsworth Department of Mathematics Strathclyde University 26 Richmond Street Glasgow G1 1XH, UK Randolph Bank Department of Mathematics University of California at San Diego La Jolla, CA 92093-0001, USA Claude Brezinski Laboratoire d’Analyse Numérique et d’Optimisation UFR de Mathématiques Pures et Appliquées Université des Sciences et Technologies de Lille 59655 Villeneuve d’Ascq Cedex, France J.J. Dongarra Computer Science Department University of Tennessee Knoxville, TN 37996-1301, USA Peter K. Jimack School of Computer Studies University of Leeds Leeds, LS2 9JT, UK Ernest Mund Service de Métrologie Nucléaire Université Libre de Bruxelles Ave. F.D. Roosevelt 50 1050 Brussels, Belgium

J.M. Sanz-Serna Facultad de Ciencias Departamento de Matemática Aplicada y Computación Universidad de Valladolid Valladolid, Spain

Editorial Board Alvin Bayliss Department of Engineering Science and Applied Mathematics Northwestern University Evanston, IL 60201, USA Stefania Bellavia Dipartimento di Energetica “Sergio Stecco” University of Florence, via C. Lombroso 6/17 50134 Firenze, Italy J.C. Butcher Department of Mathematics The University of Auckland Private Bag 92019, Auckland, New Zealand Eric de Sturler Department of Mathematics Virginia Tech 544 McBryde Hall Blacksburg, VA 24061-0123, USA Dinh Nho Hào Hanoi Institute of Mathematics Vietnam Academy of Science and Technology 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam Alan Feldstein Department of Mathematics Arizona State University Tempe, AZ 85287, USA Anne Gelb Department of Mathematics and Statistics Arizona State University Tempe, AZ 85287-1804, USA Norbert Heuer Pontificia Universidad Católica de Chile, Santiago, Chile Willem Hundsdorfer MAS Center for Mathematics and Computer Science P.O. Box 94079 1090 GB Amsterdam, The Netherlands M.Y. Hussaini Program in Computational Science and Engineering Florida State University 411 Dirac Science Center Library Tallahassee, FL 32306-3075, USA Zdzislaw Jackiewicz Department of Mathematics Arizona State University Tempe, Arizona 85287, USA

Jens Lang Department of Mathematics Darmstadt University of Technology Darmstadt 64289, Germany Yvan Notay Brussels Free University Faculty of Applied Sciences ULB, CP 165/84 Ave. F. D. Roosevelt 50 Brussels, 1050 Belgium Jorg Peters Dept of C.I.S.E. University of Florida CSE Bldg Gainesville, FL 32611-6120, USA Alfio Quarteroni Institute of Analysis and Scientific Computing, CMCS – Modelling and Scientific Computing, Ecole Polytechnique Federale de Lausanne (EPFL), Av. Piccard, Station 8, 1015 Lausanne, Switzerland G. Richter Department of Computer Science Rutgers University New Brunswick, NJ 08903, USA Adrian Sandu Department of Computer Science Virginia Polytechnic Institute Blacksburg, VA 24061, USA Roger Temam Mathematics Department Indiana University Rawles Hall, 831 East Third Street Bloomington, IN 47405, USA S. Tsynkov Department of Mathematics North Carolina State University Box 8205 Raleigh, NC 27695, USA J. Vignes Laboratoire d’Informatique de Paris 6 U. Pierre et Marie Curie 4 Place Jussieu 75230 Paris Cedex 05, France Dongbin Xiu Department of Mathematics Purdue University, West Lafayette IN 47907, USA

© 2010 IMACS. Published by Elsevier B.V. All rights reserved Publication information: Applied Numerical Mathematics (ISSN 0168-9274). For 2011, volume 61 is scheduled for publication. A combined subscription to Applied Numerical Mathematics and Mathematics and Computers in Simulation at reduced rate is available. Subscription prices are available upon request from the Publisher or from the Elsevier Customer Service Department nearest you or from this journal’s website (http://www.elsevier.com/locate/apnum). Further information is available on this journal and other Elsevier products through Elsevier’s website: (http://www.elsevier.com). Subscriptions are accepted on a prepaid basis only and are entered on a calendar year basis. Issues are sent by standard mail (surface within Europe, air delivery outside Europe). Priority rates are available upon request. Claims for missing issues should be made within six months of the date of dispatch. Orders, claims, and journal enquiries: please contact the Elsevier Customer Service Department nearest you: St. Louis: Elsevier Customer Service Department, 3251 Riverport Lane, Maryland Heights, MO 63043, USA; phone: (877) 8397126 [toll free within the USA]; (+1) (314) 4478878 [outside the USA]; fax: (+1) (314) 4478077; e-mail: [email protected]. Oxford: Elsevier Customer Service Department, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK; phone: (+44) (1865) 843434; fax: (+44) (1865) 843970; e-mail: [email protected]. Tokyo: Elsevier Customer Service Department, 4F Higashi-Azabu, 1-Chome Bldg, 1-9-15 Higashi-Azabu, Minato-ku, Tokyo 106-0044, Japan; phone: (+81) (3) 5561 5037; fax: (+81) (3) 5561 5047; e-mail: [email protected]. Singapore: Elsevier Customer Service Department, 3 Killiney Road, #08-01 Winsland House I, Singapore 239519; phone: (+65) 63490222; fax: (+65) 67331510; e-mail: [email protected]. Advertising information: If you are interested in advertising or other commercial opportunities please e-mail [email protected] and your enquiry will be passed to the correct person who will respond to you within 48 hours.

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Applied Numerical Mathematics 61 (2011) 1–23

Contents lists available at ScienceDirect

Applied Numerical Mathematics www.elsevier.com/locate/apnum

A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations Fatemeh Shakeri, Mehdi Dehghan ∗ Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15914, Iran

a r t i c l e

i n f o

Article history: Received 14 July 2009 Received in revised form 10 May 2010 Accepted 26 July 2010 Available online 1 August 2010 Keywords: Unsteady magnetohydrodynamic equations Finite volume method Spectral element method Hermit interpolation Rectangular mesh Triangular mesh

a b s t r a c t In this paper, the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular section are solved using combined finite volume method and spectral element technique, improved by means of Hermit interpolation. The transverse applied magnetic field may have an arbitrary orientation relative to the section of the pipe. The velocity and induced magnetic field are studied for various values of Hartmann number, wall conductivity and orientation of the applied magnetic field. Comparisons with the exact solution and also some other numerical methods are made in the special cases where the exact solution exists. The numerical results for these sample problems compare very well to analytical results. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction Magnetohydrodynamic (MHD) is the study of the interaction of electrically conducting fluids and electromagnetic forces. The field of MHD was initiated by Swedish physicist, Hannes Alfvén for which he received in 1970 the Nobel Prize [1]. The official birth of incompressible fluid magnetohydrodynamics is 1936–1937. In 1937, Hartmann and Lazarus [37] studied the influence of a transverse uniform magnetic field on the flow of a viscous incompressible electrically conducting fluid between two infinite parallel stationary and insulating plates. The most appropriate name for the phenomena would be MagnetoFluidMechanics, but the original name Magnetohydrodynamics is still generally used. The study of MHD developed especially after 1950, when plasma could be reproduced in the laboratory. Smith [81] obtained upper and lower bounds for the mass flow rate for the steady flow of a conducting liquid along a pipe of arbitrary cross section under the influence of a uniform transverse magnetic field. Walker [91] and Holroyd [39] also made much effort in this field. Branover and Gershon [11] reported experimental effort on high Hartman number and interaction parameter flows. In 1979, Holroyd [40] presented the results of an experimental investigation of the flow of mercury along circular and rectangular non-conducting ducts in a non-uniform magnetic field at high Hartmann number. This author [41] also carried out a theoretical and experimental study on the flow of a liquid metal along a straight rectangular duct, whose pairs of opposite walls are highly conducting and insulating, situated in a planar non-uniform magnetic field parallel to the conducting walls. Magnitudes of the flux density and mean velocity were taken to be such that the Hartmann number and interaction parameter had very large values and the magnetic Reynolds number was extremely small. MHD problems arise in a wide variety of situations ranging from the explanation of the origin of Earth’s magnetic field and the prediction of space weather to the damping of turbulent fluctuations in semiconductor melts during crystal growth

*

Corresponding author. E-mail addresses: [email protected] (F. Shakeri), [email protected], [email protected] (M. Dehghan).

0168-9274/$30.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2010.07.010

2

F. Shakeri, M. Dehghan / Applied Numerical Mathematics 61 (2011) 1–23

and even the measurement of the flow rates of beverages in the food industry. The description of MHD flows involves both the equations of fluid dynamics, the Navier–Stokes equations, and the equations of electrodynamics, Maxwell’s equations, which are mutually coupled through the Lorentz force and Ohm’s law for moving electrical conductors [21]. Magnetohydrodynamic channel flows have considerable theoretical and practical importance because of the widespread applications in designing cooling systems with liquid metals, MHD generators, accelerators, nuclear reactors, blood flow measurements, pumps and flow-meters. Due to the coupling of the equations of fluid mechanics and electrodynamics, the equations governing MHD flows are rather cumbersome and exact solutions are, therefore, available only for some simple geometries subject to simple boundary conditions [17,27,34,37,48,72]. Therefore, some numerical techniques have been used to obtain the approximate solutions for the MHD flow problems. Singh and Lal [74–78] obtained numerical solutions of the MHD flows through pipes of various cross-section for Hartmann number less than 10 using the finite difference and finite element methods. Yagawa and Masuda [96] used an incremental finite element technique to study and analyze MHD flow and its application to liquid lithium blanket design of a fusion reactor. Later, Winowich and W.F. Hughes [93] analyzed a DC electromagnetic pump with insulating side walls using the finite element method. This time they also solved the complete Navier–Stokes equation with a non-uniform applied magnetic field. The finite difference method was employed by Ramos and Winowich [66] for solving electromagnetic pump. These authors [67] also applied finite element methodology to study MHD channel flow fields as a function of the Reynolds number, electrode length and the wall conductivity. It was shown that the axial velocity profile is distorted into M-shapes by the applied electromagnetic field and that the distortion increases as the Reynolds number and the electrode length increase. Hua and Walker [42] investigated the steady, laminar flow of a liquid metal in a circular duct with electrically insulating wall in the presence of a strong transverse non-uniform field. The interaction parameter and Hartmann number were assumed to be large, whereas the magnetic Reynolds number was assumed to be small. Sterl [82] used finite difference codes to investigate the influence of Hartmann number, interaction parameter, wall conductance ratio and changing magnetic field on the flow. Sezgin and Köksal [87] used the finite element method with linear and quadratic elements to moderate Hartmann numbers (up to 100). Meir [54] dealt with magnetohydrodynamic flows in pipes with arbitrary cross-section and arbitrary wall conductivities under the influence of a transverse magnetic field. Existence, uniqueness and finite element approximation of solutions to the equations of steady-state magnetohydrodynamics with arbitrary boundary conditions which describe this phenomenon were investigated in [54]. Seungsoo and Dulikravich [71] proposed a finite difference scheme [24,25] for threedimensional unsteady MHD flow together with temperature field. They used explicit Runge–Kutta method for step-by-step computations in time. M. Hughes, Pericleous and Cross [45], considered some models for three interesting magnetohydrodynamic flows, whereby predicted numerical solutions were compared with their analytical counterparts. The mathematical formulations adopted and the analytical solutions obtained were described together with the results of the computations. These authors also [46] used two-dimensional models under externally imposed permanent magnetic field to simulate the MHD flow in macroscopic scale. Later, Suwon Cho [83] studied several analytical solutions using the two-dimensional magnetic field analysis based on an equivalent current sheet model. These theoretical studies and numerical simulations predicted serious problems linked to the transformation of the velocity profile yielding the electromagnetic driving force in the opposite direction to the fluid motion especially at turbulent flow conditions. In [33], bi-cubic B-spline finite elements, which are memory consuming, were used for solving the MHD equations. In [26] an analytic finite element method was proposed for solving the governing equations of steady magnetohydrodynamic duct flows in the range of low and moderate Hartmann numbers M < 1000. Meir and Schmidt [55] considered the steady flow of a conducting fluid, confined to a bounded region of space and driven by a combination of body forces, externally generated magnetic fields, and currents entering and leaving the fluid through electrodes attached to the surface. By means of the Biot–Savart law, they reduced the problem to a system of integrodifferential equations in the fluid region, derived a mixed variational formulation, and proved its well-posedness under a small-data assumption and then studied the finite element approximation of solutions. Sheu and Lin [73] presented the convection-diffusion-reaction model for solving unsteady MHD flow applying a finite difference method on non-staggered grids with a transport scheme in each ADI (predictor–corrector) spatial sweep. Sezgin and Han Aydın [86] used the fundamental solution of Laplace equation in the dual reciprocity boundary element method solution of uncoupled MHD equations and approximated convective terms with osculatory functions. Barrett [4] obtained solution for high values of Hartmann number by using finite element method and very fine mesh within the Hartmann layers, which is computationally very expensive, time and memory consuming. Authors of [52] presented an interpolatory formulation for moving-least-squares approximants to study two-dimensional magnetohydrodynamic flow problem which allows the direct introduction of boundary conditions, reducing the processing time and improving the condition numbers. In [85], the magnetohydrodynamic flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct, with an external magnetic field applied transverse to the flow was investigated and an analytical solution was developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind. Neslitürk and Sezgin [58,59] solved MHD flow equations in rectangular ducts by using a stabilized finite element method with residual free bubble functions for M < 1000 and general wall conductivities. Authors of [35] introduced a finite element technique for solving the Maxwell equations in the MHD limit in heterogeneous domains. Authors of [69] presented a nonlinearly implicit, conservative numerical method for integration of the single-fluid resistive MHD equations. In [98], the space–time conservation element and solution element method was applied to solve the ideal MHD equations with special emphasis on satisfying the divergence


3

free constraint of magnetic field. Han and Tang [36] presented an adaptive moving mesh algorithm for the two-dimensional ideal magnetohydrodynamics equation that utilizes a staggered constrained transport technique to keep the magnetic field divergence-free. Mignone [56] proposed a new approximate Riemann solver for the equations of magnetohydrodynamics with an isothermal equation of state. In [10], the solution of magnetohydrodynamic duct flow problems with arbitrary wall conductivity was given using the boundary element method based on the time-dependent fundamental solution. Dehghan and Mirzaei [22] proposed a meshless local Petrov–Galerkin method to obtain the numerical solution of the coupled unsteady MHD flow through a pipe of rectangular section having arbitrary conducting walls. These authors also [23] applied the meshless local boundary integral equation method to solve the coupled unsteady MHD flow with non-conducting walls. In [32], finite volume schemes for the equations of ideal magnetohydrodynamics and based on splitting these equations into a fluid part and a magnetic induction part were designed. For more studies on MHD, the interested reader is referred to [6–9,43,44,46,49,60,64,65,68,70,84,97]. This paper presents an investigation in which the spectral element method and the finite volume technique are combined for solving MHD equations. Finite volume method has a long history as a class of important numerical tools for solving differential equations. This method which investigated in the early literature [88,89] and was known as the integral finite difference method, has proved particularly popular for simulating a wide range of important applications and physical processes such as fluid mechanics, meteorology, electromagnetics, semi-conductor device simulation, mass transfer or petroleum engineering and numerous models of biological processes [2,3,5,15,16,28,38,53,61,80,92,94,99]. Finite volume approximations rely on the local conservation property expressed by the differential equation and inherit the physical conservation laws of the original problems locally, hence this method can be expected to simulate corresponding physical phenomena effectively [31,51,90]. For many physical and engineering applications, such as heat transfer, transport phenomena and flows in porous media, this numerical conservation property is crucial. Finite volume element (FVE) method [12,13,18,29,30,47,95] is a combination of the standard finite volume method and the standard Galerkin finite element method. In finite volume element method, the governing partial differential equation is posed as a local conservation law on a set of control or finite volumes that partition the problem domain. The unknown FVE solution and known data are systematically discretized by a C 0 piecewise polynomial trial space that is defined on a finite element triangulation [18]. The FVE method can be viewed as either a systematic generalization of the finite volume method or as a locally conservative (Petrov–Galerkin) variant of the standard finite element method. In this study, for the purpose of increasing the accuracy of the solution, we combine the spectral element method with the finite volume technique in a manner similar to FVE method. The spectral element method [62] is a high order weighted residual technique which combines the geometrical flexibility of the finite element method and the high accuracy of spectral method. The spectral element method provides exponential convergence for problems with smooth solutions. The structure of the remainder of this paper is as follows: In Section 2, the governing equations under consideration are described. Section 3 presents a brief description of the general triangulation and volumization of the domain problem for applying the proposed method. In Section 4, the applied procedure is discussed and a time stepping method and the obtained linear system are outlined in Section 5. Section 6 reports some numerical results and comparisons to show efficiency and applicability of the new method for the current problem. A conclusion is drawn in Section 7. 2. Mathematical formulation of MHD equations The equations governing the motion of a conducting fluid in a straight pipe of uniform cross-section in presence of transverse magnetic field are [79]

∂BZ ∂BZ ∂V Z B0 − K F ( T ) = η ∇ 2 V Z + cos θ + sin θ , ∂T μ0 ∂X ∂Y ∂V Z ∂V Z ∂BZ 1 = ∇ 2 B Z + B 0 cos θ + sin θ , ∂T μ0 σ ∂X ∂Y

ρ

where

ρ , η , σ density, viscosity and conductivity of fluid, μ0 a constant = 4π × 10−7 in MKS system, X, Y , Z T

coordinates of a point with Z -axis along axis of the pipe and X -axis parallel to the applied magnetic field, time variable, θ orientation of applied magnetic field with X -axis, B0 applied magnetic field, V Z , B Z axial velocity and induced magnetic field, − K F ( T ) pressure gradient, ∇2 is the two-dimensional Laplacian operator.

(2.1) (2.2)

4


Fig. 1. A rectangular duct flow with general BC.

The boundary conditions (BC) depend strongly on the material of the bounding walls. The general BC which are suitable in engineering practice can be formulated as [17]

∂BZ σ BZ + = 0, σ h ∂N

V Z = 0,

(2.3)

is the outward normal to the boundary of the domain, σ and h are the wall conductance and the small wall where N thickness, respectively. The initial conditions depend upon how the motion starts initially. If it is assumed that initially the fluid is at rest and the motion starts by applying the constant pressure gradient, the initial conditions become V Z ( X , Y , 0) = 0,

B Z ( X , Y , 0) = 0.

(2.4)

Introducing the non-dimensional variables and parameters by

V = x=

VZ V0 X a

,

,

V0 = y=

ρ aV 0 R= , η

Y a

K a2

η

,

, M2 =

B=

BZ V 0 μ0

B 20 a2 σ

η

R m = V 0 aμ0 σ ,

,

t=

σ η

,

λ= T V0 a

σa , σ h

,

f (t ) = F

aT V0

,

the governing equations are reduced to

∂V ∂B ∂B , − f (t ) = ∇ 2 V + M cos θ + sin θ ∂t ∂x ∂y ∂V ∂V ∂B , = ∇ 2 B + M cos θ + sin θ Rm ∂t ∂x ∂y R

(2.5) (2.6)

in Ω × [0, ∞) with boundary conditions

V = 0,

on ∂Ω, ∂B + λ B = 0, on ∂Ω, ∂ n

(2.7) (2.8)

and initial conditions

V (x, y , 0) = 0,

(x, y ) ∈ Ω,

(2.9)

B (x, y , 0) = 0,

(x, y ) ∈ Ω,

(2.10)

where Ω represents the section of the pipe in non-dimensional form with ∂Ω as the boundary. The parameters M, R and R m are called the Hartmann number, Reynolds number and magnetic Reynolds number respectively. In Fig. 1, a rectangular duct flow with the introduced BC, is presented. In the limiting cases of perfectly insulating (σ = 0, λ = ∞) and conducting (σ = ∞, λ = 0) walls, the BC become V = B = 0 and V = ∂∂ nB = 0, respectively.


5

Fig. 2. The control volume corresponding to the point z (the colored polygon).

3. General triangulation and volumization In order to introduce the fundamental concepts of the proposed method, let Th denotes a triangulation of the domain Ω , partitioning of Ω into triangular or rectangular elements, assumed to be regular in the usual sense [20,50]. The mesh parameter h is the maximum of the diameters hk over all elements and T h is the p-fold refinement of Th that connects p

interpolation (nodal) points defined on each element in Th in the manner of [57]. Let Mh = {mi : i ∈ I } and M h = {n j : j ∈ J } be the sets of all nodal points of Th and T h , respectively and M h ( T ) = p

p

{n j : j ∈ J , n j ∈ T } be the vertices of T ∈ T h , where I and J are suitable index sets.

p

p

Now, we can construct a dual partition D h = { D z : z ∈ M h } corresponding to the triangulations T h , respectively in p

p

p

the following way [18,19]. Let z T be the circumcenter or barycenter of T ∈ T h . We connect z T with line segments to the p

midpoints of the edges of T , thus we partition T into some quadrilaterals K z , z ∈ M h ( T ). Then with each vertex z ∈ M h , p

p

we associate a control volume D z which is a closed polygon and consists of the union of the subregions K z , sharing the vertex z (Fig. 2), so there is a one-to-one correspondence between the interpolation points and finite volumes. 4. Methodology fundamentals The finite volume-spectral element (FVSE) method for solving MHD equations is based on the fact that these equations arise from the conservation laws. Finite volume approximations rely on the local conservation property expressed by the differential equation. The FVSE method is based on the Petrov–Galerkin formulation in which the solution space consists of continuous piecewise polynomial functions over the primal partition, Th , and the test space consists of piecewise constant functions over the dual partition, D h . The test space essentially conserves the local conservation property of the method. The approximate spaces we use are

p

Vh := v ∈ C 0 (Ω): v | T ∈ P p ( T ), ∀ T ∈ Th , W h := w ∈ L 2 (Ω): w | D z ∈ P 0 ( D z ), ∀ D z ∈ D h , p

p

where P p ( K ) is the set of polynomials of degree p within each spectral element or control volume K and in the cases of rectangular and triangular partitions is specified as

P p ( K ) = span xi y j : 0 i , j p , (x, y ) ∈ K ,

(4.1)

and

P p ( K ) = span xi y j : 0 i + j p , (x, y ) ∈ K , respectively. Obviously, W h = span{χz : z ∈ M h }, where

χz (x, y ) =

p

1, 0,

p

(4.2)

χz is the characteristic function of volume D z defined by

if (x, y ) ∈ D z , otherwise,

and Vh = span{ψz : z ∈ M h }, where ψz is the nodal basis function associated with the node z will be defined later in this p

section. Given an interpolation node z ∈ M h , integrating (2.5) and (2.6) over the associated control volume D z and using the p

Green’s formula, we obtain

6


R Dz

∂V dD = ∂t

∇ V . n dΓ + Dz

∂ Dz

∂B dD = ∂t

Rm

∂B ∂B M cos θ + sin θ ∂x ∂y

Dz

∂V ∂V M cos θ + sin θ ∂x ∂y

∇ B . n dΓ +

+ f (t ) dD ,

(4.3)

dD ,

(4.4)

Dz

∂ Dz

denotes the unit outward normal vector on ∂ D z , the boundary of D z . where n Now, the FVSE method is to find V p , B p ∈ Vh such that

R Dz

Rm

∂Vp dD = ∂t

∂Bp ∂Bp M cos θ + sin θ ∂x ∂y

∇ V p . n dΓ + Dz

∂ Dz

∂Bp dD = ∂t

Dz

∇ B p . n dΓ +

∂Vp ∂Vp M cos θ + sin θ ∂x ∂y

+ f (t ) dD ,

(4.5)

dD .

(4.6)

Dz

∂ Dz

The FVSE method is viewed as a perturbation of the spectral element method with the help of an interpolation operator I h∗ : Vh → W h defined by p

I h∗ u =

u ( z)χz .

z∈M h

p

Then, the discrete FVSE problem is then written as follows: Find V p , B p ∈ Vh such that for all w ∈ Vh

a V p , B p , I h∗ w = − f , I h∗ w , b V p , B p , I h∗ w = 0.

(4.7)

Here, a(., ., I h∗ .), b(., ., I h∗ .) and

∗

a u, v , Ih w =

( f , I h∗ .) are defined as follows:

z∈M h

p

+

∇ u . n I h∗ w dΓ

∂ Dz

M cos θ

z∈M h D p z

∗

b u, v , Ih w =

z∈M h

p

+

∗

f , Ih u =

∂v ∂v + sin θ ∂x ∂y

−R

∂u ∗ I h w dD , ∂t

∀u , v , w ∈ Vh ,

(4.8)

∇ v . n I h∗ w dΓ

∂ Dz

M cos θ

z∈M h D p z

f I h∗ u dΩ,

∂u ∂u + sin θ ∂x ∂y

− Rm

∂v ∗ I h w dD , ∂t

∀u , v , w ∈ Vh ,

∀u ∈ Vh .

(4.9)

(4.10)

Ω

FVSE procedure is usually easier to implement than the spectral element procedure and offers most of the advantages of flexibility for handling complicated domain geometries. More important, the test space W h ensures the local conservation p

of the numerical flux over each computational element. Thus, this method is highly desirable in computational conservation laws. For the unsteady MHD flow through a pipe of rectangular section Ω = I x × I y , I x = [ax , b x ], I y = [a y , b y ] which we consider here, we describe non-uniform rectangular and also triangular partitions of Ω , briefly in the next section. 4.1. Rectangular mesh Firstly, we consider the rectangular partition of Ω . In order to take the advantage of the properties of spectral methods, N +1 we have divided Ω into N x × N y non-overlapping rectangular elements using Gauss–Lobatto–Legendre (GLL) points {xi }i =x1


7

Fig. 3. The filled squares and the filled squares together with the filled circles represent the nodal points related to the Th and T h meshes, respectively. p

The control volume associated to each point is denoted by a black lined rectangle. N y +1

and { y j } j =1 in the x and y directions, respectively. GLL points are placed at zeros of the completed Lobatto polynomials [63] defined on the interval ζ ∈ [−1, 1] as c 2 Lm +1 (ζ ) = (1 − ζ ) L m (ζ ),

(ζ ) denotes the derivative of the mth Legendre polynomial. For getting these points in the arbitrary interval [a, b ], where L m the shifted Legendre polynomials are used [14]. Regarding (4.1), the total number of nodal points in each element in this case is ( p + 1)2 , hence once the base triangulation Th is defined, ( p + 1) × ( p + 1) interpolation points are considered in p +1 p +1 each element [xi , xi +1 ] × [ y j , y j +1 ] using again GLL points {xri }r =1 and { y sj }s=1 and therefore each rectangle in Th is further subdivided into p × p non-overlapping rectangles which form T h , the p-fold refinement of Th . Note that considering the p +1

nodal points as described above, we have x1i = xi , xi Let s− 12

yj

(xri ,

y sj )

= y sj −

p

be an interpolation point of T h . Consider

k sj−1 2

p

s+ 12

, yj

= y sj +

k sj 2

p +1

= xi +1 , y 1j = y j , y j hri

=

xri +1

−

xri ,

k sj

= y j +1 . r − 12

= y sj+1 − y sj , xi

= xri −

hri −1 , 2

r + 12

xi

= xri +

hri 2

,

, then

r − 12 r + 12 s− 1 s+ 1 , y ∈ yj 2, yj 2 , Λrs , xi i j = (x, y ): x ∈ xi is a control volume or dual element of node (xri , y sj ). For boundary nodes, their control volumes should be modified correspondingly. For example,

1 1+ 12 1+ 1 , y ∈ y 11 , y 1 2 . Λ11 11 = (x, y ): x ∈ x1 , x1 The control volumes Λrs , i = 1, . . . , N x + 1, j = 1, . . . , N y + 1, r = 1, . . . , p + 1 and s = 1, . . . , p + 1, form the dual partition ij D h . In Fig. 3, the domain problem initially has been divided into 4 × 4 rectangles using filled squares corresponding 5 × 5 p

GLL points (Th ) and then each element has been partitioned into 2 × 2 elements using 3 × 3 GLL points (T h ). Also the p

control volume associated to each point, according to previous paragraph, has been denoted. One may consider T h as the tensor product of Γx and Γ y , the partitions of the intervals I x and I y , respectively, which p

are another representations of the nodal points xri and y sj as follows σ =b , Γσ : aσ = β1σ < β2σ < β3σ < · · · < β M σ σ

where σ = x, y and M σ = N σ p + 1. p Let θk,q (ζ ), k = 1, . . . , N σ and q = 1, . . . , p + 1 be the Lagrange interpolation polynomials of degree p in the interval q p +1 (σk , σk+1 ), corresponding to the point σk , through the p + 1 GLL points, {σkl }l=1 , in this interval, defined by p

θk,q (ζ ) =

1

(ζ 2 − 1) Lk , p (ζ )

p ( p + 1) L k, p (σ q )(ζ − σ q ) k k

,

8


σ = x, y and Ln ,m denotes the derivative of the mth shifted Legendre polynomial in the (σn , σn+1 ). We may then

where

p

p

write down discrete representations V N x N y and B N x N y for the velocity and induced magnetic field, respectively, as follows: p V N x N y (ξ,

η, t ) =

My Mx

y

(4.11)

y

(4.12)

v i j (t )ψix (ξ )ψ j (η),

i =1 j =1

p

B N x N y (ξ, η, t ) =

My Mx

b i j (t )ψix (ξ )ψ j (η),

i =1 j =1

where ψkσ is the global interpolation function corresponding to βkσ which according to the spectral element method [63], is p

continuous piecewise polynomial function by piecing together the local basis functions θi , j s and is denoted as follows. Let 1 k N σ and 1 k p + 1 such that βkσ = σkk : if k = 1 and k = 1,

⎧ p ⎪ ⎨ θk −1, p +1 (ζ ), σk −1 < ζ < σk , ψkσ (ζ ) = θ p (ζ ), σk < ζ < σk +1 , ⎪ ⎩ k ,1 0, otherwise,

(4.13)

if k = p + 1 and k = M σ ,

⎧ p ⎪ ⎨ θk , p +1 (ζ ), σk < ζ < σk +1 , σ ψk (ζ ) = θ p ⎪ k +1,1 (ζ ), σk +1 < ζ < σk +2 , ⎩ 0, otherwise,

else

ψkσ (ζ ) =

(4.14)

p

θk ,k (ζ ), σk < ζ < σk +1 , 0,

(4.15)

otherwise.

Regarding the above notations and also (4.5) and (4.6) with D z = Λrs , i = 1, . . . , N x + 1, j = 1, . . . , N y + 1, r = 1, . . . , p + 1 ij and s = 1, . . . , p + 1, we have

R

My Mx ∂ v kl k =1 l =1

∂t

y

ψkx (ξ )ψl (η) dΛ =

My Mx

v kl (t )

k =1 l =1

Λrs ij

y ∇ ψkx (ξ )ψl (η) . n dΓ

∂Λrs ij

+ M cos θ

My Mx

k =1 l =1

+ M sin θ

My Mx

y ψkx (ξ )ψl (η) dΛ

Λrs ij

y ψkx (ξ ) ψl (η) dΛ

bkl (t )

k =1 l =1

Λrs ij

+ f (t )

bkl (t )

dΛ,

(4.16)

Λrs ij

Rm

My Mx ∂ bkl k =1 l =1

∂t

Λrs ij

y ψkx (ξ )ψl (

η ) dΛ =

My Mx

bkl (t )

k =1 l =1

+ M cos θ

y ∇ ψkx (ξ )ψl (η) . n dΓ

∂Λrs ij My Mx

v kl (t )

k =1 l =1

+ M sin θ

My Mx k =1 l =1

y ψkx (ξ )ψl (η) dΛ

Λrs ij

v kl (t ) Λrs ij

y ψkx (ξ ) ψl (η) dΛ,

(4.17)


9

Fig. 4. The filled squares represent the nodal points in parent element.

where (ψiσ ) is the derivative of ψiσ with respect to σ , σ = x, y. To facilitate the computation of the mentioned integrations, we can map the element K = {(x, y ) | a1 x b1 , a2 y b2 } onto a parent element

Kˆ = (ξ, η) −1 ξ,

η1 ,

achieved using the following mapping

ϕ : K −→ Kˆ ,

ϕ (x, y ) =

2x − a1 − b1 2 y − a2 − b2 b 1 − a1

,

b 2 − a2

(4.18)

.

4.2. Triangular mesh In this case, we partition the domain Ω into N triangular elements. For simplicity, we map each physical triangle from the xy plane to a right isosceles (parent) triangle in the ξ η parametric plane where ξ, η > −1 and ξ + η < 0 and then using Jacobian matrix and obtained computations for parent element, we get the computations for physical triangle [63]. Considering (4.2), the total number of nodal points in the case of triangular mesh in each element is l = 12 ( p + 1)( p + 2). To generate the nodal points on the parent triangle, consider a one-dimensional master grid v i , i = 1, 2, . . . , p + 1, where v i s positioned at the zeros of the p + 1 degree completed Lobatto polynomial. The nodes are then identified by the coordinates (ξk , ηq ) = ( v k , v q ), where k = 1, 2, . . . , p + 1 and q = 1, 2, . . . , p + 2 − k. In Fig. 4, the parent element and nodal points for p = 4 are shown. Also we assign to each index (k, q), k = 1, 2, . . . , p + 1, q = 1, 2, . . . , p + 2 − k, the node number k−2 i = q + s=0 ( p + 1 − s). p The local interpolation function θk,q corresponding to nodal point (ξk , ηq ), is a pth degree polynomial in ξ and η , required to satisfy the l interpolation conditions p

θk,q (ξk , ηq ) = δkk δqq ,

k = 1, 2, . . . , p + 1, q = 1, 2, . . . , p + 2 − k .

(4.19)

In the most general approach, the local interpolation functions are expressed as linear combinations of a set of l independent polynomials that forms a complete base of the pth order expansions in the ξ η plane, φ j (ξ, η), for j = 1, 2, . . . , l, i.e. p

θk,q (ξ, η) =

l

j j =1 ck,q φ j (ξ,

η), where ckj,q comprise a set of l expansion coefficients for node (ξk , ηq ). Enforcing the Kroneck-

er’s delta condition (4.19), we obtain the linear system V.c = ei , where i is the node number corresponding to (k, q), V is the generalized Vandermonde matrix whose (r , s) entry is the value of φs at the rth node, c = [ck1,q ck2,q . . . ckl−,q1 ckl ,q ] T and p

ei is a l × 1 vector whose ith entry is 1 and zero elsewhere. Solving this system, the local interpolation function θk,q is identified. To construct the control volume corresponding to node (ξk , ηq ), for q = p + 2 − k, consider the control volume in the same manner as rectangular partition, for q = p + 2 − k, k = 1 and k = p + 1, let

ηq − ηq−1 ξk − ξk−1 Λkq = (ξ, η): ηq − ξ, ξk − η, ξ + η < 0 , 2

2

(4.20)

10


Fig. 5. A global interpolation function corresponding to ith point, located on the boundary of an element of Th before applying Hermit interpolation.

for q = p + 2 − k and k = 1, let

Λkq = (ξ, η): ηq −

ηq − ηq−1 2

ξ, ξk η, ξ + η < 0 ,

(4.21)

and for q = p + 2 − k and k = p + 1, let

ξk − ξk−1 Λkq = (ξ, η): ηq ξ, ξk − η, ξ + η < 0 .

(4.22)

2

Such as previous section, for the nodal points located at the boundary of an element, the union of the interpolation functions corresponding to adjacent elements and for the interior functions yield global M nodes, the interpolation themselves, p p M interpolation functions. Then we can consider V N (ξ, η, t ) = i =1 v i (t )ψi (ξ, η), and B N (ξ, η, t ) = i =1 b i (t )ψi (ξ, η), as approximations of V and B, respectively, where ψi , is the global interpolation function corresponding to ith point and M is the total number of nodal points in the whole domain. p

Notice that Eqs. (4.16) and (4.17) are satisfied for triangular mesh by replacing V N x N y (ξ, η, t ) = y p ψix (ξ )ψ j ( ) and B N x N y (ξ, , t ) M i =1 b i (t )ψi (ξ, ), respectively.

η

η

=

M x M y i =1

y b (t )ψix (ξ )ψ j ( j =1 i j

η) with

p V N (ξ,

η, t ) =

M

i =1

M x M y i =1

v i (t )ψi (ξ, η) and

v (t ) × j =1 i j p B N (ξ, , t ) =

η

η

Remark. It is expected that increasing the number of elements in Th , increases the accuracy of the solution, but the numerical results show that the error in the boundary of elements of Th grows by increasing the number of elements in Th . Because regarding the global interpolation functions as described before, causes sharp points to be formed at the boundary of elements of Th which tends to effect the differentiability of the global basis functions and results some errors in these points. For the purpose of handling this problem, we use a Hermit interpolation at a strip of width 2 around the boundary of elements of Th to slight the behavior of the basis functions in these points and make them differentiable. The is considered such small that has a little effect on the interpolation basis functions. The numerical results in our simulations show that smaller gives more accurate results, especially when p increases, but notice that this reduction of the size of , continues until no underflow error in computations occurs. A global interpolation function before and after applying this improvement for the case of rectangular mesh which is product of global interpolation functions in one dimension, are displayed in Figs. 5 and 6, respectively. 5. Time discretization In this section, we describe the time discretization for the case of rectangular partition. The time discretization for the triangular mesh is quite similar and performed with small changes only in notations. The time discretization, applied here, is performed with the Crank–Nicolson discretization, which is an implicit method. The main reason for choosing this method is the high order of convergence. Also theoretically, the Crank–Nicolson method is unconditionally stable. In this technique, a time step, t is selected and then the differential equations (4.16) and (4.17) at time t + 12 t are evaluated. The time derivative is approximated with a centered finite difference. For the rest terms, the average of times t and t + t is considered. Therefore Eqs. (4.16) and (4.17) can be written as:


11

Fig. 6. A global interpolation function corresponding to ith point, located on the boundary of an element of Th after applying Hermit interpolation. Table 1 Comparison of velocity field of Shercliff’s problem at M = 20 using MLPG method [22], FVE method [12,13,18,29] and FVSE method with p = 5 in rectangular partition.

(x, y )

Exact value

FVE method with Nx = N y = 7

FVE method with N x = N y = 10


FVSE method with Nx = N y = 7

FVSE method with N x = N y = 10

MLPG method with N = 1681

(0.00, 0.00) (0.25, 0.00) (0.50, 0.00) (0.00, 0.25) (0.25, 0.25) (0.50, 0.25) (0.00, 0.50) (0.25, 0.50) (0.50, 0.50) (0.00, 0.75) (0.50, 0.75)

0.049918641 0.049880236 0.049760102 0.049662783 0.049570651 0.049299034 0.047716857 0.047452918 0.046677531 0.037657703 0.036166028

0.060535196 0.049914241 0.042771026 0.059340793 0.049190306 0.042257986 0.056345686 0.047031664 0.040559079 0.040699955 0.030529006

0.050154180 0.049908275 0.049399719 0.049763373 0.049474403 0.048822959 0.047917937 0.047508053 0.046447175 0.037972159 0.036241337

0.049949831 0.049931721 0.049810629 0.049673904 0.049653541 0.049311106 0.047654403 0.047395908 0.046513455 0.037475349 0.036002088

0.049904402 0.049869511 0.049777626 0.049652578 0.049563709 0.049319260 0.047704620 0.047443914 0.046693784 0.037646269 0.036174089

0.049904635 0.049866591 0.049745566 0.049646821 0.049555008 0.049282439 0.047697841 0.047434150 0.046657809 0.037637608 0.036145676

0.049921141 0.049883013 0.049764512 0.049684382 0.049593666 0.049327163 0.047785558 0.047520638 0.046745300 0.037763549 0.036262629

Table 2 CPU time for solving Shercliff’s problem at M = 20 using MLPG method [22], FVE method [12,13,18,29] and FVSE method with p = 5 in rectangular partition. FVE method with Nx = N y = 7






17.375000000 s

27.218750000 s

128.125000000 s

36.046875000 s

126.765625000 s

919.846000000 s

Table 3 Velocity field of Shercliff’s problem at M = 20 using FVSE method with N x = N y = 10 and different values of p in rectangular partition.

(x, y )

Exact value

p=3

p=4

p=5

(0.00, 0.00) (0.25, 0.00) (0.50, 0.00) (0.00, 0.25) (0.25, 0.25) (0.50, 0.25) (0.00, 0.50) (0.25, 0.50) (0.50, 0.50) (0.00, 0.75) (0.50, 0.75)

0.049918641 0.049880236 0.049760102 0.049662783 0.049570651 0.049299034 0.047716857 0.047452918 0.046677531 0.037657703 0.036166028

0.049897425 0.049864621 0.049668870 0.049675969 0.049590600 0.049249341 0.047756709 0.047499074 0.046659524 0.037710746 0.036190621

0.049894502 0.049853383 0.049723799 0.049631527 0.049536675 0.049255528 0.047672973 0.047406598 0.046623227 0.037619383 0.036120422

0.049904635 0.049866591 0.049745566 0.049646821 0.049555008 0.049282439 0.047697841 0.047434150 0.046657809 0.037637608 0.036145676

Table 4 CPU time for solving Shercliff’s problem at M = 20 using FVSE method with N x = N y = 10 and different values of p in rectangular partition. p=3

p=4

p=5

36.468750000

43.406250000 s

126.765625000 s

12


Table 5 Comparison of velocity field of Shercliff’s problem at M = 5 using MLPG method [22], FVE method [12,13,18,29] and FVSE method with p = 5 in triangular partition.

(x, y )

Exact value

FVE method N = 98

FVE method N = 162

FVE method N = 722

FVSE method N = 98

FVSE method N = 162

MLPG method N = 441

(0.00, 0.00) (0.25, 0.00) (0.50, 0.00) (0.00, 0.25) (0.25, 0.25) (0.50, 0.25) (0.00, 0.50) (0.25, 0.50) (0.50, 0.50) (0.25, 0.75) (0.50, 0.75) (0.75, 0.75)

0.171601814 0.168372009 0.155787639 0.164754886 0.161621571 0.149458308 0.141207698 0.138504000 0.128048264 0.089937761 0.083498372 0.064544256

0.170217438 0.165892367 0.154134853 0.158711395 0.154749438 0.143480875 0.130773635 0.127904263 0.119247020 0.079360255 0.074544113 0.063581548

0.170829389 0.166395134 0.154428858 0.159481686 0.155395527 0.143952256 0.131991731 0.128900469 0.119616812 0.080439469 0.075135557 0.060219517

0.171508731 0.166994125 0.153954518 0.160365891 0.156257834 0.143932112 0.132898732 0.129751728 0.119841829 0.080865884 0.075216935 0.059579241

0.171600899 0.168369637 0.155783037 0.164755121 0.161621098 0.149455924 0.141209328 0.138505348 0.128048405 0.089940573 0.083502002 0.064442661

0.171556364 0.168326676 0.155742364 0.164718369 0.161585326 0.149422849 0.141184658 0.138481830 0.128027365 0.089930550 0.083492387 0.064547312

0.170849580 0.167642203 0.155128896 0.164090329 0.161035982 0.148874152 0.140772330 0.138115184 0.127831234 0.089835973 0.083378769 0.064740823

Table 6 CPU time for solving Shercliff’s problem at M = 5 using MLPG method [22], FVE method [12,13,18,29] and FVSE method with p = 5 in triangular partition. FVE method with N = 98

FVE method with N = 162


FVSE method with N = 98



3.828125000 s

8.718750000 s

134.343750000 s

113.968750000 s

132.171875000 s

31.628000000 s

Table 7 Velocity field of Shercliff’s problem at M = 5 using FVSE method with N = 162 and different values of p in triangular partition.

(x, y )

Exact value

FVSE method with p=3



(0.00, 0.00) (0.25, 0.00) (0.50, 0.00) (0.00, 0.25) (0.25, 0.25) (0.50, 0.25) (0.00, 0.50) (0.25, 0.50) (0.50, 0.50) (0.25, 0.75) (0.50, 0.75) (0.75, 0.75)

0.171601814 0.168372009 0.155787639 0.164754886 0.161621571 0.149458308 0.141207698 0.138504000 0.128048264 0.089937761 0.083498372 0.064544256

0.171361449 0.168054414 0.155169239 0.164501303 0.161333434 0.148947405 0.140937387 0.138240774 0.127716857 0.089742452 0.083310281 0.064580340

0.171525725 0.168300692 0.155724385 0.164681586 0.161546525 0.149391160 0.141145483 0.138439836 0.127985406 0.089896087 0.083451611 0.064493341

0.171556364 0.168326676 0.155742364 0.164718369 0.161585326 0.149422849 0.141184658 0.138481830 0.128027365 0.089930550 0.083492387 0.064547312

Table 8 CPU time for solving Shercliff’s problem at M = 5 using FVSE method with N = 162 and different values of p in triangular partition.

M y Mx

p=3

p=4

p=5

14.828125000 s

60.640625000 s

132.171875000 s

i j ,rs

R ωHkl

1

M y Mx

i j ,rs

R ωHkl

1

Rmω

i j ,rs Hkl

M y Mx k =1 l =1

i j ,rs

−

1 2

i j ,rs

R m ωHkl

i j ,rs Fkl

1

2

bnkl+1

i j ,rs

−

i j ,rs

M cos(θ)Skl

v nkl +

+ Fkl 2

1

2

k =1 l =1

=

v nkl+1 −

+ Fkl

k =1 l =1 M y Mx

2

k =1 l =1

=

i j ,rs

− Fkl

1 2

1 2

bnkl +

i j ,rs

M cos(θ)Skl

M

i j ,rs n+1 bkl

1 2

i j ,rs cos(θ)Skl

+ sin(θ)Kkl

2

i j ,rs + sin(θ)Kkl v nkl+1

2

(5.1)

i j ,rs n v kl

+ sin(θ)Kkl

1

+ Li j ,rs f n ,

+ sin(θ)Kkl

i j ,rs

M cos(θ)Skl

i j ,rs n bkl

1

− Li j ,rs f n+1

,

(5.2)


13

Table 9 Velocity field of Shercliff’s problem at M = 20 using FVSE method and rectangular partition with N x = N y = 10, p = 5 and different values of interpolation.

in Hermit

(x, y )

Exact value

= .005

= .001

= .00001

(0.00, 0.00) (0.25, 0.00) (0.50, 0.00) (0.75, 0.00) (0.00, 0.25) (0.25, 0.25) (0.50, 0.25) (0.75, 0.25) (0.00, 0.50) (0.25, 0.50) (0.50, 0.50) (0.75, 0.50) (0.00, 0.75) (0.25, 0.75) (0.50, 0.75) (0.75, 0.75)

0.049918641 0.049880236 0.049760102 0.049227495 0.049662783 0.049570651 0.049299034 0.048549629 0.047716857 0.047452918 0.046677531 0.045177836 0.037657703 0.037296934 0.036166028 0.033933507

0.042932354 0.042315264 0.040363575 0.036801070 0.041969000 0.041306934 0.039245376 0.035566689 0.038520710 0.037778903 0.035515679 0.031627987 0.028674921 0.028063657 0.026148522 0.022733414

0.048337155 0.048161076 0.047586800 0.046255595 0.047905480 0.047674085 0.046947189 0.045405510 0.045572049 0.045183013 0.044003650 0.041812838 0.035486822 0.035052056 0.033682656 0.031057899

0.049904635 0.049866591 0.049745566 0.049220916 0.049646821 0.049555008 0.049282439 0.048540801 0.047697841 0.047434150 0.046657809 0.045165251 0.037637608 0.037277029 0.036145676 0.033919548

Fig. 7. The contour plots of velocity (left) and induced magnetic field (right) for λ = ∞, M = 20 with N x = N y = 14 and p = 5 in rectangular partition.

where i, r, j and s are indices such that (xri , y sj ) is an interior node of Ω , v nkl = v kl (nt ), bnkl = bkl (nt ), f n = f (nt ),

ω=

1

t , and i j ,rs Hkl

y

ψkx (ξ )ψl (η) dΛ,

=

(5.3)

Λrs ij i j ,rs

Fkl

y ∇ ψkx (ξ )ψl (η) . n dΓ,

=

(5.4)

∂Λrs ij i j ,rs Skl

=

y ψkx (ξ )ψl (η) dΛ,

(5.5)

y ψkx (ξ ) ψl (η) dΛ,

(5.6)

Λrs ij i j ,rs

Kkl

= Λrs ij

14


Fig. 8. The contour plots of velocity (left) and induced magnetic field (right) for λ = ∞, M = 300 with N x = N y = 14 and p = 5 in rectangular partition.

Fig. 9. Plot of the velocity along the x-axis ( y = 0) with N x = N y = 14, p = 5 and for θ = 0, λ = ∞ and various values of M.

Li j ,rs =

dΛ.

(5.7)

Λrs ij y

If (xri , y sj ) = (βkx , βl ) ∈ ∂Ω , in the case of the Dirichlet boundary condition (2.7) (or the boundary condition (2.8) when

λ = ∞), it is not necessary to carry out the corresponding integration on Λrs in (5.1) ((5.2)) and we set v kl (t ) = 0 (bkl (t ) = 0) ij in the representation (4.11) ((4.12)). But considering the boundary condition (2.8), when λ = ∞, Eq. (5.2) is reduced to My Mx

i j ,rs

R ωHkl

1 2

i =1 j =1

=

My Mx i =1 j =1

where

i j ,rs

− Qkl i j ,rs

R ωHkl

1

1 2

i j ,rs

+ Qkl 2

i j ,rs

+ Pkl

1

bnkl+1 −

i j ,rs

− Pkl 2

1 2

i j ,rs

M cos(θ)Skl

bnkl +

1 2

i j ,rs

M cos(θ)Skl

i j ,rs n+1 v kl

+ sin(θ)Kkl

i j ,rs n v kl ,

+ sin(θ)Kkl

(5.8)


15

Fig. 10. Plot of the induced magnetic field along the x-axis ( y = 0) with N x = N y = 14, p = 5 and for θ = 0, λ = ∞ and various values of M.

Fig. 11. Sparseness structure of coefficient matrix for λ = ∞, M = 300 with N x = N y = 14, p = 5 in rectangular partition.

i j ,rs

Qkl

y ∇ ψkx (ξ )ψl (η) . n dΓ,

=

(5.9)

∂Λrs −∂Λrs ∩∂Ω ij ij i j ,rs

Pkl

=λ

y

ψkx (ξ )ψl (η) dΓ.

(5.10)

∂Λrs ∩∂Ω ij

Eqs. (5.1) and (5.2) (or (5.8)) constitute a system of linear algebraic equations of the unknowns v nkl+1 and bnkl+1 . Notice

0 0 = bkl = 0. Then, solving the obtained linear system that at first, using the initial conditions (2.9) and (2.10), we have v kl repeatedly, we get v nkl and bnkl , for n = 1, 2, . . . , until the steady state is achieved.

6. Case studies In the following test problems, we present our procedure by taking a square pipe |x| 1, | y | 1. The wall conductivity, λ, is taken uniform over the entire boundary. The computations have been carried out for various values of θ , λ and M. For all cases, we have chosen R = R m = 1. Also for the transient flow with constant pressure gradient considered here, we have f (t ) = 1. However, for pulsating flows f (t ) may be taken as a periodic function of time. The numerical results are obtained at steady state, which is considered when the absolute difference between the solutions of two consecutive time

16


Fig. 12. Sparseness structure of coefficient matrix for λ = ∞, M = 20 with N = 162, p = 5 in triangular partition.

Fig. 13. The contour plots of velocity (left) and induced magnetic field (right) for λ = 0, M = 20, with N x = N y = 18, p = 4.

levels is less than 10−6 . Also the numerical calculations are performed with t = 0.01. (All numerical simulations are run on a standard desktop computer, Pentium IV 2.8 GHz.) 6.1. Problem 1: duct with insulating walls, λ = ∞ and horizontal magnetic field, θ = 0 Shercliff [72] obtained the exact solution for the flow in an insulated square duct with an applied magnetic field parallel to one pair of sides. In the current paper, this solution is selected as a benchmark problem which can be defined as the solution of (2.5) and (2.6) when θ = 0, further B = V = 0 along the channel boundary, i.e. λ = ∞. Tables 1–2 and 5–6, display the numerical results and CPU time obtained by Meshless Local Petrov–Galerkin (MLPG) [22], FVE [12,13,18,29,30,47] and FVSE methods for solving MHD equations in rectangular and also triangular partitions (in the MLPG method, N is the number of randomly located nodes in the domain Ω , for more details, see [22]). We can see from these tables that our steady-state solutions by FVSE for the velocity and the induced magnetic field agree very well with the exact solution given by Shercliff [72] and is more accurate than FVE and also MLPG methods, even when FVE method is applied with N x = N y = 30 (N = 722) for which the CPU time is approximately equal to the FVSE method’s CPU time with N x = N y = 10 (N = 162) in rectangular (triangular) mesh. In Tables 3 and 7, the exact value of velocity and also FVSE approximation with N x = N y = 10 (N = 162) in rectangular (triangular) partition for different values of p are reported. The CPU time computed in applying this procedure is shown in


17

Fig. 14. Plot of the velocity along the x-axis ( y = 0) with N x = N y = 14, p = 5 for M = 20, θ = 0 and various values of λ.

Fig. 15. Plot of the induced magnetic field along the x-axis ( y = 0) with N x = N y = 14, p = 5 for M = 20, θ = 0 and various values of λ.

Tables 4 and 8. In Table 9, the effect of different values of in constructing Hermit interpolation in applying FVSE method is presented. The contour graphs of velocity and induced magnetic field for small and large values of Hartmann number M = 20 and M = 300 with N x = N y = 14 and p = 5 in rectangular partition are plotted in Figs. 7 and 8, respectively. These figures show that the velocity is symmetric with respect to both x- and y-axes. The induced magnetic field is antisymmetric with respect to y-axis and hence the current lines change their direction in the left and right parts of the ducts. The graphs for triangular mesh in this case and also other cases are similar to the rectangular partition. The velocity and induced magnetic field profiles along the x-axis, in the y = 0 plane of the duct, are plotted in Figs. 9 and 10, respectively for θ = 0, λ = ∞ and different values of Hartmann number M. It is noted from these figures that as M increases, the velocity becomes uniform and almost stagnant at the center of the duct and the boundary layer formation starts for both the velocity and the induced magnetic field. The velocity decreases taking a uniform value at the center of the duct and V has its maximum value through the center. This flattening tendency is observed in the magnetic field also. These are the well known characteristic of MHD flow. The location of nonzero entries in the coefficient matrix in the obtained linear algebraic system for λ = ∞, M = 300 (M = 20) with N x = N y = 14 (N = 162), p = 5 in rectangular (triangular) partition is presented in Fig. 11 (Fig. 12). For other test problems, the sparseness structure of the coefficient matrix is similar to this case.

18


Fig. 16. The contour plots of velocity (left) and induced magnetic field (right) M = 200, λ = ∞, θ = π4 with N x = N y = 14 and p = 5.

Fig. 17. The contour plots of velocity (left) and induced magnetic field (right) M = 200, λ = ∞, θ = π3 with N x = N y = 14 and p = 5.


19

Fig. 18. Plot of the velocity along the x-axis ( y = 0) for M = 20, λ = ∞ and various values of θ with N x = N y = 14 and p = 5.

Fig. 19. Plot of the induced magnetic field along the x-axis ( y = 0) for M = 20, λ = ∞ and various values of θ with N x = N y = 14 and p = 5.

6.2. Problem 2: duct with arbitrary wall conductivity, λ, 0 λ < ∞ and horizontal magnetic field, θ = 0 In this problem, at first, we consider the unsteady MHD flow in a duct with the perfect conductivity on the walls of the duct, i.e. λ = 0. The FVSE method is employed to the governing equations at M = 20, with N x = N y = 18 and p = 4. The contour graphs are displayed in Fig. 13. One can observe that when λ = 0, which is the high conducting wall case, the induced magnetic field contours are perpendicular to the walls. To see the effect of increase in λ, we present Figs. 14 and 15 for M = 20, θ = 0 and various values of conductivity parameter λ. As λ increases, the graphs along the x-axis ( y = 0), converge to the case λ = ∞ and show the behavior of solution of MHD flow with insulated walls. 6.3. Problem 3: duct under oblique magnetic field, θ > 0 Here, the MHD flow problems (2.5) and (2.6) by taking externally applied magnetic field making a positive angle θ with the x-axis are considered. The contour graphs for M = 200, λ = ∞, θ = π4 and θ = π3 with N x = N y = 14, p = 5 are plotted in Figs. 16 and 17, respectively. The boundary layers are concentrated near the corners in the direction of the applied oblique magnetic field for both the velocity and induced magnetic field. Figs. 18 and 19 are made with the purpose of demonstrating the effect of various values of θ along the x-axis ( y = 0), for M = 20, λ = ∞. As can be seen from these figures, the absolute value of the induced magnetic field along the x-axis is reduced when θ is increased from 0 to π2 .

20


Table 10 Comparison of velocity field of Shercliff’s problem at M = 0 using FVE method [12,13,18,29] and FVSE method with p = 4 in rectangular partition.

(x, y )

Exact value

FVE method with Nx = N y = 7





(0.00, 0.00) (0.25, 0.00) (0.50, 0.00) (0.75, 0.00) (0.00, 0.25) (0.25, 0.25) (0.50, 0.25) (0.75, 0.25) (0.00, 0.50) (0.25, 0.50) (0.50, 0.50) (0.75, 0.50) (0.00, 0.75) (0.25, 0.75) (0.50, 0.75) (0.75, 0.75)

0.294685413 0.278882332 0.229339626 0.139729128 0.278882332 0.264148031 0.217799304 0.133327705 0.229339629 0.217799304 0.181144632 0.112736689 0.139729128 0.133327705 0.112736685 0.072819791

0.302207785 0.271056627 0.231701020 0.134706103 0.271056627 0.243847323 0.209248350 0.122734219 0.231701020 0.209248350 0.180457147 0.107135393 0.134706103 0.122734219 0.107135393 0.066055943

0.284537549 0.270541100 0.225332898 0.137869228 0.270541100 0.257379628 0.214822788 0.131999258 0.225332898 0.214822788 0.180701477 0.112813110 0.137869228 0.131999258 0.112813110 0.073174537

0.292268617 0.277749251 0.227641472 0.138616943 0.277749251 0.264140337 0.217023600 0.132722756 0.227641472 0.217023600 0.179895157 0.111849409 0.138616943 0.132722756 0.111849409 0.072124775

0.293524315 0.277822912 0.228399600 0.139165561 0.277822912 0.263184058 0.216940686 0.132813676 0.228399600 0.216940686 0.180367959 0.112252718 0.139165561 0.132813676 0.112252718 0.072497677

0.294639646 0.278837322 0.229310695 0.139718764 0.278826724 0.264093068 0.217760813 0.133311067 0.229284974 0.217744528 0.181102961 0.112715493 0.139689585 0.133288009 0.112704285 0.072803408

Table 11 CPU time for solving Shercliff’s problem at M = 0 using FVE method [12,13,18,29] and FVSE method with p = 4 in rectangular partition. FVE method with Nx = N y = 7





13.296875000 s

23.546875000 s

46.062500000 s

25.218750000 s

49.703125000 s

Table 12 Comparison of velocity field of Shercliff’s problem at M = 0 using FVE method [12,13,18,29] and FVSE method with p = 4 in triangular partition.

(x, y )

Exact value






(0.00, 0.00) (0.25, 0.00) (0.50, 0.00) (0.75, 0.00) (0.00, 0.25) (0.25, 0.25) (0.50, 0.25) (0.75, 0.25) (0.00, 0.50) (0.25, 0.50) (0.50, 0.50) (0.75, 0.50) (0.00, 0.75) (0.25, 0.75) (0.50, 0.75) (0.75, 0.75)

0.294685413 0.278882332 0.229339626 0.139729128 0.278882332 0.264148031 0.217799304 0.133327705 0.229339629 0.217799304 0.181144632 0.112736689 0.139729128 0.133327705 0.112736685 0.072819791

0.272737607 0.270070641 0.232178096 0.147922581 0.246426412 0.246761771 0.214940575 0.138844587 0.193414412 0.195047613 0.174818965 0.116039194 0.112446646 0.113918187 0.103958632 0.077761183

0.276831690 0.274342165 0.236691537 0.151786413 0.250426204 0.249691638 0.217414055 0.141155358 0.196930994 0.197288581 0.174774201 0.116176105 0.114695307 0.115420152 0.104121521 0.073605998

0.283129869 0.280684993 0.242644202 0.156145942 0.256324801 0.254652832 0.221439015 0.143927349 0.201775194 0.200917054 0.176534715 0.117148959 0.117599170 0.117417161 0.104909256 0.072498974

0.294674456 0.278880091 0.229340906 0.139730493 0.278880005 0.264143559 0.217799796 0.133319897 0.229340630 0.217799601 0.181146698 0.112721197 0.139730163 0.133319626 0.112721037 0.073013319

0.294685929 0.278888301 0.229347874 0.139736015 0.278888173 0.264155549 0.217809201 0.133333345 0.229347469 0.217808893 0.181154245 0.112738381 0.139735503 0.133332849 0.112738061 0.072811937

6.4. Problem 4: duct with M = 0, λ, θ = arbitrary In this case, the velocity and induced magnetic field, are calculated for M = 0, i.e. hydrodynamic case in which results obviously will be independent of λ and θ . The exact solution of this problem has been given by Singh and Lal [76]. In Tables 10–13, the exact value of velocity for this case, FVE and FVSE approximations and CPU time for rectangular and triangular meshes are reported. The results for both methods, with N x = N y = 7 and N x = N y = 10 (N = 98 and N = 162) in rectangular (triangular) mesh are presented. In these two cases, the accuracy of FVSE method is higher than FVE method, but the FVE method uses less CPU time. Also there exists another column with N x = N y = 20 (N = 512) for FVE method for which the CPU time is approximately equal to FVSE method with N x = N y = 10 (N = 162) in rectangular (triangular) partition, but as can be seen, FVE’s accuracy has not yet achieved to FVSE’s accuracy.


21

Table 13 CPU time for solving Shercliff’s problem at M = 0 using FVE method [12,13,18,29] and FVSE method with p = 4 in triangular partition. FVE method with N = 98





5.500000000 s

8.031250000 s

55.359375000 s

42.281250000 s

56.218750000 s

7. Conclusion In this study, a FVSE method is developed for solving MHD duct problem with various values of θ , orientation of applied magnetic field with x-axis, λ, the wall conductivity and M, the Hartmann number. The effects of various values of these parameters are visualized in terms of graphics showing the characteristics of MHD duct flow. Comparisons with two other numerical methods in the cases for which the exact solution exists show that this technique is more accurate and agrees very well with the exact solution. Acknowledgements The authors are very grateful to the three reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper very much. The authors are very much thankful to the Editor, Professor Spencer Sherwin for his useful comments. Also we would like to thank Editor-in-Chief, Professor Robert Beauwens for managing the review process for this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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Adaptive sparse grid algorithms with applications to electromagnetic scattering under uncertainty Meilin Liu a,b,1 , Zhen Gao b,c,1 , Jan S. Hesthaven b,∗ a b c

College of Information Science and Technology, Nanjing University of Aeronautics and Astronautics, China Division of Applied Mathematics, Brown University, USA Research Center for Applied Mathematics, Ocean University of China, China

a r t i c l e

i n f o

Article history: Received 2 January 2010 Received in revised form 9 August 2010 Accepted 9 August 2010 Available online 14 August 2010 Keywords: Sparse grids Stochastic collocation Adaptivity Maxwell’s equations

a b s t r a c t We discuss adaptive sparse grid algorithms for stochastic differential equations with a particular focus on applications to electromagnetic scattering by structures with holes of uncertain size, location, and quantity. Stochastic collocation (SC) methods are used in combination with an adaptive sparse grid approach based on nested Gauss–Patterson grids. As an error estimator we demonstrate how the nested structure allows an effective error estimation through Richardson extrapolation. This is shown to allow excellent error estimation and it also provides an efficient means by which to estimate the solution at the next level of the refinement. We introduce an adaptive approach for the computation of problems with discrete random variables and demonstrate its efficiency for scattering problems with a random number of holes. The results are compared with results based on Monte Carlo methods and with Stroud based integration, confirming the accuracy and efficiency of the proposed techniques. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction With the increasing need to quantify the impact of random input parameters on different types of physical systems or engineering applications, often described by stochastic ordinary or partial differential equations, comes the need for new and efficient computational techniques to deal with such problems. The classic Monte Carlo based method are increasingly inadequate and a number of alternatives have begun to emerge. It is reasonable to categorize the majority of these methods into two groups: sampling based statistical methods and probabilistic techniques. In the first category one finds the classic Monte Carlo (MC) method [4] with the clear advantage of being simple and non-intrusive, e.g., one needs only a deterministic solver. The simplicity, however, comes at the cost of very slow convergence as O ( M −1/2 ) where M is the number of samples. This quickly becomes prohibitive even if reasonable accuracy is required, in particular if the interest is on higher moments such as variance/sensitivity. A notable exception to this is for very high dimensional problems where the advantage of the dimensionally independent convergence rate eventually becomes important. However, we shall not consider this limit here. To accelerate convergence of the MC method, several techniques have been proposed, e.g., Latin hypercube sampling [15], quasi-MC (QMC) method [5], and the Markov chain MC (MCMC) [6] method. However, additional restrictions are often imposed by these methods and their applicability is limited.

* 1

Corresponding author. E-mail address: [email protected] (J.S. Hesthaven). The two lead authors have contributed equally and substantially to the completion of the published work.


M. Liu et al. / Applied Numerical Mathematics 61 (2011) 24–37

25

An alternative to sampling based techniques has been recently received substantial attention. These methods, known as Stochastic Galerkin or Polynomial Chaos (PC) methods [10,21], are probabilistic in nature and are based on a generalization of the Wiener–Hermite PC expansion [20]. In this approach, the randomness is represented by the Wiener expansion and the unknown expansion coefficients are found by a Galerkin procedure in the inner product associated with the random variables using in the Wiener expansion. Substantial recent work has confirmed the accuracy and efficiency of this approach, in particular for problem with low to moderate dimensionality and for problems with sufficient smoothness in observation space, resulting in very efficient representations through the Wiener expansion. However, a substantial disadvantage of the Galerkin approach lies in the need to have to develop entirely new software to solve the large coupled equations resulting from this procedure. This represents a significant problem as validated existing software cannot be used directly to model the impact of randomness and uncertainty. To address this shortcoming of an otherwise successful approach, several authors have proposed a slight modification of this traditional approach. As mentioned above, the bottleneck in the stochastic Galerkin approach is the creation of a new large coupled system through the required inner product. It is natural to attempt to satisfy the high-dimensional problem in a collocation fashion instead, resulting in a large number of decoupled small problems, much in the sense of an MC approach. However, in the collocation approach, the sampling points are deterministic and associated with integration formulas for the evaluation of high-dimensional integrals in contrast MC based techniques where the sampling points are drawn randomly from some a priori distribution. This approach, now known as stochastic collocation, was first proposed by Tatang et al. [17] and more recently revisited and extended in [22] and subsequently considered in more detail by numerous authors, see [24] for a recent review. A clear advantage of this approach over the stochastic Galerkin formulation is its non-intrusive nature, enabling one to use validated software much in the same way as for MC based techniques. A central component of the efficiency and accuracy of these techniques is the construction of efficient and accurate integration methods for high-dimensional problems. In [22,24] several options are discussed in detail, including the Stroud’s cubature points [16], resulting in an efficient approach at moderate accuracy, and sparse grids constructed through Smolyak’s algorithm [14] combined with the Clenshaw–Curtis integration method [3]. This latter approach improves accuracy but is costly due to the moderate accuracy of the quadrature. In this work we discuss the tradeoff between accuracy and computational efficiency in a few different ways. We first consider the use of hierarchical Gauss–Patterson integration formulas as a more accurate alternative to the Clenshaw–Curtis nodes. A complementary discussion of some of this can be found in [9]. To further decrease computational cost we propose to use Richardson extrapolation between the levels to estimate errors and ultimately predict the results of an increased accuracy at very little additional cost. Most of the past work discussed in the above focuses on problems with continuous random variables. In this work we also discuss basic approaches for the adaptive solution of problems with discrete random variables and demonstrate how to use a priori information effectively. Throughout, we illustrate the efficiency and accuracy of the methods on a set of benchmark problems, often referred to the Genz problems [7,8]. However, we conclude this work with the consideration of problems of electromagnetic scattering in which Maxwell’s equations are solved in the time-domain using a discontinuous Galerkin method. It is important to note, however, that the adaptive techniques proposed here are problem independent and can be applied to general stochastic problems, solved using a variety of computational techniques. What remains of the paper is organized as follows. In Section 2 we provide a brief overview of polynomial chaos techniques with a focus on stochastic collocation methods. This sets the stage for Section 3 where we discuss high-dimensional integration schemes and introduce Gauss–Patterson integration methods in combination with the Smolyak construction. Section 4 discusses in detail the strategies for adaptivity and the use of Richardson extrapolation as an error estimator in this context. We also briefly discuss ideas that allow the efficient adaptive solution of problems in which discrete random variables are used to model the random behavior. In Section 5 we provide a brief overview of the application cases and the computational technique used to solve Maxwell’s equations and then illustrate some additional results in support of the generality of the proposed techniques. Section 6 contains a few concluding remarks. 2. Stochastic collocation methods Let us adopt the notation of [22]. (Ω, A, P ) is a complete probability space, where Ω is the event space, A ∈ 2Ω the σ algebra, and P the probability measure. Assume a d-dimensional bounded domain D ⊂ R d (d = 1, 2, 3), with boundary ∂ D, ¯ → R, such that for P -almost everywhere and focus on the following problem: find a stochastic function, u ≡ u (ω, x) : Ω × D ω ∈ Ω , the following equation holds,

L(ω, x; u ) = f (ω, x),

x ∈ D,

(1)

subject to the boundary condition

B(ω, x; u ) = g (ω, x),

x ∈ ∂ D,

(2)

where x = (x1 , . . . , xd , t ), L is a differential operator, and B is a boundary operator. Note that we do not designate between spatial and temporal dimensions at this stage to keep the notation simple.

26


We assume that the randomness can be represented by p independent variables with zero mean and unit variance, each depending on the random event ω . There are several ways to achieve this depending on details of the problem [24]. We assume validity of the standard technique of a Karhunen–Loéve expansion to express the randomness as

Ξ (ω) = μ0 +

p

λk ψk ξk (ω),

k =1

where μ0 represents the mean of the random field and ψk are the orthogonal eigenfunctions associated with the eigenvalues, λk , of the correlation function of the random process being represented, and ξk are the random variables (see e.g. [10,24] for details). Representing the randomness in this generic form in Eqs. (1)–(2) yields a ( p + d)-dimensional differential equation in strong form as

L(ξ, x; u ) = f (ξ, x),

(ξ, x) ∈ Γ × D,

subject to the boundary condition

B(ξ, x; u ) = g (ξ, x),

(ξ, x) ∈ Γ × ∂ D,

where Γ is the p-dimensional random space. To account for the impact of the uncertainty, it is natural to consider moments of the solutions over the probabilistic space. In other words, we need to evaluate multi-dimensional integrals of the form

I[ f ] =

f (ξ ) dμ(ξ ). Γ

The simplest way to achieve this is through a Monte Carlo approach like

I[ f ]

M 1

M

f (ξm )

m =1

with the M instances, ξm , being drawn from the distribution, μ(ξ ). As mentioned previously, the disadvantage of this approach is its low convergence rate which, however, is independent on the dimension p of the random space. Realizing that all we need is to be able to evaluate integrals accurately, it seems reasonable to utilize more accurate integration techniques. At least for problems of moderate dimensionality one would expect these to be superior in terms of accuracy vs cost. This line of arguments were first explored in [17] for relatively simply ordinary differential equations and discussed in much more detail in [22]. We refer to these and [24] for further aspects of this. The essence of the stochastic collocation approach is to abandon the random sampling approach and consider the use of more advanced integration approached and, in this work, adaptive hierarchical integration techniques. In other words we shall solve the deterministic problems

L(ξk , x; u ) = f (ξk , x),

x ∈ D,

with boundary condition

B(ξk , x; u ) = g (ξk , x),

x ∈ ∂ D.

where ξk ∈ Γ are specific instances chosen with an integration formula in mind. Clearly, an objective in identifying this integration approach is to minimize the number of samples to achieve a given accuracy in evaluating the integral. 3. Integration for high-dimensional problems For the multi-dimensional integration, we consider a number of different approaches, the simplest of which is the Stroud [16] cubature points. These are useful when computing integrals of the form

I[ f ] =

f (x) dx.

(3)

[−1,1] p

This set of cubature points based on ( p + 1) points is exact for polynomials of degree two, and are given as

I[ f ]

n i =1

ωi f (xi ),

(4)


27

where the n = p + 1 cubature points xi = (x1i , x2i , . . . , xni ) are given by

x2r i

=

2

−1 x2r = i

cos

3 2 3

sin

2r (i − 1)π n+1

2r (i − 1)π

(5)

,

,

n+1

√

for r = 1, . . . , n/2 . If n is odd, xni = (−1)(i −1) / 3. The weights in (4) are all equal to 2n /(n + 1). Similarly, we have the Stroud-3 method based on 2p points which is exact for polynomials of degree three:

I[ f ]

n

ωi f (xi ),

(6)

i =1

where the n = 2p cubature points xi = (x1i , x2i , . . . , xni ) are now defined by

−1 x2r = i

x2r i

=

2

cos

3 2 3

sin

(2r − 1)i π n

(2r − 1)i π n

,

(7)

,

for r = 1, . . . , n/2 . The weights in (6) are all equal to 2n /2n. It can be shown [3,23] that the Stroud-2 and Stroud-3 methods use the minimal number of points for their corresponding integration accuracy. The very simple schemes have recently been extended to general weights in [23]. While the Stroud schemes are efficient and may suffice to compute the expectation, their limited accuracy is often a problem. The most straightforward way to extend the many known one-dimensional integration methods to higher (p) dimensions is through the use of simple tensor products. However, this quickly becomes prohibitive with the number of samples growing like N p for a quadrature of order N used in p dimensions. A valuable and often superior alternative to this is the use of sparse grid methods of which the most notable ones are those based on the Smolyak construction. In [14], Smolyak proposed a construction of sparse multivariate quadrature formulas based on sparse tensor products of one-dimensional quadrature formulas. Let us consider the numerical integration of functions f (x) over a p-dimensional unit hypercube Γ := [−1, 1] p ,

I [ f ] :=

f (x) dx, Γ p

p

p

by a sequence of nl -point quadrature formulas with level l ∈ N and nl < nl+1 , n

p Ql

f :=

p

l

ωli f (xli )

(8)

i =1

using the weights

ωli and abscissas xli . Moreover, we define the underlying grids of a quadrature formula by

p p Γl := xli ∈ [−1, 1] p : 1 i nl .

(9)

Now, define the difference quadrature as

k1 f := Q k1 − Q k1−1 f

with Q 01 f := 0.

(10)

Smolyak’s construction for the integration of p-dimensional functions f is p

Q l f :=

|k|1 l+ p +1

k11 ⊗ · · · ⊗ k1d f

(11)

where l ∈ N and k ∈ N p . An essential feature of this construction is that the sparse quadrature formulas are nested if the corresponding one-dimensional quadrature nodes are nested. Notably, this rules out classic Gauss quadratures which are not nested and the impact of this is illustrated in Fig. 1 and will be discussed shortly.

28


Fig. 1. Two-dimensional Smolyak based sparse grids based on Clenshaw–Curtis (left), Gauss–Patterson (middle) and Gauss–Legendre (right) for 6 levels in the hierarchical integration.

Table 1 The number of grid points required in a two-dimensional Smolyak sparse grid based Clenshaw–Curtis (C-C), Gauss–Patterson (G-P) or the Gauss–Legendre (G-L) integration nodes. Level

C-C

G-P

G-L

1 2 3 4 5 6

5 13 29 65 145 321

5 17 49 129 321 769

5 21 73 221 609 1573

3.1. Gauss–Patterson quadrature rules Seeking nested one-dimensional integration formulas, simple trapezoidal rules immediately come to mind. However, the limited accuracy of these makes this a less interesting choice. A more appropriate, and widely used, approach is based on the Clenshaw–Curtis rule [3] which is exact for polynomials of order n when using n + 1 points. This is considerably better than the second order accuracy of the trapezoidal rules but falls short of the 2n + 1 polynomial exactness of the Gaussian quadrature. The natural question to raise is whether there are nested quadratures which are better than the Clenshaw–Curtis rules, but perhaps not quite as good as the classic quadratures. This question was first addressed by Kronrod [3] who extended an n-point Gauss–Legendre quadrature formula by n + 1 points such that the quadrature formulas completed the polynomial degree of the exactness with degree 3n + 1 (n even) or 3n + 2 (n odd). Patterson iterated Kronrod’s scheme recursively and obtained a sequence of nested quadrature formulas with maximal degree of exactness. We refer to [3] for a discussion of the details of this construction. For f ∈ C r , the error bounds of integration f using Gauss–Patterson sparse grid formula will be [9]

1

E f = O 2−lr , l

where l refers to the number of levels in the hierarchical construction. When considering the efficiency of the integration measured through polynomial exactness, it is well known that using a quadrature with n points, the Clenshaw–Curtis is exact for polynomials up to order n − 1 and the Gauss–Legendre quadrature is exact for orders up to 2n − 1. For the general Gauss–Patterson rule, one can show exactness up to order (3n − 1)/2, confirming that this is truly a compromise between the two alternatives [3]. The nested structure of the Gauss–Patterson quadrature grids in combination with the Smolyak’s construction results in a natural hierarchical structure for computing the integrals. Hence, to improve the accuracy one needs only compute those new additional grids required to increase from level l to level l + 1. This is an important property, in particular for highdimensional problems. For the one-dimensional Clenshaw–Curtis rule, the number of points grows like 2l−1 + 1, whereas the growth for the Gauss–Patterson rule is 2l − 1 since the rule is based on the Gauss quadrature. Hence, when comparing cost of the two methods, it is most appropriate to compare the Clenshaw–Curtis rule at level l with the Gauss–Patterson rule at level l − 1. When the dimensionality of the problem increases this becomes more pronounced because the number of quadrature points grows as O (2l ld−1 ) in the sparse grid [2]. This is illustrated in Table 1 listing the number grid points required for the two-dimensional example with an increasing number of levels. The importance of the nested structure is evident for all levels but the first one. As a further illustration of this aspect, we plot in Fig. 1, the Smolyak based sparse grids Clenshaw– Curtis, Gauss–Patterson and Gauss–Legendre formules for two variables and with 6 levels in the Clenshaw–Curtis based scheme and 5 levels in the two other cases. The final question to address is whether the Gauss–Patterson based approach, with its improved accuracy but with more quadrature points at a given level, is competitive with the more traditional Clenshaw–Curtis scheme when one compares cost vs accuracy, i.e., to achieve a given accuracy in the integral, which of the two schemes require the least number of function evaluations.


29

Fig. 2. Accuracy vs level for integration of the test-function for dimensions p = 2, 5, 10.

Fig. 3. Cost vs accuracy for integration of the test function for dimensions p = 2, 5, 10.

To address this question, we consider the test function [8] for x ∈ [0, 1] p

f (x1 , . . . , x p ) = cos 2πω1 +

p

c i xi ,

i =1

where ω1 and c = (c 1 , . . . , c p ) are randomly generated and the c i ’s sum to 9. In Fig. 2 we show the convergence of the different high-dimensional integration schemes discussed previously. We note that the accuracy of the Gauss–Patterson scheme is comparable to that of the scheme based on the full Gauss integration. We also note that the sparse grid based on the Clenshaw–Curtis formula experiences convergence problems for the highdimensional case. This is a known problem [9] and is likely associated with the kink-phenomena discussed in detail in [19]. It is noteworthy that there are numerous other examples of high-dimensional cases where the Clenshaw–Curtis formula converges as expected. A more important question is, however, which of these methods are preferred when taking into account the work needed to achieve a specific accuracy. In Fig. 3 we address this question by showing the time taken to achieve a specific accuracy. From this, it is clear that the sparse grid scheme based on the Gauss–Patterson scheme is superior except when a very moderate accurate is needed in which case the Stroud based schemes may well be preferred. Similar conclusions have been reached by considering other test functions and we will therefore focus on the Gauss–Patterson based sparse grid schemes going forward. This conclusion was also reached in [9] although the problems being considered were different and less focused on the cost vs accuracy considerations introduced here. 4. Error estimation and adaptivity in the sparse grid construction With the computational work growing exponentially with the dimensionality, it is essential that we seek to minimize the cost by carefully using the required degrees of freedom only where they are needed. This suggests that an adaptive approach is warranted and therefore the need to formulate an effective error estimator. The hierarchical nature of the Smolyak based integration immediately suggests that some kind of extrapolation be use to predict the value of the integral at the next level using results obtained at the previous levels [1]. We here propose the use of Richardson extrapolation to achieve this. It is known that for functions f ∈ C r , the error in the Smolyak algorithm is | E lp f | = O (n−r / p (log n)( p −1)(r / p +1) ) when

the integration utilize n = 2l l p −1 as the number of integration point sparse grid at level l to evaluate a p-dimensional integral [13]. With this knowledge of the local error behavior as a function of the level in the Smolyak grid, we can apply classic Richardson extrapolation in the following way.

30


Assume we have computed an estimate to the integral, I [ f ], at two different levels. We then know that

I [ f ]l = I [ f ] + C rp 2l l p −1 l +1

I[ f ]

= I[ f ] +

C rp

−r / p

l +1

2

log 2l l p −1

(l + 1)

( p −1)(r / p +1)

p −1 −r / p

l

log 2 (l + 1)

,

p −1

( p −1)(r / p +1)

.

Through simple algebra one recovers that

I[ f ] =

α1 I ( f )l+1 − α2 I ( f )l , α1 − α2

where

(−r / p ) l ( p −1) ( p −1)·(r / p +1) · log 2 · l ,

( p −1)·(r / p +1)

(l+1)

(− r / p ) α2 = 2 · (l + 1)( p −1) · log 2(l+1) · (l + 1)( p −1) .

α1 = 2l · l( p−1)

This improved estimate of the integral depends on r which we recall is a measure of the smoothness of f ∈ C r . This is generally unknown. However, extensive tests have shown that the efficiency of the error estimator has limited sensitivity to this number and we have found that taking it to values of 2–4 generally yields excellent results as we will illustrate shortly. In some cases, there may be benefits to estimating a different value and this can simply be done by evaluating the accuracy/efficiency of the extrapolated values at the coarse grid levels and identify the optimal value of r. In our approach we will not compute the next level in the Smolyak grid if the extrapolation result is close to the result at the current level to within a given tolerance. This is vastly advantageous over previously used methods which simply compare solutions at the two levels, often resulting in having to compute an additional level at substantial cost – typically the cost of a new level is comparable to the combined cost of all previous levels. Instead of this, we use the Richardson extrapolation to estimate the result at the next level, hence dramatically reducing the overall cost without impacting the accuracy. To test the validity of this approach as a way to accelerate the adaptive sparse grid algorithm and reduce the computational cost we consider a set of high-dimensional test functions proposed in [7,8]. These function are all defined on [0, 1] p and we seek to integrate them as accurately as possible. The functions being considered have different characteristics: 1. Oscillatory:

f 1 (x) = cos 2πω1 +

p

c i xi ;

(12)

i =1

2. Product peak:

f 2 (x) =

p

c i−2 + (xi − ωi )2

− 1

;

(13)

i =1

3. Corner peak:

f 3 (x) =

1+

p

−(d+1) c i xi

;

(14)

i =1

4. Gaussian:

f 4 (x) = exp −

p

c i2 (xi

2

− ωi )

;

(15)

i =1

5. Continuous:

f 5 (x) = exp −

p

c i | xi − ωi | ;

(16)

i =1

Different test functions can be obtained by varying the parameters c = (c 1 , . . . , c p ) and ω = (ω1 , . . . , ω p ). The parameters ωi act as shift parameters, and the difficulty of the functions is increasing with c i > 0. We test the integration with the dimension p = 10 and use parameters c i such that p

ci = b j ,

i =1

where b j depends on the family f j and is given by Table 2.

(17)


31

Table 2 Parameter b j for the five different test functions, f j . j

1

2

3

4

5

bj

9 .0

7.25

1.85

7.03

20.4

Table 3 Integrals computed for functions f 1 – f 4 for p = 10 using sparse grid (SG) and Richardson extrapolation (RE) at levels one to six. Note that the integral was also computed at level zero to enable the extrapolation but is not shown due to its poor accuracy. Level 1

Level 2

Level 3

Level 4

Level 5

Level 6

SG RE

−0.1191 –

−0.1378 −0.1191

−0.1359 −0.1378

−0.1360 −0.1359

−0.1360 −0.1360

−0.1360 −0.1360

f2

SG RE

4.1472e–06 –

4.3598e–06 4.1472e–06

4.3452e–06 4.3601e–06

4.3458e–06 4.3451e–06

4.3458e–06 4.3458e–06

4.3458e–06 4.3458e–06

f3

SG RE

0.0014 –

0.0017 0.0014

0.0018 0.0017

0.0018 0.0018

0.0018 0.0018

0.0018 0.0018

f4

SG RE

0.3970 –

0.4085 0.3970

0.4091 0.4085

0.4091 0.4091

0.4091 0.4091

0.4091 0.4091

f1

Table 4 Integrals computed for function f 5 for dimensions p = 2, 5, 10 using sparse grid (SG) and Richardson extrapolation (RE) at levels zero to seven. Level 0

Level 1

Level 2

Level 3

Level 4

Level 5

Level 6

Level 7

p=2

SG RE

0.0029 –

0.0208 –

0.0201 0.0214

0.0452 0.1337

0.0326 0.0099

0.0330 0.0336

0.0338 0.0349

0.0338 0.0338

p=5

SG RE

0.0047 –

0.0114 –

0.0102 0.0114

0.0066 0.0112

0.0068 0.0063

0.0073 0.0056

0.0072 0.0078

0.0072 0.0072

p = 10

SG RE

0.0092 –

0.0069 –

0.0021 0.0069

0.0034 0.0021

0.0031 0.0035

0.0034 0.0030

0.0035 0.0033

0.0035 0.0035

In Table 3 we illustrate the accuracy of the extrapolation for p = 10 for the first four test functions above. It is evident that the extrapolation works very well and offers an accurate estimate of the integral at the next level once it is reasonably well approximated at previous levels. We have used r = 3 as an estimator of the smoothness in this example but similar results very obtained with r = 4, confirming the relative insensitivity of the estimate on the performance of the scheme. The final test function, f 5 , is more challenging and in fact violates the smoothness assumption required to derive the error estimates for the Richardson extrapolation. We show in Table 4 the results for the extrapolation for different dimensions and observe that in spite of the more complex function, the results confirm the accuracy of the proposed approach for local error estimation and, thus, enables accurate adaptive integration. Naturally, the exact same idea can be explored as a dimensional error estimator to enable anisotropic adaptivity, although we have not explored this further in this work. Note that in the above results, we used r = 3 for all tests, confirming the lack of sensitivity to this parameter. 4.1. Predictive sampling for discrete variables For discrete random variables, sparse grids and extrapolation are not directly applicable and we need to seek an alternative approach. We will, in this work, simply use the knowledge of the a priori density of the discrete random variables. Let us, for instance, assume a discrete random variable k with a density

f (k; λ) = e −λ

λk , k!

k = 0, 1 , 2, . . .

(18)

where the mean and variance of k are: μk = k¯ = λ > 0 and σk2 = λ, respectively. We use the a priori density to select which samples to compute first. Hence, if k = 2 is the most likely case, this will be computed first, followed by the next most likely value of the discrete variable. The convergence of the integral is used to decide whether additional instances are needed. We will demonstrate the computational advantage of this simple idea in the following section. 5. Applications to electromagnetic scattering problems To evaluate the benefits of the techniques discussed in the above for problems of a more practical character, we consider electromagnetic scattering by a two-dimensional cylinder with holes where the size, the location, and ultimately, the number of holes, are considered as uncertain and described by continuous and discrete random variables.

32


The physical model is Maxwell’s equations

∂E = ∇ × H, ∂t ∂H μ = −∇ × E, ∂t

(19) (20)

in the general three-dimensional domain Λ without sources. E is electric field intensity, H is the magnetic field, are the permittivity and the permeability of the domain. As an output measure of interest we use the Radar Cross Section (RCS) defined as

|F(φ)|2 RCSdb (φ) = 10 log 2π , | E i |2

and μ

(21)

where Ei is the incident field and F(φ) is a function of E and H, computing the scattered far field as a function of the polar angle, φ . In this particular case, F(φ), is near-to-far-field transformation along some closed contour [18]. 5.1. Nodal discontinuous Galerkin finite element method Discontinuous Galerkin methods are a general and flexible way of solving Maxwell’s equations. We follow the formulation in [11] and refer to [12] for a more general and in depth account of these techniques. We assume the computational domain Ω is approximated by K elements D k as

Λ∼ =

K

Dk ,

(22)

k =1

where D k is a two-dimensional simplex. In each element, we approximate E and H by Lagrange polynomials as

[Eh , Hh ] =

N

E(x j , t ), H(x j , t ) l j (x),

(23)

j =1

where x j are the interpolation points. The number of nodes N is given as

N=

(n + 1)(n + 2) 2

for the nth order polynomial in two dimensions. We insert the approximate solution into Maxwell’s equation and require that the local residual is orthogonal to all nth order polynomials. Integrating by part twice, this yields the scheme

∂ Eh ˆ k × Hk − H∗ li (x) dx, − ∇ × Hh li (x) dx = − n ∂t

Dk

μ Dk

(24)

∂ Dk

∂ Hh + ∇ × Eh li (x) dx = ∂t

ˆ k × Ek − E∗ li (x) dx, n

(25)

∂ Dk

ˆ is the unit outward normal vector where [E∗ , H∗ ] denotes the numerical flux of the corresponding vector quantities and n along ∂ D k . The numerical flux is responsible for the coupling of the elements, for the stability of the scheme, and for the imposition of boundary conditions. We use a standard upwind flux with the explicit form given in [11]. We use a low storage Runge–Kutta scheme [11] for the temporal integration. It should be noted that the method used to solve Maxwell’s equations in this work is less important since the focus is on the efficiency of the techniques dealing with the stochastic elements of the problem. The computational setup of the problem is a plane wave that impinges on a 2-D cylinder from some specific direction. The cylinder has an uncertain number of holes in it and their sizes and locations may also be uncertain. Examples of meshes of the computational model with different number of holes, hole sizes and locations are shown in Fig. 4. The bistatic RCS has an exact solution for the case of a plane wave impinging on a 2-D cylinder without holes. We consider the case with ka = π where a is the radius and k = 2π /λ is the wave number associated with wavelength of the incoming wave. We employ a perfectly matched layer (PML) to absorb the reflected wave in our computation [18]. The comparison of exact solution and numerical solution is shown in Fig. 5, confirming the accuracy of the solver. A complete analysis of the solution approach can be found in [11].


33

Fig. 4. Examples of meshes of the cylinder with different number of holes, hole size, and hole location. (a) has one hole, (b) three holes, (c) four holes, and (d) six holes.

Fig. 5. Comparison between numerical and exact bistatic RCS for a ka = π metallic cylinder.

5.2. Low-dimensional examples To verify our algorithm, we first compare the results using the three methods for low-dimensional problems, i.e., we restrict the problem to having one or two random parameters, all assumed to be uniformly distributed random variables. The three methods are a standard Monte Carlo (MC) method, Stroud’s method at second and third order, and the sparse grid method. For all case we recover the mean of the RCS as

RCS =

Q i =1

ωi RCSi ,

(26)

34


Fig. 6. Mean RCS computed using four different sampling methods.

Fig. 7. Mean RCS and sensitivity for a two-parameter random problem computed using the adaptive sparse grid method.

and the variance of RCS as

var(RCS) =

Q

2

ωi RCSi − RCS ,

(27)

i =1

where the number of terms, Q , and the integration weights, ωi , depend on the specific integration technique used. We first consider the problem with one random parameter, taken to be a cylinder with one hole of random size. The size π , π ) in the polar angle. In Fig. 6 we show the mean RCS computed using of the hole is assumed to be in the range of ( 12 6 the four methods. The level of the sparse grid method is 6 (31 quadrature points) and the Monte Carlo method employs 2000 uniformly distributed points. The difference between level 5 and level 6 results is of order 10−3 . The Stroud method of degrees 2 and 3 yields the same results since they are identical in this simple case. We observe excellent agreement between the four methods and, in particular, observe excellent agreement between the sparse grid result and the MC computation. Next, we consider 2 random parameters; the hole size and the angle of the incidence plane wave. The incident wave impinges from the left on the cylinder and changes in the range of ( 35 π , 37 π ). The RCS mean and it plus/minus one 36 36 standard deviation computed using the sparse grid method are presented in Fig. 7, illustrating the value of these techniques by enabling the computation of sensitivities of output measures of interest. We use 5 levels in the sparse grid computation and estimate the error to be of the order of 10−3 based on the extrapolation of the RCS.


35

Fig. 8. RCS computed for cylinders illustrated in Fig. 4, with 2 (left), 3 (middle), and 4 (right) holes with uniformly distributed hole size and angle of incidence of the illuminating wave.

Fig. 9. Poisson density associated with the number of holes in the cylinder.

5.3. Example of a higher-dimensional problem To simulate a more realistic random wave problem, we need to consider higher-dimensional problems. We achieve this in two different ways. In the first example we assume the number of holes is deterministic but that their size and illumination are uniformly distributed random variables. In Fig. 8 we present the results for the cylinder with 2 to 4 holes obtained with results obtained using the sparse grid method with 5 levels in the integration. We note in particular the impact of the number of holes on the sensitivity of the RCS. To further add to the complexity, let us also assume that the number holes is a discrete random variable with a Poisson probability distribution, illustrated in Fig. 9. We assume that the number of holes is ranging between one and nine but that their location is fixed. The RCS mean and it sensitivity computed with the adaptive sparse grid method is presented in Fig. 10. We use this case to demonstrate the value of the approach discussed in Section 4.1. To illustrate this, we compute the L 2 -error of the RCS with different numbers of holes and list the error as the number of holes increases. Here the reference solution is assumed to be the one where all variations of one to nine holes are accounted for, i.e., for each discrete number of holes, the randomness is in the hole size is accounted for as discussed previously and the moments in the discrete variables are computed subsequently. The results, simply reflecting the error in the L 2 -error or the mean on the RCS, are shown in Table 5. As expected, we quickly see convergence with as little as three to four holes and can terminate the computation and, thus saving substantially in the overall computational cost.

36


Fig. 10. RCS mean and sensitivity for problem with random number of holes and uniformly distributed hole size.

Table 5 The L 2 -error of the mean RCS as a function of the number of holes, assuming that the number of holes is a Poisson distributed random discrete variable. Hole number

L 2 -error of the mean RCS

1 2 3 4 5 6

4.2345 2.4427 1.0191 0.4481 0.1181 0

6. Concluding remarks In the paper we have discussed the development of adaptive sparse grid methods in the context of stochastic collocation methods for solving partial differential equations with uncertainty. The emphasis has been on identifying methods which delivers maximum accuracy at minimal cost. We find that the combination of Gauss–Patterson quadratures and Smolyak sparse grid constructions is an effective way to reach this and results in a computationally robust approach. In particular we confirm, in agreement with related work, that the widely used Clenshaw–Curtis Smolyak based approach may have problems with convergence for certain high-dimensional test functions. No such problem was observed with the Gauss–Patterson based scheme. Similar behavior has also been observed by other authors [9] but the detailed study of the cost vs accuracy question is a new direction of research. Furthermore, we demonstrate how the strict hierarchical structure of the Smolyak sparse grid lends itself very well to the use of Richardson extrapolation as an efficient estimator for the computed value of the integral at the next level. This is important as it allows for significant savings in the overall computational expense. We demonstrated that this is a robust and general approach as illustrated for several standard test functions. This suggests an accurate and fast adaptive sparse grid scheme and we illustrated its use and flexibility by considering electromagnetic scattering problems with the scatterer having randomized holes. We also demonstrate a simple but effective way to deal with discrete random variables in an error controlled manner. In combination with the adaptive sparse grid approach, this yields substantial computational savings. An immediate next step is to look more carefully at the theory of the overall procedure and, in particular, the accuracy of the Richardson extrapolation procedure. We hope to report on this in the near future. Acknowledgement The first author acknowledges the support of the China Scholarship Committee and National Science Foundation of China (No. 60771017) for this research. The second author also acknowledges the support of China Scholarship Committee and National Science Foundation of China (No. 2008633049) for this research. The last author acknowledges partial support by OSD/AFOSR FA9550-09-1-0613, and by NSF, and DoE.


37

References [1] H. Bungartz, M. Griebel, U. Rüde, Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems, Comput. Methods Appl. Mech. Engrg. 116 (1994) 243–252. [2] H. Bungartz, M. Griebel, Sparse grids, Acta Numer. 13 (2004) 147–269. [3] P.J. Davis, P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975. [4] G. Fishman, Monte Carlo, Concepts, Algorithms, and Applications, Springer-Verlag, New York, 1996. [5] B. Fox, Strategies for Quasi-Monte Carlo, Kluwer, Dordrecht, The Netherlands, 1999. [6] D. Gamerman, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Chapman and Hall, London, 1997. [7] A.C. Genz, Testing multidimensional integration routines., in: B. Ford, J.C. Rault, F. Thomasset (Eds.), Tools, Methods, and Languages for Scientific, Engineering Computation, North-Holland, Amsterdam, 1984, pp. 81–94. [8] A.C. Genz, A package for testing multiple integration subroutines, in: P. Keast, G. Fairweather (Eds.), Numerical Integration, Kluwer, Dordrecht, 1987, pp. 337–340. [9] T. Gerstner, M. Griebel, Numerical integration using sparse grid, Numer. Algorithms 18 (1998) 209–232. [10] R.G. Ghanem, P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991. [11] J.S. Hesthaven, T. Warburton, Nodal high-order methods on unstructured grids I. Time-domain solution of Maxwell’s equations, J. Comput. Phys. 181 (2002) 186–221. [12] J.S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer-Verlag, New York, 2007. [13] E. Novak, K. Ritter, The curse of dimension and a universal method for numerical integration, in: G. Nurnberger, J.W. Schmidt, G. Walz (Eds.), Multivariate Approximation and Splines, Birkhauser, Boston/Basel, 1997, pp. 177–188. [14] S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of funcitions, Dokl. Akad. Nauk SSSR 4 (1963) 240–243. [15] M. Stein, Large sample properties of simulations using Latin hypercube sampling, Technometrics 29 (1987) 143–151. [16] A. Stroud, Remarks on the disposition of points in numerical integration formulas, Math. Comp. 11 (1957) 257–261. [17] M.A. Tatang, W.W. Pan, R.G. Prinn, G.J. McRae, An efficient method for parametric uncertainty analysis of numerical geophysical model, J. Geophys. Res. 102 (1997) 21925–21932. [18] A. Taflove, Computational Electrodynamics – The Finite-Difference Time-Domain Method, Aztech House, Boston, 1995. [19] J.A.C. Weideman, L.N. Trefethen, The kink phenomenon in Fejer and Clenshaw–Curtis quadrature, Report no. 6/16, Oxford University Computing Laboratory, 2006. [20] N. Wiener, The homogeneous chaos, Amer. J. Math. 60 (1938) 897–936. [21] D. Xiu, G.E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput. 24 (2002) 619–644. [22] D. Xiu, J.S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput. 27 (2005) 1118–1139. [23] D. Xiu, Numerical integration formulas of degree two, Appl. Numer. Math. 58 (2008) 1515–1520. [24] D. Xiu, Fast numerical methods for stochastic computations: A review, Commun. Comput. Phys. 5 (2009) 242–272.




On Numerov’s method for a class of strongly nonlinear two-point boundary value problems ✩ Yuan-Ming Wang a,b,∗ a

Department of Mathematics, East China Normal University, Shanghai 200241, People’s Republic of China Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science, E-Institute of Shanghai Universities, Shanghai Normal University, Shanghai 200234, People’s Republic of China b

a r t i c l e

i n f o

Article history: Received 15 September 2009 Received in revised form 15 June 2010 Accepted 14 August 2010 Available online 18 August 2010 Keywords: Strongly nonlinear two-point boundary value problem Numerov’s method Fourth-order accuracy Monotone iterations Upper and lower solutions

a b s t r a c t The purpose of this paper is to give a numerical treatment for a class of strongly nonlinear two-point boundary value problems. The problems are discretized by fourthorder Numerov’s method, and a linear monotone iterative algorithm is presented to compute the solutions of the resulting discrete problems. All processes avoid constructing explicitly an inverse function as is often needed in the known treatments. Consequently, the full potential of Numerov’s method for strongly nonlinear two-point boundary value problems is realized. Some applications and numerical results are given to demonstrate the high efficiency of the approach. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction Nonlinear two-point boundary value problems arise from many fields of applied sciences and have been investigated extensively in the literature both analytically and numerically (cf. [1,3,7,31–33]). In this paper, we seek high-accuracy numerical solution of the following strongly nonlinear two-point boundary value problem:

⎧ ⎨ ⎩

−

d dx

k (u )

u (0) = α ,

du dx

= f (x, u ),

0 x 1,

(1.1)

u (1) = β,

where α , β are given constants, and the functions f (x, u ) and k(u ) (which, in general, are nonlinear in u) are prescribed smooth functions of their respective arguments. The consideration of problem (1.1) is motivated by some heat-conduction problems and diffusion problems. For example, if the ends of a rod are kept at given temperature and the thermal conductivity is temperature dependent then the steadystate temperature distribution u (x) in the rod is governed by the above problem (1.1), and in this case, k(u ) is the thermal conductivity and f (x, u ) is the internal source that may also be temperature dependent. Various aspects of such heatconduction problems, such as the qualitative analysis of the equations and the computation of the solutions, have been

✩ This work was supported in part by the National Natural Science Foundation of China, No. 10571059, E-Institutes of Shanghai Municipal Education Commission, No. E03004, the Natural Science Foundation of Shanghai, No. 10ZR1409300 and Shanghai Leading Academic Discipline Project No. B407. Address for correspondence: Department of Mathematics, East China Normal University, Shanghai 200241, People’s Republic of China. E-mail address: [email protected].

*


Y.-M. Wang / Applied Numerical Mathematics 61 (2011) 38–52

39

investigated in the literature (cf. [6,20,25,26,29–33]). For some other applications of problem (1.1) in the theory of diffusion we see [1,10–12,20,31–34] and the references therein. As we know, it is difficult to give the analytical solution of problem (1.1), even if the function f (·, u ) is linear in u. Therefore, numerical methods are of considerable practical interest. If k(u ) ≡ 1, problem (1.1) is reduced to a semilinear problem. For solving such a semilinear problem, various numerical methods have been developed in the literature. These methods include, for example, finite difference methods in [4,7,15,18,23,43–46], Petrov-Galerkin methods in [21], variation methods in [14], shooting methods in [22,36], spline methods in [5,16,24] and multiderivative methods in [41]. In the context of finite difference discretizations, one of the well-known methods is Numerov’s method (cf. [3,23,28]). Because Numerov’s method possesses fourth-order accuracy and a compact property, it has attracted considerable attention and has been extensively used in practical computations (cf. [2,3,13,15,18,19,37,43–46]). The compact property of Numerov’s method means that the difference stencil in this method only utilizes mesh points directly adjacent to the mesh point at which a difference approximation is being made. This property makes that the boundary conditions can be easily treated in the same manner as in the standard second-order method. In other words, Numerov’s method requires only a regular threepoint difference stencil similar to that used in the standard second-order method but still possesses fourth-order accuracy. For the developments of Numerov’s method, we refer to the survey paper [4]. If k(u ) depends on u, problem (1.1) becomes more complicated due to the nonlinearity of the function k(u ). In this case, Numerov’s method cannot be directly applied, even if k(u ) is linear in u. However, in the article [44], the author showed that after a proper transformation of (1.1), one can use Numerov’s method to find indirectly an accurate numerical solution of (1.1). The numerical method proposed there consists of three steps: Step 1. Using a transformation T (see (2.1)) to transform (1.1) into a semilinear problem; Step 2. Applying Numerov’s method to solve the resulting semilinear problem; Step 3. Making use of the inverse T −1 to obtain the numerical solution of problem (1.1). Although the above method possesses the same accuracy as Numerov’s method regardless of the strong nonlinearity of problem (1.1), it is only valid for the situation when the inverse T −1 can be constructed explicitly. This limits its applications since the construction of the inverse T −1 is not always easy to do. The purpose of this paper is to search a new technique for developing a direct method that avoids constructing the inverse T −1 and directly offers the numerical solution of (1.1) but still maintains the same accuracy as Numerov’s method. Specifically, our approach here is to formulate (1.1) as a coupled system of a semilinear two-point boundary value problem and a nonlinear functional equation and then to discretize the coupled system by Numerov’s method. To solve the resulting discrete problem we develop a linear monotone iterative algorithm by the method of upper and lower solutions and its associated monotone iterations. This approach makes it possible to compute accurately the numerical solution of (1.1) by using Numerov’s method without finding the inverse T −1 . Consequently, the full potential of Numerov’s method for strongly nonlinear two-point boundary value problem (1.1) is realized. The outline of the paper is as follows: In the next section, we formulate the strongly nonlinear problem (1.1) as a coupled system of a semilinear two-point boundary value problem and a nonlinear functional equation and then apply Numerov’s method to discretize the coupled system. Section 3 is devoted to a linear monotone iterative algorithm for the resulting discrete problem. It is shown that using an upper solution and a lower solution as initial iterations the iterative algorithm yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the resulting discrete problem. The convergence of the method is investigated in Section 4 where the fourth-order accuracy of the method is proven. In Section 5, we give some applications to several model problems arising from heat-conduction, population dynamics and chemical engineering. Some numerical results are presented to demonstrate the monotone property of the iterative algorithm and the high accuracy of the numerical solution. The final section is for some concluding remarks. 2. Numerov’s method Let I be an interval in R such that α , β ∈ I . Throughout this paper, we assume that the nonlinear functions f (·, u ) and k(u ) satisfy the following basic hypothesis:

(H )

f (·, u ) and k(u ) are C 1 -functions of u , and there exists a positive constant k0 such that k(u ) k0 > 0 for all u ∈ I .

In order to apply Numerov’s method to (1.1), we form the problem as a coupled system of a semilinear two-point boundary value problem and a nonlinear functional equation. Let δ ∈ I . Define

u v = T (u ) =

k(s) ds,

∀u ∈ I .

δ

Using the transformation T we transform problem (1.1) into the following coupled system:

(2.1)

40


⎧ ⎨ ⎩

−

d2 v dx2

= f (x, u ),

v (0) = T (α ),

T u (x) = v (x),

0 x 1,

(2.2)

v (1) = T (β).

It is clear that u (x) is a solution of (1.1) in I (i.e., u (x) ∈ I for each x) then (u (x), v (x)) = (u (x), T (u (x))) is a solution of the coupled system (2.2) and vice versa. As compared with the treatment in [44], a new feature of the above system is that it avoids the inverse T −1 . This makes it possible to compute accurately the numerical solution of (1.1) without finding the inverse T −1 . Let γ (x) be any nonnegative function satisfying

γ (x) − f u (x, u )/k0 for all 0 x 1 and u ∈ I ,

(2.3)

where f u = d f /du. Then we have

γ (x)k(u ) + f u (x, u ) 0 for all 0 x 1 and u ∈ I .

(2.4)

In other words, the function

g (x, u ) = γ (x) T (u ) + f (x, u )

(0 x 1)

(2.5)

is monotone nondecreasing in u for u ∈ I . By adding γ (x) v (x) on both sides of the first equation in (2.2) and using the relation (2.1), we obtain the following equivalent system:

⎧ ⎨ ⎩

−

d2 v dx2

+ γ (x) v (x) = g (x, u ),

v (0) = T (α ),

T u (x) = v (x),

0 x 1,

(2.6)

v (1) = T (β).

Our Numerov’s scheme for the numerical solution of (2.2) is based on the above equivalent form (2.6) (for the reason why we don’t discretize (2.2) directly, see Remark 2.1). Let h = 1/ N be the mesh size, and let xi = ih (0 i N ) be the mesh points. For convenience, we introduce the finite difference operators δh2 and Ph as follows:

δh2 u (xi ) = u (xi −1 ) − 2u (xi ) + u (xi +1 ), h 2

Ph u (xi ) =

12

1 i N − 1,

u (xi −1 ) + 10u (xi ) + u (xi +1 ) ,

1 i N − 1.

(2.7)

Using the following Numerov’s formula (cf. [4,28])

δh2 u (xi ) = Ph u (2) (xi ) + O h6 , where u (2) (xi ) =

2

d u ( xi ) , dx2

1 i N − 1,

(2.8)

we have from (2.6) that

−δh2 v (xi ) + Ph γ (xi ) v (xi ) = Ph g xi , u (xi ) + O h6 , v (x0 ) = T (α ),

T u (xi ) = v (xi ),

v (x N ) = T (β).

1 i N − 1,

(2.9)

After dropping the O (h6 ) term, we derive a finite difference scheme of (2.6) as follows:

−δh2 v i + Ph (γi v i ) = Ph g i (u i ), v 0 = T (α ),

T (u i ) = v i ,

1 i N − 1,

v N = T (β),

(2.10)

where u i and v i represent the approximations of u (xi ) and v (xi ), respectively, and

γi = γ (xi ),

g i (u i ) = g (xi , u i ).

To write (2.10) in a vector form, we let J = ( J i , j ) and B = ( B i , j ) be the ( N − 1)-order symmetric tridiagonal matrices with the following elements:

J i , i = 2, B i , i = 5 /6 ,

J i ,i −1 = J i ,i +1 = −1, B i ,i −1 = B i ,i +1 = 1/12,

1 i N − 1,

and let Γ be the diagonal matrix given by

Γ = diag(γ1 , γ2 , . . . , γ N −1 ). Also we define the following ( N − 1)-dimensional column vectors:

(2.11)


U = (u 1 , u 2 , . . . , u N −1 )T ,

41

V = ( v 1 , v 2 , . . . , v N −1 )T ,

T

G (U ) = g 1 (u 1 ), g 2 (u 2 ), . . . , g N −1 (u N −1 ) ,

T

T (U ) = T (u 1 ), T (u 2 ), . . . , T (u N −1 ) ,

T

H = T (α ) + h2 g (0, α ) − γ0 T (α ) /12, 0, . . . , 0, T (β) + h2 g (1, β) − γ N T (β) /12 .

(2.12)

Then system (2.10) can be written in the vector form:

J + h 2 B Γ V = h 2 B G (U ) + H ,

(2.13)

T (U ) = V .

This is a coupled system of a nonlinear algebraic equation and a nonlinear functional equation. To analyze it and develop a linear monotone iterative algorithm for its solution, we review some well-known properties about the matrices J and B. For two constants M and M satisfying M M > −π 2 we define

⎧ 12 ⎪ ⎪ ⎪ M, ⎪ ⎪ ⎪ ⎪ ⎨ 1, h( M , M ) = 12 12 ⎪ min { , (1 + πM2 ) }, ⎪ ⎪ π2 M ⎪ ⎪ ⎪ ⎪ ⎩ 12 (1 + M ), π2 π2

M > −8, M > 0, M > −8, M 0, (2.14)

M −8, M > 0, M −8, M 0.

Lemma 2.1. (See Lemma 2.3 of [45] or Lemma 3.1 of [46].) Let M = diag( M 1 , . . . , M N −1 ) be a diagonal matrix and let M and M be two constants such that

−π 2 < M M i M ,

i = 1, 2, . . . , N − 1.

(2.15)

Then the inverse ( J + h2 B M )−1 exists and is nonnegative provided that h < h( M , M ). The following lemma will be used in Section 4 to prove the convergence of the resulting finite difference scheme (2.10) (or (2.13)). Lemma 2.2. (See Lemma 2.4 of [45].) Let M, M and M i (i = 1, 2, . . . , N − 1) be the given constants satisfying (2.15), and let M = diag( M 1 , . . . , M N −1 ). Also let Z and R be the vectors in R N −1 such that

J + h2 B M Z = R .

(2.16)

Then when h < h( M , M ),

⎧ −2 ⎪ M 0, ⎨ ( R ∞ /8)h , Z ∞ ( R ∞ /(8 + M ))h−2 , −8 < M < 0, ⎪ ⎩ ( R ∞ /(2π (1 − θ)))h−2 , −π 2 < M −8, h H ,

(2.17)

where H < h( M , M ) and

θ =−

M

π 2 (1 − π 2 H 2 /12)

< 1.

Remark 2.1. The formulation of the system (2.13) is based on the equivalent form (2.6). This system can be clearly written as

J V = h2 BF (U ) + H ,

(2.18)

T (U ) = V , where

T

F (U ) = f 1 (u 1 ), f 2 (u 2 ), . . . , f N −1 (u N −1 ) ,

f i (u i ) = f ( xi , u i )

(i = 1, 2, . . . , N − 1).

(2.19)

System (2.18) is the discretization of the original problem (2.2) by Numerov’s method. As compared with (2.18), an important property of (2.13) is that the function G (U ) in this system is monotone nondecreasing in U (see (2.4)). This property, as we shall see later, makes it more convenient to design a linear monotone iterative algorithm for the computation of the solutions. This is our main reason for considering (2.13) instead of (2.18).

42


Remark 2.2. Much research has been done for the numerical solutions of the initial value problems (IVPs) related to the second-order ordinary differential equations. Various numerical methods have appeared in the literature, including the linear multistep methods and hybrid methods which are often used in practical applications (see [8,9,17,27,35,37–40,42]). A detailed survey for the recent developments of these methods is given in [9]. It is interesting to examine if the techniques in these methods can be applied to present two-point boundary value problems. Further investigations will be made in the future. 3. Linear monotone iterative algorithm To develop a linear monotone iterative algorithm for the solutions of (2.13) we use the method of upper and lower solutions and its associated monotone iterations. A pair of ordered upper and lower solutions of (2.13) are defined as follows: Definition 3.1. A vector (U˜ , V˜ ) ∈ R N −1 × R N −1 is called an upper solution of (2.13) if

J + h2 B Γ V˜ h2 B G (U˜ ) + H ,

(3.1)

T (U˜ ) V˜ .

Similarly, (Uˆ , Vˆ ) ∈ R N −1 × R N −1 is called a lower solution of (2.13) if it satisfies the above inequalities in the reversed order. A pair of upper and lower solutions (U˜ , V˜ ) and (Uˆ , Vˆ ) are said to be ordered if (U˜ , V˜ ) (Uˆ , Vˆ ). In the above definition, inequalities between vectors are in the sense of componentwise. It is clear that every solution of (2.13) is an upper solution as well as a lower solution. For any W in R N −1 , we denote w i the i-th component of W . Given a pair of ordered upper and lower solutions (U˜ , V˜ ) and (Uˆ , Vˆ ), we define the sectors

S = (U , V ) ∈ R N −1 × R N −1 ; (Uˆ , Vˆ ) (U , V ) (U˜ , V˜ ) , uˆ i , u˜ i = {u i ∈ R; uˆ i u i u˜ i },

(3.2)

and define the interval I0 by

I0 = min uˆ i , max u˜ i . i

i

We assume that I0 ⊂ I throughout the paper, where I is defined in hypothesis ( H ). To compute the solutions of (2.13) we use the following linear iterative scheme:

J + h2 B Γ V (m+1) = h2 B G U (m) + H ,

Γ (m) U (m+1) = Γ (m) U (m) − T U (m) + V (m+1) ,

(3.3)

m = 0, 1 , 2 , . . . ,

where the initial iteration (U (0) , V (0) ) is either (U˜ , V˜ ) or (Uˆ , Vˆ ), and

(m) (m) Γ (m) = diag γ1 , . . . , γ N −1 , (m)

The functions u i

γi(m) = max k(s); u (i m) s u (i m) .

(3.4)

s

, u (i m) in the definition of γi(m) are the respective components of U (m) and U (m) which are obtained from

(3.3) with (U (0) , V (0) ) = (U˜ , V˜ ) and (U (0) , V (0) ) = (Uˆ , Vˆ ), respectively.

To show that the sequences given by (3.3) are well-defined it is crucial that the sequences {U (m) } and {U (m) } possess the property U (m) U (m) for every m. It is clear from (3.4) that

Γ (m) U − U − T (U ) + T U 0 whenever U (m) U U U (m) .

(3.5)

Moreover, by the monotone nondecreasing property of g (x, u ) in u,

G (U ) G U

whenever Uˆ U U U˜ .

(3.6) that the inverse ( J + h B Γ )−1 exists and is a nonnegative 2

We observe from Lemma 2.1 and the nonnegative property of Γ matrix provided that h < h(0, Γ ∞ ). These properties lead to the following well-defined and monotone properties of the sequences from (3.3). Lemma 3.1. Let hypothesis ( H ) be satisfied and let h < h(0, Γ ∞ ). Also let (U˜ , V˜ ) and (Uˆ , Vˆ ) be a pair of ordered upper and lower solutions of (2.13). Then the sequences {(U (m) , V (m) )}, {(U (m) , V (m) )} and {Γ (m) } given by (3.3) and (3.4) with (U (0) , V (0) ) = (U˜ , V˜ )


43

and (U (0) , V (0) ) = (Uˆ , Vˆ ) are all well-defined and possess the monotone property

(Uˆ , Vˆ ) U (m) , V (m) U (m+1) , V (m+1) U (m+1) , V (m+1) U (m) , V (m) (U˜ , V˜ )

(3.7)

for every m = 0, 1, 2, . . . . Proof. Let m = 0 in (3.3). Since U (0) = U˜ , U (0) = Uˆ and U˜ Uˆ , the elements

γi(0) of the diagonal matrix Γ (0) are well-

defined. By hypothesis ( H ), the inverse (Γ (0) )−1 exists and is a nonnegative matrix. Hence the first iterations and (U (1) , V (1) ) exist and satisfy the relation

(U (1) , V (1) )

⎧ ⎪ J + h2 B Γ V (0) − V (1) = J + h2 B Γ V˜ − h2 B G (U˜ ) − H , ⎪ ⎨ J + h2 B Γ V (1) − V (0) = h2 B G (Uˆ ) + H − J + h2 B Γ Vˆ , ⎪ ⎪ ⎩ J + h2 B Γ V (1) − V (1) = h2 B G (U˜ ) − G (Uˆ ) .

(3.8)

By (3.1) and (3.6), the right-hand side of the above relation is nonnegative, and thus by the nonnegative property of ( J + h2 B Γ )−1 , V (0) V (1) V (1) V (0) . Knowing this relation we have from (3.1), (3.3) and (3.5) that

Γ (0) U (0) − U (1) = T (U˜ ) − V (1) T (U˜ ) − V˜ 0, Γ (0) U (1) − U (0) = − T (Uˆ ) + V (1) − T (Uˆ ) + Vˆ 0, Γ (0) U (1) − U (1) = Γ (0) (U˜ − Uˆ ) − T (U˜ ) + T (Uˆ ) + V˜ − Vˆ 0. In view of (Γ (0) )−1 0, U (0) U (1) U (1) U (0) . This proves that (3.7) holds for m = 0. Assume, by induction, that for some m 1, (U (m) , V (m) ), (U (m) , V (m) ), (U (m−1) , V (m−1) ) and (U (m−1) , V (m−1) ) exist and (3.7) holds when m is replaced by m − 1. It follows from Uˆ U (m) U (m) U˜ that Γ (m) is well-defined and its inverse (Γ (m) )−1 is nonnegative. This ensures that (U (m+1) , V (m+1) ) and (U (m+1) , V (m+1) ) exist. By (3.3) and (3.6),

J + h2 B Γ

2

2

J + h BΓ J + h BΓ

(m−1)

V (m) − V (m+1) = h2 B G U (m−1) − G U (m) V

(m+1)

V

(m+1)

−V

(m)

−V

(m+1)

2

(m)

0,

=h B G U −G U 0, (m) (m) 2 =h B G U −G U 0.

The nonnegative property of ( J + h2 B Γ )−1 leads to that V (m) V (m+1) V (m+1) V (m) . This relation together with (3.3) and (3.5) yields

Γ (m) U (m) − U (m+1) Γ (m−1) U (m−1) − U (m) − T U (m−1) + T U (m) 0, Γ (m) U (m+1) − U (m) Γ (m−1) U (m) − U (m−1) − T U (m) + T U (m−1) 0, Γ (m) U (m+1) − U (m+1) Γ (m) U (m) − U (m) − T U (m) + T U (m) 0 which implies that U (m) U (m+1) U (m+1) U (m) . The conclusion of the lemma follows from the principle of induction.

2

In view of the monotone property (3.7), the limits

lim U (m) , V (m) = (U , V ),

m→∞

lim U (m) , V (m) = (U , V )

m→∞

(3.9)

exist and satisfy

(Uˆ , Vˆ ) U (m) , V (m) U (m+1) , V (m+1) (U , V ) (U , V ) U (m+1) , V (m+1) U (m) , V (m) (U˜ , V˜ ), m = 0, 1, 2, . . . .

(3.10)

The following theorem shows that the limits (U , V ) and (U , V ) are the maximal and minimal solutions of (2.13) in S , respectively, in the sense that if (U , V ) is any solution of (2.13) in S then (U , V ) (U , V ) (U , V ). Theorem 3.1. Let the conditions in Lemma 3.1 hold. Then the sequences {(U (m) , V (m) )} and {(U (m) , V (m) )} converge monotonically to the maximal solution (U , V ) and the minimal solution (U , V ) of (2.13) in S , respectively. Moreover, the relation (3.10) holds. Proof. Let

γ i = maxs {k(s); u i s u i } and Γ = diag(γ 1 , . . . , γ N −1 ), where u i and u i are the respective components of

the limits U and U in (3.9). Then by the definition of Γ (m) , Γ (m) Γ (m+1) Γ for every m, and so the sequence {Γ (m) } converges as m → ∞. Letting m → ∞ in (3.3) shows that both (U , V ) and (U , V ) are solutions of (2.13) in S . We next show

44


that these solutions are the maximal and minimal solutions of (2.13) in S . Let (U , V ) be any solution of (2.13) in S . Then by (2.13) and (3.3),

J + h2 B Γ

V (m+1) − V = h2 B G U (m) − G (U ) ,

Γ (m) U (m+1) − U = Γ (m) U (m) − U − T U (m) + T (U ) + V (m+1) − V , Since (U , V ) ∈ S , the above relation for m = 0 gives

J + h2 B Γ

m = 0, 1 , 2 , . . . .

(3.11)

V (1) − V = h2 B G (U˜ ) − G (U ) 0,

Γ (0) U (1) − U = Γ (0) (U˜ − U ) − T (U˜ ) + T (U ) + V (1) − V V (1) − V .

(3.12)

This yields (U (1) , V (1) ) (U , V ) by using the nonnegativity of ( J + h2 B Γ )−1 and (Γ (0) )−1 . A similar argument gives (U (1) , V (1) ) (U , V ). It follows from an induction argument that (U (m) , V (m) ) (U , V ) (U (m) , V (m) ) for every m. This shows (U , V ) (U , V ) (U , V ), which proves the maximal and minimal property of (U , V ) and (U , V ). 2 It is obvious from the maximal and minimal property of (U , V ) and (U , V ) that if (U , V ) = (U , V ) (≡ (U ∗ , V ∗ )) then S . To ensure (U , V ) = (U , V ), it is necessary to impose some additional conditions on f (·, u ). To give such a sufficient condition, we define

(U ∗ , V ∗ ) is the unique solution of (2.13) in

f u ( xi , ξ ) , − k(η) ξ,η∈uˆ i ,u˜ i

σ = max max i

σ = min i

f u ( xi , ξ ) − k(η) ξ,η∈uˆ i ,u˜ i min

(3.13)

and assume

σ > −π 2 ,

h < h(σ , σ ).

(3.14)

Theorem 3.2. Let the conditions in Lemma 3.1 hold. If, in addition, condition (3.14) holds, then the sequences {(U (m) , V (m) )} and {(U (m) , V (m) )} converge monotonically from above and below, respectively, to the unique solution (U ∗ , V ∗ ) of (2.13) in S . Proof. It suffices to show (U , V ) = (U , V ), where (U , V ) and (U , V ) are the respective limits of the sequences {(U (m) , V (m) )} and {(U (m) , V (m) )}. By (2.13) (or (2.18)),

J ( V − V ) = h 2 B F (U ) − F (U ) ,

(3.15)

T (U ) − T (U ) = V − V . Applying the mean-value theorem we obtain

T (U ) − T (U ) = T u Θ (U − U ),

F (U ) − F (U ) = F u (Θ)(U − U ),

(3.16)

where

F u (Θ) = diag f u (x1 , θ1 ), . . . , f u (x N −1 , θ N −1 ) ,

(θ1 , θ2 , . . . , θ N −1 ) ∈ Uˆ , U˜ ,

T u Θ = diag k θ1 , . . . , k θ N −1 ,

θ1 , θ2 , . . . , θ N −1 ∈ Uˆ , U˜ .

(3.17)

Using the relation (3.16) in (3.15) leads to

J − h2 BF u (Θ) T u Θ

−1

( V − V ) = 0.

(3.18)

The condition (3.14) implies that the matrix − F u (Θ)( T u (Θ ))−1 satisfies the condition of Lemma 2.1, and therefore by this lemma, the inverse ( J − h2 BF (Θ)( T (Θ ))−1 )−1 exists. It follows from (3.18) that V = V . This implies T (U ) − T (U ) = 0 u

u

which ensures U = U . This proves (U , V ) = (U , V ).

2

Remark 3.1. (a) If k(u ) ≡ 1 then all the conclusions in Theorems 3.1 and 3.2 hold true for the semilinear problem (1.1). In this situation, the iteration process (3.3) is reduced to that in [43] and the uniqueness condition (3.14) becomes the one in [46]. (b) For each m, the iterations V (m) and U (m) from (3.3) can be computed, respectively, by solving a tridiagonal system of linear algebraic equations and a diagonal system of linear algebraic equations, and thus (3.3) provides a linear monotone iterative algorithm. Remark 3.2. We see from the monotone property (3.10) that for each m, (U (m) , V (m) ) is an upper bound of the maximal solution (U , V ), while (U (m) , V (m) ) gives a lower bound of the minimal solution (U , V ). Moreover, these bounds are improved, step-by-step, as m increases. If (U , V ) = (U , V ) (≡ (U ∗ , V ∗ )) then they become improved upper and lower bounds of (U ∗ , V ∗ ). This demonstrates the superiority of the monotone convergence over the ordinary convergence.


45

The linear iterative scheme (3.3) can be replaced by the following form

J + h2 B Γ V (m+1) = h2 B G U (m) + H ,

Γ (0) U (m+1) = Γ (0) U (m) − T U (m) + V (m) ,

(3.19)

m = 0, 1 , 2 , . . .

which results in less computations in the process of iterations but still maintains the monotone convergence (3.10) of the sequences. The latter result follows the same line as in the proofs of Lemma 3.1 and Theorem 3.1. However, the sequence given by (3.19) converges slower than the one by (3.3). Specifically, we have the following comparison result. Theorem 3.3. Let the conditions in Lemma 3.1 hold, and let (U˜ , V˜ ) and (Uˆ , Vˆ ) be a pair of ordered upper and lower solutions of (2.13). Denote by {(U (m) , V (m) )} and {(U (m) , V (m) )} the sequences from (3.3) with (U (0) , V (0) ) = (U˜ , V˜ ) and (U (0) , V (0) ) = (Uˆ , Vˆ ), and by

{(U ∗ , V ∗ )} and {(U ∗ , V ∗ )} the sequences from (3.19) with (U ∗ , V ∗ ) = (U˜ , V˜ ) and (U ∗ , V ∗ ) = (Uˆ , Vˆ ). Then we have (m)

(m)

(m)

(m)

(0)

U (m) , V (m) U (∗m) , V (∗m) ,

(0)

U (m) , V (m) U (∗m) , V (∗m) ,

(0)

(0)

m = 0, 1, 2, . . . .

(3.20)

Proof. By (3.3) and (3.19),

J + h2 B Γ

V (∗m+1) − V (m+1) = h2 B G U (∗m) − G U (m) ,

(3.21)

and

Γ (0) U (∗m+1) − U (m+1) = Γ (0) U (∗m) − U (m) + Γ (0) − Γ (m) U (m) − U (m+1) − T U (∗m) + T U (m) + V (∗m) − V (m+1) .

(3.22)

Since Γ (0) Γ (m) , U (m) U (m+1) and V (m) V (m+1) , the equality (3.22) implies

Γ (0) U (∗m+1) − U (m+1) Γ (0) U (∗m) − U (m) − T U (∗m) + T U (m) + V (∗m) − V (m) .

(3.23)

In view of (U (0) , V (0) ) = (U ∗ , V ∗ ) = (U˜ , V˜ ), we have from the above relations (3.21) and (3.23) for m = 0 that (0)

J + h2 B Γ

(0)

Γ (0) U (∗1) − U (1) 0.

V (∗1) − V (1) = 0,

The nonnegative property of ( J + h2 B Γ )−1 and (Γ (0) )−1 yields that V (1) = V ∗ and (3.23) we get from (3.5) and (3.6) that

(1)

J + h2 B Γ

(2)

Γ (0) U (∗2) − U (2) 0.

V (∗2) − V (2) 0,

This proves V (2) V ∗

(1)

and U (1) U ∗ . Using this result in (3.21)

(2)

and U (2) U ∗ . An induction argument shows that

U (m) , V (m) U (∗m) , V (∗m) ,

m = 0, 1 , 2, . . . .

Another inequality in (3.20) can be proved in the same manner.

2

Remark 3.3. The comparison result in Theorem 3.3 implies that with the same initial iteration, which is either an upper solution or a lower solution, the sequence given by (3.3) converges faster than the one by (3.19). 4. Convergence of the method In this section, we deal with the convergence of finite difference scheme (2.10) (or (2.13)), and show its fourth-order accuracy. Let (u (xi ), v (xi )) be the value of the solution of (2.2) at xi , and let (u i , v i ) stand for the solution of (2.10). Since, by (2.2), (2.5) and (2.10),

g xi , u ( xi ) = γ ( xi ) v ( xi ) + f xi , u ( xi ) ,

g i (u i ) = γi v i + f (xi , u i ),

we have from (2.9) and (2.10) that the solutions (u (xi ), v (xi )) and (u i , v i ) satisfy, respectively, the systems

−δh2 v (xi ) = Ph f xi , u (xi ) + O h6 , v (x0 ) = T (α ),

and

1 i N − 1,

v (x N ) = T (β),

−δh2 v i = Ph f (xi , u i ), v 0 = T (α ),

T u (xi ) = v (xi ),

T (u i ) = v i ,

1 i N − 1,

v N = T (β).

Based on (4.1) and (4.2), we show the convergence of (u i , v i ) to (u (xi ), v (xi )) as h → 0.

(4.1)

(4.2)

46


Theorem 4.1. Assume that u i , u (xi ) ∈ Λ ⊂ I for each i and some interval Λ in R, and assume

f u (x, ξ )

< π 2,

k(η)

0 x 1, (ξ, η) ∈ Λ × Λ.

(4.3)

Then for sufficiently small h,

maxu (xi ) − u i ch4 ,

max v (xi ) − v i ch4 ,

i

(4.4)

i

where c is a positive constant independent of h. Proof. Let e i = u (xi ) − u i and e i = v (xi ) − v i . A subtraction of the corresponding equations in the systems (4.1) and (4.2) and using the mean-value theorem lead to

−δh2 e i = Ph f u (xi , ξi )e i + O h6 ,

e

0

= e

N

k ξi e i = e i ,

1 i N − 1,

= 0,

(4.5)

where ξi and ξi are two intermediate values between u (xi ) and u i . Define

T

E = e 1 , e 2 , . . . , e N −1 ,

E = (e 1 , e 2 , . . . , e N −1 )T ,

F u (Ξ ) = diag f u (x1 , ξ1 ), f u (x2 , ξ2 ), . . . , f u (x N −1 , ξ N −1 ) ,

T u Ξ = diag k ξ1 , k ξ2 , . . . , k ξ N −1 . Then the vector form of (4.5) is given by

JE = h2 BF u (Ξ ) E + O h6 ,

(4.6)

T u Ξ E = E . This yields

J − h2 BF u (Ξ ) T u Ξ

−1

E = O h6 .

(4.7)

By (4.3), the matrix − F u (Ξ )( T u (Ξ ))−1 satisfies the condition (2.15). An application of Lemma 2.2 to (4.7) shows that for sufficiently small h, E ∞ c 1 h4 where c 1 is a positive constant independent of h. Hence E ∞ c 1 ( T u (Ξ ))−1 ∞ h4 1 4 c 1 k− 0 h . This proves the estimate (4.4). 2 Remark 4.1. (a) Theorem 4.1 shows that as h → 0, the solution (u i , v i ) of (2.10) converges to the solution (u (xi ), v (xi )) of (2.2) with the convergence order O (h4 ) in the L ∞ -norm. (b) If k(u ) ≡ 1, the convergence result in Theorem 4.1 coincides with that in [45] for the semilinear problem. 5. Applications and numerical results In this section, we apply the results of the previous sections to several model problems arising from heat-conduction, population dynamics and chemical engineering. We present some numerical results to demonstrate the monotone property of the linear iterative algorithm (3.3) and the high accuracy of the numerical solution. All computations are carried out by using a MATLAB subroutine on a Pentium-4 computer with 2G memory. Since the solution of (1.1) is usually nonnegative for practical problems, we assume the boundary values α 0 and β 0 in this section. Also we assume that the functions f (·, u ) and k(u ) are C 1 -functions of u, and there exists a positive constant ρ max{α , β} such that k(u ) k0 > 0 for all u ∈ [0, ρ ]. This implies that the functions f (x, u ) and k(u ) satisfy hypothesis ( H ) with the set I = [0, ρ ]. For this situation, we always take δ = 0 in the definition of transformation T . Under these assumptions, the zero vector (Uˆ , Vˆ ) = (0, 0) ∈ R N −1 × R N −1 is a lower solution of (2.13) if

f (0, α ) 0,

f (1, β) 0,

f (x, 0) 0 (0 < x < 1).

(5.1)

To construct an upper solution of (2.13), we let (U˜ , V˜ ) = (ρ E , T (ρ )E ), where E = (1, 1, . . . , 1)T ∈ R N −1 . A simple calculation shows that this vector is an upper if

JT (ρ )E h2 BF (ρ E ) + H , where F (u ) is defined by (2.19). The above inequality is satisfied by any

ρ satisfying


⎧ f (x, ρ ) 0 (0 x 1), ⎪ ⎪ ⎨ ρ

α ⎪ ⎪ ⎩

k(s) ds max 0

β 2

(5.2)

2

k(s) ds + h f (0, α )/12, 0

47

k(s) ds + h f (1, β)/12 . 0

We now consider three examples to illustrate the applications of the above construction and the results in the previous sections. Example 5.1. Consider the heat transfer in a rod with the given temperatures α 0 and β 0 at two ends. If the thermal conductivity is considered temperature dependent and the form of the source function is based on the so-called Boltzmann fourth-power law, the equation governing the steady-state temperature distribution u (x) in the rod is given by (1.1) with

f (x, u ) = σ a4 (x) − u 4 (x) ,

(5.3)

where σ is a positive constant and a(x) 0 denotes the surrounding temperature (see [20,30]). Assume that a(0) α and a(1) β , and let a max0x1 a(x). Since f u (x, u ) = −4σ u 3 , the function γ (x) in (2.3) can be chosen as γ (x) ≡ γ = 4σρ 3 /k0 . In this case, the function g (x, u ) in (2.5) and the difference scheme (2.10) are, respectively, reduced to

g (x, u ) = γ T (u ) + f (x, u ), and

(5.4)

−δh2 v i + γ Ph v i = Ph g i (u i ), v 0 = T (α ),

T (u i ) = v i ,

1 i N − 1,

(5.5)

v N = T (β).

The vector form of (5.5) is given by

J + γ h2 B V = h2 BG(U ) + H ,

(5.6)

T (U ) = V ,

where H = ( T (α ) + h2 ( g (0, α ) − γ T (α ))/12, 0, . . . , 0, T (β) + h2 ( g (1, β) − γ T (β))/12)T . Since a(0) α , a(1) β and a(x) 0, the condition (5.1) is satisfied by the function f (x, u ) in (5.3). This ensures that (Uˆ , Vˆ ) = (0, 0) is a lower solution of (5.6). On the other hand, the condition (5.2) for the present problem holds if

⎧ ρ a, ⎪ ⎪ ⎨ ρ

α β 4 4 2 4 2 4 ⎪ k(s) ds max k(s) ds + h σ a (0) − α /12, k(s) ds + h σ a (1) − β /12 . ⎪ ⎩ 0

0

(5.7)

0

This requirement can be satisfied by a sufficiently large ρ or by taking mesh size h suitably small. Therefore, (U˜ , V˜ ) = (ρ E , T (ρ )E ) and (Uˆ , Vˆ ) = (0, 0) form a pair of ordered upper and lower solutions of (5.6). With respect to this pair, the constants σ and σ in (3.13) is given by

σ = 4σρ 3 max k−1 (η),

σ = 0.

η∈0,ρ

(5.8)

Let h < min{ 12/γ , 12/σ }. Using (U (0) , V (0) ) = (ρ E , T (ρ )E ), (U (0) , V (0) ) = (0, 0) and the functions in (5.3) and (5.4) in the linear iterative scheme (3.3) with Γ = γ I (where I denotes the identity matrix), we can compute the sequences {(U (m) , V (m) )} and {(U (m) , V (m) )}. By Theorem 3.2, these sequences converge monotonically to the unique solution (U ∗ , V ∗ ) of (5.6) in S where

S = (U , V ) ∈ R N −1 × R N −1 ; (0, 0) (U , V ) ρ E , T (ρ )E .

(5.9)

Since f u (x, u ) 0 for all u 0 and all 0 x 1, the condition (4.3) is trivially satisfied. By Theorem 4.1, the solution (u ∗i , v ∗i ) of (5.5) converges to the continuous solution (u ∗ (xi ), v ∗ (xi )) of (2.2), (5.3) at every mesh point xi with the convergence order O (h4 ), as h → 0. To give some numerical results, we consider a special case of the above model problem where α = β = 0, σ = 1 and

k (u ) =

1 1 + u3

,

1/4

π 2 (sin π x + sin4 π x + 3 sin2 π x cos2 π x) a(x) = + sin4 π x (1 + sin3 π x)2

.

(5.10)

48


Fig. 1. The monotone convergence of the sequences at xi = 0.5 for Example 5.1.

Table 1 Numerical solution (U ∗ , V ∗ ) of Example 5.1. xi

u ∗i

v ∗i

0.1 0.2 0.3 0.4 0.5

0.30901814073003 0.58778747153044 0.80901821366068 0.95105728817513 1.00000076672804

0.30677611425942 0.56098264809778 0.72574878708737 0.81026505270532 0.83564923162852

It can be checked that u = sin π x is a solution of this example, and the transformation T in (2.1) is given by

u v = T (u ) = 0

√ 2u − 1 (1 + u )2 1 3π ds = ln 2 + √ arctan √ + . 3 6 18 1+s u −u+1 3 3 1

1

(5.11)

We see that it is very difficult to obtain the inverse T −1 . Since 0 a(x) 1.8 for all 0 x 1, the constants appeared above may be taken as

k 0 = 1/ 1 + ρ 3 ,

a = ρ = 1.8,

γ = σ = 4ρ 3 1 + ρ 3 .

(5.12)

With Γ = γ I in the linear iterative scheme (3.3), we compute the sequences {(U (m) , V (m) )} and {(U (m) , V (m) )} with (U (0) , V (0) ) = (ρ E , T (ρ )E ) and (U (0) , V (0) ) = (0, 0) for various values of h. In all the numerical computations, the basic feature of monotone convergence of the sequences was observed. Let h = 1/40. In Fig. 1, we present some numerical results of these sequences at xi = 0.5, where the solid line denotes the sequence {U (m) } or { V (m) } and the dash-dot line stands for the sequence {U (m) } or { V (m) }. As expected from our theoretical analysis in Theorem 3.2, the monotone convergence of the sequences holds and their common limit (U ∗ , V ∗ ) is the unique solution of (5.6) in S given by (5.9). More numerical results of (U ∗ , V ∗ ) at various xi are explicitly given in Table 1. To demonstrate the accuracy of the numerical solution (U ∗ , V ∗ ), we calculate the order of maximum error which is defined by

orderα (h) = log2

errorα (h)

errorα (h/2) erroru (h) = maxu ∗i − u ∗ (xi ), i

,

α = u, v ,

error v (h) = max v ∗i − v ∗ (xi ), i

(5.13)

where (u ∗i , v ∗i ) is the i-th component of (U ∗ , V ∗ ) and (u ∗ (xi ), v ∗ (xi )) is the continuous solution of (2.2), (5.3) at xi . The termination criterion of iterations is given by

(m) U − U (m)

∞

+ V (m) − V (m) ∞ < 10−13 .

(5.14)

In Table 2, we list the maximum errors erroru (h) and error v (h) as well as the orders of them for various values of h. The data in this table demonstrate that the numerical solution (U ∗ , V ∗ ) has the fourth-order accuracy. This coincides with the analysis very well.


49

Table 2 The accuracy of the numerical solution (U ∗ , V ∗ ) of Example 5.1. h

erroru (h)

orderu (h)

error v (h)

order v (h)

1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512

2.81395566738641e–002 1.24652816805571e–003 9.06433894494185e–005 5.51294410744418e–006 3.42257746477337e–007 2.13555905181906e–008 1.33351585329677e–009 8.10035372111884e–011

4.49661221054842 3.78156986465225 4.03930692923534 4.00966787470430 4.00239749843280 4.00130691012906 4.04110625795553

2.04635438672907e–002 1.18007005963311e–003 7.73722447332537e–005 4.70594950063852e–006 2.97095818502235e–007 1.85340933711586e–008 1.15732573524596e–009 7.04389879757628e–011

4.11611159332585 3.93091257322739 4.03925838585124 3.98548563403334 4.00267485004741 4.00131466422679 4.03827698371076

Example 5.2. Our second example is a logistic equation in population dynamics problems where the diffusion is density dependent. This equation in the one-dimensional domain [0, 1] is governed by (1.1) with

f (x, u ) = σ u (1 − u ) + q(x),

(5.15)

where σ is a positive constant and q(x) 0 is a possible internal source (see [7,11]). For this example, the function γ (x) in (2.3) can be chosen as γ (x) ≡ γ = σ (2ρ − 1)/k0 . In this case, the function g (x, u ) and the difference scheme (2.13) are reduced to (5.4) and (5.6), respectively, where f (x, u ) is replaced by (5.15). On the other hand, the conditions (5.1) and (5.2) for this example are, respectively, reduced to

σ α (1 − α ) + q(0) 0, and

σ β(1 − β) + q(1) 0,

(5.16)

⎧ σρ (1 − ρ ) + q(x) 0 (0 x 1), ⎪ ⎪ ⎨

α ρ β 2 2 ⎪ k(s) ds max k(s) ds + h f (0, α )/12, k(s) ds + h f (1, β)/12 . ⎪ ⎩ 0

0

(5.17)

0

We now consider the above problem with

k(u ) = arctan(1 + u ),

α = β = 0, σ = 2 and

q(x) = 2 arctan(1 + z) −

(2x − 1)2 − 2z(1 − z), 2 + 2z + z2

(5.18)

where z = x(1 − x). In this case, u = z is a solution of the problem, and the transformation T in (2.1) is given by

u v = T (u ) =

arctan(1 + s) ds = (1 + u ) arctan(1 + u ) −

1 2

ln 2 + 2u + u 2 −

π 4

√ + ln 2.

(5.19)

0

Clearly, it is difficult to find the inverse T −1 explicitly. With respect to the above parameters, the condition (5.16) is obviously satisfied, and the condition (5.17) holds by ρ = 2. This implies that the pair (U˜ , V˜ ) = (2E , T (2)E ) and (Uˆ , Vˆ ) = (0, 0) are a pair of ordered upper and lower solutions. Since ρ = 2 and arctan(1 + u ) π /4 for all u 0, we have k0 = π /4 and γ = 24/π . Using (U (0) , V (0) ) = (2E , T (2)E ) and (U (0) , V (0) ) = (0, 0), we compute the corresponding sequences {(U (m) , V (m) )} and {(U (m) , V (m) )} from (3.3) with Γ = γ I for the present problem, where γ = 24/π . Let h = 1/40. Some numerical results of these sequences at xi = 0.5 are plotted in Fig. 2, where the solid line denotes the sequence {U (m) } or { V (m) } and the dashdot line stands for the sequence {U (m) } or { V (m) }. As expected from our theoretical analysis in Theorem 3.2, the monotone convergence of the sequences holds and their common limit (U ∗ , V ∗ ) is the unique solution of the present problem in S given by (5.9) where ρ = 2. The maximum errors erroru (h) and error v (h) of the numerical solution (U ∗ , V ∗ ) as well as the orders of them for various values of h are presented in Table 3. In our computations, the termination criterion of iterations is still determined by (5.14). We see from Table 3 that the numerical solution (U ∗ , V ∗ ) has the fourth-order accuracy. Example 5.3. Our last example is given by (1.1) with

f (x, u ) = σ (1 − u ) exp −ν /(1 + u ) + q(x),

(5.20)

where σ and ν are positive constants, and q(x) is a nonnegative function. This problem arises from a chemical reactor with first-order reaction (cf. [10,12,20,31,32,34]). To find γ (x) in (2.3) we observe from (5.20) that

f u (x, u ) = −σ exp −ν /(1 + u )

1−

ν (1 − u ) (1 + u )2

.

(5.21)

50


Fig. 2. The monotone convergence of the sequences at xi = 0.5 for Example 5.2. Table 3 The accuracy of the numerical solution (U ∗ , V ∗ ) of Example 5.2. h

erroru (h)

orderu (h)

error v (h)

order v (h)

1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512

1.58684599293107e–004 1.01956076546672e–005 6.41229860487425e–007 4.01376447634050e–008 2.50951628921747e–009 1.56848256605002e–010 9.50076128880539e–012 5.40345546085064e–013

3.96014245092926 3.99096234202004 3.99781374138991 3.99947474754768 3.99996791204800 4.04518256310889 4.13608892220934

1.42185276064216e–004 9.13580885505971e–006 5.74577388973152e–007 3.59655524528879e–008 2.24866553155678e–009 1.40544686999533e–010 8.51330117512816e–012 4.84223772190262e–013

3.96009579872077 3.99095934055636 3.99781355033980 3.99947477046760 3.99996826760600 4.04516643685877 4.13597285658087

This implies that − f u (x, u ) σ (1 + ν ) for all u 0, and therefore, γ (x) can be chosen as γ (x) ≡ γ = σ (1 + ν )/k0 . Then the function g (x, u ) and the difference scheme (2.13) for this example are reduced to (5.4) and (5.6), respectively, where f (x, u ) is replaced by (5.20). To accommodate the exact solution of u = 1 − x2 /2, we let

k (u ) =

1 + u2 ,

q(x) =

1 + z2 − √

x2 z 1 + z2

− σ (1 − z) exp −ν /(1 + z) ,

The above choice implies that α = 1, β = 1/2, k0 = 1 and q(x) 0 for all x ∈ [0, 1] if transformation T in (2.1) is given by

v = T (u ) =

u

1 + s2 ds =

1 u 1 + u 2 + lnu + 1 + u 2 . 2

z=1−

x2 2

(5.22)

. √

σ 3 exp(ν /2)/ 5. Moreover, the

(5.23)

0

As in the previous examples, we can prove that for any

√

√

√

σ + 2 exp(ν /2) 13 2 21 5 , ρ max , , σ 12 40 ˆ ˆ the pair (U˜ , V˜ ) = (ρ E , T (ρ )E ) and √ (U , V ) = (0, 0) are a pair of ordered upper and lower solutions of the present problem. Let σ = 1, ν = 0.5 and ρ = 1 + 2 exp(1/4). We compute the corresponding numerical solution (U ∗ , V ∗ ) by the linear iterative scheme (3.3) with the initial iterations (U (0) , V (0) ) = (ρ E , T (ρ )E ) and (U (0) , V (0) ) = (0, 0). The termination criterion is the same as in Example 5.1. Fig. 3 shows the monotone convergence of the sequences {(U (m) , V (m) )} and {(U (m) , V (m) )} at xi = 0.5, where the solid line denotes the sequence {U (m) } or { V (m) } and the dash-dot line stands for the sequence {U (m) } or { V (m) } as before. The data in Table 4 show the maximum errors erroru (h) and error v (h) of the numerical solution (U ∗ , V ∗ ) as well as the orders of them for various values of h. We see that the fourth-order accuracy of the numerical solution (U ∗ , V ∗ ) is demonstrated. 6. Concluding remarks In this paper, we gave a numerical treatment for a class of strongly nonlinear two-point boundary value problems by Numerov’s method. The main contribution is to develop precisely a technique for computing accurately the numerical


51

Fig. 3. The monotone convergence of the sequences at xi = 0.5 for Example 5.3. Table 4 The accuracy of the numerical solution (U ∗ , V ∗ ) of Example 5.3. h

erroru (h)

orderu (h)

error v (h)

order v (h)

1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512

2.54113143148826e–005 1.68847875603451e–006 1.05727220867102e–007 6.62888643976345e–009 4.14343670485096e–010 2.58856269752528e–011 1.61481938931729e–012 1.14130926931466e–013

3.91167506288797 3.99730525735689 3.99543651708451 3.99986674925372 4.00060471650773 4.00270654236108 3.82261112679379

3.23999380957840e–005 2.16726185686689e–006 1.35707057924428e–007 8.48539494224809e–009 5.30407606724737e–010 3.31368266159871e–011 2.06989980711114e–012 1.42330591756945e–013

3.90204567565112 3.99730581562862 3.99937013513628 3.99980843898597 4.00059411265700 4.00080260662483 3.86224325092020

solutions of the problems by Numerov’s method without finding any inverse function. We generalized the method of upper and lower solutions to a coupled system of a nonlinear algebraic equation and a nonlinear functional equation. A linear monotone iterative algorithm was presented for the solutions of the resulting discrete problems. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration. The proposed method complements the known method in [44] where it is necessary to find explicitly an inverse function, and so the full potential of Numerov’s method for strongly nonlinear two-point boundary value problems is realized. The numerical results coincide with the analysis very well. We conclude by taking note that the present study serves as an initial exploration of the application of higher-order compact finite difference methods to solve strongly nonlinear differential equations. The analysis is limited to a ordinary differential equation. The extension to partial differential equations is an interesting issue, and more challenging problems concerning this extension will be studied in the future. Acknowledgements The author would like to thank the referees for their valuable comments and suggestions which improved the presentation of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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Higher order optimization and adaptive numerical solution for optimal control of monodomain equations in cardiac electrophysiology Chamakuri Nagaiah ∗ , Karl Kunisch Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36, Graz, A-8010, Austria

a r t i c l e

i n f o

Article history: Received 14 December 2009 Received in revised form 29 July 2010 Accepted 9 August 2010 Available online 27 August 2010 Keywords: Reaction–diffusion equations Monodomain model PDE constraint optimization Adaptive FEM Newton’s optimization algorithm

a b s t r a c t In this work adaptive and high resolution numerical discretization techniques are demonstrated for solving optimal control of the monodomain equations in cardiac electrophysiology. A monodomain model, which is a well established model for describing the wave propagation of the action potential in the cardiac tissue, will be employed for the numerical experiments. The optimal control problem is considered as a PDE constrained optimization problem. We present an optimal control formulation for the monodomain equations with an extra-cellular current as the control variable which must be determined in such a way that excitations of the transmembrane voltage are damped in an optimal manner. The focus of this work is on the development and implementation of an efficient numerical technique to solve an optimal control problem related to a reaction–diffusions system arising in cardiac electrophysiology. Specifically a Newton-type method for the monodomain model is developed. The numerical treatment is enhanced by using a second order time stepping method and adaptive grid refinement techniques. The numerical results clearly show that super-linear convergence is achieved in practice. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction Optimization problems governed by partial differential equations arise in many application areas of natural science and engineering. These problems, now called PDE constrained optimization problems, due to their problem size and/or complexity still they present a significant challenge to efficient numerical realization. One such interesting application is the optimal control of the bidomain model equations in cardiac electrophysiology. The bidomain model [8,18,22] describes the electrical behavior of the cardiac tissue by a reaction–diffusion system coupled with an ordinary differential equations which model the ionic currents associated with the reaction terms. The equations model the fact that currents leaving the extracellular space by traversing the cellular membranes are the sources of the intracellular current density and vice versa. That is, currents leaving the intracellular space are acting as sources of the extracellular current density. The numerical solution of the bidomain equations is computationally expensive, see [25,24], due to the high spatio-temporal resolution required to resolve the fast transients and steep gradients governing wavefront propagation in the heart. Assuming that the anisotropy ratios of the two spaces are equal leads to a reduced model, referred to as the monodomain model, which can be solved at a much lower computational expense by avoiding the time consuming solution of an elliptic PDE [17]. Under many circumstances of practical relevance the monodomain model can be considered to approximate the bidomain model fairly well [19,15]. Cardiac arrhythmia refers to sudden, irregular patterns in the heart rhythm that may cause the heart to stop beating completely or slow down to the point where life is unsustainable. This must be treated immediately to avoid sudden cardiac

*

Corresponding author. E-mail addresses: [email protected] (C. Nagaiah), [email protected] (K. Kunisch).


54

C. Nagaiah, K. Kunisch / Applied Numerical Mathematics 61 (2011) 53–65

death. The only way to reestablish a normal rhythm is to apply a strong electrical shock, a process called defibrillation. During defibrillation extracellular currents are injected via electrodes to establish an extracellular potential distribution which acts to reduce the complexity of the activity. This is achieved either by extinguishing all electrical activity, i.e. the entire tissue returns to its quiescent state, or gradients in V m are smoothed out to drive the system to a less heterogeneous state which reduces the likelihood of triggering new wavefronts via “break” mechanisms when switching off the applied field. In the context of optimally control cardiac arrhythmias, it is essential to determine the control response to an applied electric field as well as the optimal extracellular current density that acts to damp gradients of transmembrane voltage in the system. The main objective of this work is the development and computer implementation of an efficient numerical technique to solve optimal control problems for monodomain equations. The optimal control approach is based on minimizing a properly chosen cost functional J ( v , I e ) depending on the extracellular current I e as input and on the transmembrane potential v as one of the state variables. First- and second-order derivatives of the reduced cost are derived using a Lagrangian based formalism. Numerical technique for solving the optimal control problems requires combining a discretization technique with an optimization method. A traditional method to solve such problems is by using the optimize before discretize approach. In this case one expresses the continuous optimality system first before discretizing it. A second approach is to first discretize the differential equations and the cost and to solve the resulting nonlinear programming method with an efficient method. Clearly it is desirable to follow an approach were these two strategies commute or at least lead to very similar results. For the problem class under consideration here we carried out a detailed comparison for the 1D case in Nagaiah et al. [13]. In this present work we stick to the optimize before discretize technique. In the numerical simulations there are many important factors which put a high demand on the computing time. These include different length and time scales of the reaction terms, strong nonlinearities caused by ionic currents, anisotropy related to the fiber orientation, and rapid changes of the potentials [7,24]. We have chosen the finite element method for the spatial and higher order linearly implicit Runge–Kutta time stepping methods for the temporal discretization. The mesh has to be adapted, during the primal and dual solves, at each time step in all regions in order to preserve the accuracy of the solution. To control the spatial discretization error, a-posteriori error estimators are computed to steer the mesh improvement by refinement and coarsening in each time step during the primal and dual solves in optimization algorithm. The adaptivity strategy is based on the ZZ-error estimator [26] in our work. Various other forms of adaptive mesh refinement techniques were applied successfully for excitable media by varying the spatial or temporal resolution or both [5,7,12,21]. At present our numerical experiments are based on allowing adaptivity in the spatial grids, while the time step is kept constant during the primal and dual solves. More details will be given in Section 4.3. Our numerical realization is based on the public domain software package DUNE [1]. The article is organized as follows: in the next section the governing equations for the action potential and the behavior of the ionic current variables using ionic models are described. In Section 3 the control problem is posed for the monodomain equations and the first and second order derivatives of the reduced cost are characterized. Also a brief description of a Newton’s algorithm is given. The numerical approach for solving the primal and the adjoint state equations is presented in Section 4. Numerical results for two test cases are discussed in Section 5. Finally concluding remarks are given. 2. The governing equations The monodomain model consists in a parabolic reaction–diffusion equation for the transmembrane potential v coupled with a system of ODEs for the gating variables. We set Q = Ω × [0, T ], where Ω ⊂ Rd , d = 2, denotes the cardiac tissue sample domain with Lipschitz boundary ∂Ω . The system is given by

∂v = ∇ · σ¯ i ∇ v − I ion ( v , w ) + I e (x, t ) in Q , ∂t ∂w = g ( v , w ) in Q , ∂t

(1) (2)

where v : Q → R is the transmembrane voltage, w : Q → Rn represents the ionic current variables, σ¯ i : Ω → Rd×d is the intracellular conductivity tensors, I e (x, t ) is an extracellular current density stimulus, I ion ( v , w ) is the current density flowing through the ionic channels and the function g ( v , w ) determines the evolution of the gating variables. The above mentioned Eq. (1) is a parabolic type equation and Eq. (2) is a set of ordinary differential equations which can be solved independently for each node. In the absence of a conductive bath both intracellular and extracellular domains are electrically isolated along the tissue boundaries and homogeneous Neumann boundary conditions are appropriate to reflect this fact. The initial values of transmembrane voltage and ion current variables are prescribed by given constant values. Here the initial and boundary conditions are chosen as

σ¯ i ∇ v · η = 0 on ∂Ω × [0, T ], v (x, 0) = v 0 and w (x, 0) = w 0

(3) on Ω,

(4)


55

where v 0 : Ω → Rd denotes the initial transmembrane potential and w 0 : Ω → Rd is the initial ionic current variables at time t = 0. Ionic model In our numerical experiments, we considered a phenomenological model, namely a variant of the Fitzhugh–Nagumo (FHN) model [20], which is constructed to reproduce the macroscopically observed behavior of the cells. The ionic activity is modeled by an ordinary differential equations. In this case, the I ion ( v , w ) term is a cubic polynomial function in terms of the transmembrane potential v and linear in terms of the gating variable w:

v

I ion ( v , w ) = G v 1 −

g ( v , w ) = η2

v vp

v th

1−

v

+ η1 v w ,

vp

(5)

− η3 w ,

(6)

where G , η1 , η2 , η3 are positive real coefficients, v th is a threshold potential and v p the peak potential. 3. Optimal control problem In this section we specify the optimal control problem that is under consideration. We consider

(P)

min J ( v , I e ),

(7)

e ( v , w , I e ) = 0 in Q ,

where v and w are the state variables, and I e is the control variable of the optimal control problem. Here Q = Ω × (0, T ) denotes the space-time cylinder and the coupled systems of PDE and ODE constraints is expressed as e ( v , w , I e ) = 0, where

e( v , w , I e ) =

∇ · (σ¯ i ∇ v ) −

∂v ∂t

∂w ∂t

− I ion ( v , w ) + I e (x, t )

− g(v , w )

(8)

.

The initial conditions in (4) are explicitly enforced and the homogeneous Neumann boundary condition in (3) will be realized in the variational setting of the FE discretization. We refrain here from entering a function space setting. This requires independent investigations which will be reported elsewhere. The control variable I e is chosen such that it is nontrivial only on the control domain Ωcon of Ω , i.e. I e : Ωcon × (0, T ) → R, and I e is extended by zero on (Ω \ Ωcon ) × (0, T ). It will not be necessary to introduce extra notation for this purpose. It can be argued that for each I e ∈ L 2 ((0, T ) × Ωcon ; R) there exists a unique ( v , w ) ∈ L 2 (0, T ; H 1 (Ω)) ∩ C (0, T ; L 2 (Ω)) ∩ 4

L 4 ( Q ) × L 2 (0, T ; H 1 (Ω)) ∩ C (0, T ; L 2 (Ω)) with ut , w t ∈ L 2 (0, T ; H 1 (Ω)∗ + L 3 ( Q )) such that e ( v ( I e ), w ( I e ), I e ) = 0, see [4]. With v ( I e ) thus defined we introduce the reduced cost functional

ˆJ ( I e ) := J v ( I e ), I e .

(9)

Next we turn to the choice of the cost functional. For this purpose we introduce the observation domain Ωobs ⊂ Ω . The control objective consists in dampening out the excitation wave of the transmembrane voltage Ωobs . We therefore set

J (v , Ie ) =

1

T

| v |2 dΩobs + α

2 0

Ωobs

| I e |2 dΩcon dt ,

(10)

Ωcon

where α is the weight of the cost of the control, which is used to determine the influence of the corresponding components, Ωobs is the observation domain. The inclusion of the tracking type term in the cost functional serves code-validation purposes, see Nagaiah et al. [14] for more numerical results and further discussion. To derive the first derivative of the reduced cost we use the Lagrange functional

T L( v , w , I e , p , q ) = J ( v , I e ) +

T ∂v ∂w − g ( v , w ) q dΩ dt + − I ion ( v , w ) + I e p dΩ dt . ∇ · σ¯ i ∇ v − ∂t ∂t

0 Ω

0 Ω

The first order optimality system is obtained by formally setting the partial derivatives of L equal to 0. We find

Lv :

v |Ωobs + ∇ · σ¯ i ∇ p + pt − ( I ion ) v p − g v q = 0,

(11)

Lw :

−( I ion ) w p − qt − g w ( v , w )q = 0,

(12)

where the subscripts v and w denote the partial derivatives with respect to these variables. Further we obtain the

56


terminal conditions: boundary conditions:

p ( T ) = 0 q ( T ) = 0,

(13)

σ¯ i ∇ p · η = 0 on ∂Ω × [0, T ],

and the optimality condition:

L I e : α I e + p = 0,

(14)

on Ωcon .

(15)

The first order necessary optimality conditions consist of the coupled system of primal equations (1)–(2), adjoint equations (11)–(12), together with initial conditions (4), terminal conditions (13) and the optimality condition (15). 3.1. Newton’s method In this subsection, we present the inexact Newton method for solving the reduced optimization problem. In our case it is infeasible to set up the Hessian matrix. Rather we explain the necessary steps for the computation of “the action of the Hessian of the reduced cost” on a given vector. Once this is achieved approximate Hessian directions of appropriately discretized problems can be computed. This procedure was advocated e.g. in [9], Hinze et al. [10] for solving optimization problem governed by evolutionary partial differential equations. We use the Lagrangian functional based approach to derive the second derivative of the reduced cost functional. Proceeding in formal terms it can be expressed as follows [10]:

ˆJ = L I e I e + δ u ∗ Lu I e + L I e u δ u + δ u ∗ Luu δ u , 1 −1 where δ u = −e − u e I e . In this case e u is only formal since it requires to invert a differential operator. Here, for brevity, we denote u = ( v , w ) and y = (u , I e ). Here and below L and its derivatives are evaluated at ( v , w , I e ). We now introduce the matrix operator T ( I e ) and the second derivative of L as follows,

T (Ie ) =

1 −e − u e Ie

and

Id I e

Lyy =

Luu

Lu I e

LIe u

LIe Ie

,

where Id I e is the identity operator in the control space. From these quantities the representation of the Hessian can be constructed by using the following formula

J ( I e ) = T ∗ ( I e )L y y T ( I e ),

(16)

where T ∗ ( I e ) is the adjoint operator to T ( I e ). Now we carry out these steps for the calculation of the second derivative of the reduced cost functional associated to the monodomain problem. In this process we have to evaluate the first derivatives, i.e. the sensitivities. We calculate these derivatives w.r.t. state and control variables as follows:

e u (δ v , δ w , δ I e ) = and

e I e (δ I e ) =

δ Ie

∇ · (σ¯ i ∇δ v ) − δ v t − [ I ion ] v δ v − [ I ion ] w δ w

δ w t − ηv 2p δ v + η2 η3 δ w

(17)

(18)

0

where

[ I ion ] v =

G v p v th

( v th − v )( v p − v ) − v ( v p − v ) − v ( v th − v ) + η1 w ,

[ I ion ] w = η1 v . The computation of the second derivative operator L applied to the vector (δ v , δ w , δ I e ) can be expressed as

⎛

⎞ δ v |Ω obs − [ I ion ] v v p − η1 δ wp ⎠ −η1 δ vp L y y (δ v , δ w , δ I e ) = ⎝ αδ I e

2G where [ I ion ] v v = δ v v − ( v p − v ) − ( v th − v ) . v p v th

(19)

We note that in order to obtain the action of the Hessian on a given vector, one linearized primal problem and one linearized adjoint equation have to be solved. The basic steps to compute the action of the Hessian are summarized next. 1. Compute the first derivative ˆJ (un ) = α I e + p, which requires one primal and one dual solve. 2. In each step iteratively evaluate the action of ˆJ (un ) on δn which is done using the following sequence of computations.


57

(a) solve the linearized primal equation for δ v , δ w using δ I ek

∇ · (σ¯ i ∇δ v ) − (δ v t + [ I ion ] v δ v + [ I ion ] w δ w )

δ w t − ηv 2p δ v + η2 η3 δ w

=

−δ I ek

0

;

(b) evaluate ( z1 , z2 ) := L y y (δ v , δ w , δ I e ) from Eq. (19); (c) solve the adjoint equation with ( z1 , z2 ) as r.h.s. i.e.

∇ · σ¯ i ∇ w 1 + w 1t − [ I ion ] v w 1 − ηv 2p w 2 −[ I ion ] w w 1 − w 2t + η2 η3 w 2

=

z1 z2

;

(d) finally compute the action of ˆJ (uk ) on δ I ek , i.e. − w 1 + α δ I e . In this way one can compute the action of the Hessian of the reduced cost. Consider the system of step (7) in Algorithm 1, given in Appendix A. Its dimension is that of the control space dimension. To evaluate this step one must use an iterative method, e.g. a linear conjugate gradient method. 4. Numerical discretization For the approximation of the optimality system (1)–(2) and (11)–(12) we use a finite element method for the spatial discretization. This results in an initial value problem for a system of ordinary differential equations. The time discretization is based on an explicit Euler method for the ODE equations and a linearly implicit Runge–Kutta method for the parabolic equations. 4.1. Space discretization using FEM In this subsection we give a brief description of the piecewise linear finite element discretization to solve the monodomain equations. We commence with the primal problem in variational form: find v : [0, T ] → H 1 (Ω) and w : [0, T ] → L 2 (Ω) such that for a.e. t ∈ (0, T )

∇ · σ¯ i ∇ v , ϕ =

∂v + ∂t

∂w , ϕ = g ( v , w ), ϕ . ∂t

I ion ( v , w ) − I e , ϕ

for all ϕ ∈ H 1 (Ω)

Q

(20)

Let V h ⊂ H 1 (Ω) be the finite-dimensional subspace of piecewise Nlinear basis functions with respect to the spatial grid. The approximate solution v is expressed in the form v(t ) = i =0 vi ωi where {ωi }iN=1 denote the basis functions. This results in the following system of nonlinear ordinary differential equations:

M

∂v = −Ai v − Iion (v, w) + Ie , ∂t

(21)

∂w = g(v, w), ∂t v(0) = 0, w(0) = 0, where Ai is the stiffness and M is the mass matrix, and Iion and Ie are vectors defined by Iion = { I ion , ω

{ I e , ω j } Nj=1 , respectively.

(22) (23) N j } j =1

and Ie =

Space discretization of dual problem We use an analogous derivation as for the primal problem and obtain the following semi-discrete form of the dual equations:

M

∂p = −Mobs v + Ai p + M(Iion ) v p + Mqg v , ∂t

∂q T = −gw ( v , w )q − ( I ion ) w p, ∂t p( T ) = 0, q( T ) = 0,

(24) (25) (26)

where Mobs is a mass matrix in observation domain. To solve the semi-discretized primal and the dual problems (21)–(23) and (24)–(26), we first approximate the ODE system solution at the current time step. This gives the ionic current variable update w while solving the primal problem,

58


which is subsequently used to update Eq. (21). Analogously, the adjoint variable update q is used in Eq. (24). In our numerical computations the primal problem is solved by decoupling the system as follows:

step-1:

wn+1 = wn + tg vn , wn ,

∂v = −Ai vn − Iion vn , wn+1 − Ie . ∂t

(27)

n

step-2:

M

(28)

Analogously, the dual problem is decoupled as follows:

step-1:

qn = (1 − t η2 η3 )qn+1 + t η1 vn+1 pn+1 ,

∂ pn = − Mobs vn+1 − Ai pn + M(Iion ) v pn + g v Mqn+1 . step-2: M ∂t

(29) (30)

4.2. Time discretization using linearly implicit RK methods In this subsection we give a brief description of the time discretization for solving the systems of ordinary differential equations which we now express in the following form:

M

∂u = F(u), ∂t

u t 0 = u0 .

(31)

To solve (31), we introduce discrete steps in the time interval [0, T ]:

0 = t 0 , t 1 , . . . , tn = T , which are not necessarily equidistant. We further set τ i = t i +1 − t i and denote by ui the numerical solution at time t i . For time discretization linearly implicit Runge–Kutta methods, specifically Rosenbrock methods, are used. These belong to a large class of methods which try to avoid the nonlinear system and replace it by a sequence of linear ones. For the construction of the Jacobian matrix we used exact derivatives of the vector F(u). For our computations the ROS2 method was employed which has two internal stages to solve in each iteration see [11]. After the time discretization one ends up with a system of linear equations. For solving this linear system the BiCGSTAB method with ILU preconditioning is used. 4.3. Spatial grid adaptivity The adaptive mesh refinement(AMR) algorithm uses a hierarchy of properly nested levels. It automatically refines or coarsens the mesh to achieve a solution having a specified accuracy in an optimal fashion. Here we used the AMR technique based on the Z 2 error indicator of Zienkiewicz and Zhu [26]. See also [23] for a more detailed description of error estimators. Here we will recall the Z 2 error indicator. We denote by W h the space of all piecewise linear vector-fields and set X h := W h ∩ C (Ω, R2 ). Denote by v and v h the unique solution of problems (1 ) and (20), respectively. In this case G v h − ∇ v h L 2 ( T ) can be used as an error estimator, where G v h is an easily computable approximation of ∇ v h . Let G v h ∈ X h be the ·,· h -projection of ∇ v h onto X h . It can computed by a local averaging of ∇ v h | T (xi ) as follows

G v h (xi ) =

|T | ∇ v h |T (xi ). |D x|

(32)

T ⊂Dx

Here, D x is the union of the triangles having x as a vertex. We finally set the error indicator locally and globally respectively as follows

η Z ,T := G v h − ∇ v h L 2 (T ) , and

η Z :=

(33)

1/2

η

2 Z ,T

.

(34)

T ∈Th

The Z 2 indicator η Z , T is an estimate for ∇ v th (·, t i ) − ∇ v t (·, t i ) L 2 ( T ) , see Verfürth [23] for complete details. Let λ( T ) ∈ N0 be the refinement level of triangle T ∈ T , λmax ∈ N0 be a given maximum refinement level, and φ1 , . . . , φλmax be given real numbers satisfying 0 φ1 · · · φλmax . We set φ0 = 0 and φλmax = ∞. With the choice of φ1 , . . . , φλmax one controls the structure of the grid. If we set φ1 = · · · = φλmax = 0 this leads to a uniform triangulation of level λmax . We set φ1 = · · · = φλmax = 10−2 in our numerical computations. Here we used the scaled indicator

φT := η Z ,T / | T |.

Suppose that an initial coarse triangular grid is constructed using a grid generator. A triangle T is marked for

(35)


59

Fig. 1. Control and excitation region at the cardiac domain.

1. refinement if φ T > φλ( T ) and λ( T ) < i for i = 0, . . . , λmax , φλ(T ) and λ( T ) > i for i = 0, . . . , λmax , 2. coarsening if φ T < 100 where φ T is calculated according to Eq. (35). So far we explained adaptivity for the direct problem. Let us turn to the optimal control problem next. One prominent approach in the literature is the dual weighted residual methodology as presented in [3] as a technique to derive a posteriori error estimates for both time dependent and stationary problems governed by partial differential equations. Another approach uses grid adaptivity based on a posteriori error estimate of the cost functional [2,6]. In our computational work we proceeded as follows. We used AMR in space only and assume that, during the optimization iterations, the control variable lives always on the coarse grid dimension where we start the simulations. This can be related to the fact that in practice in is difficult to manipulate the control mechanism spatially. The initial coarse grid, denoted by L0 is constructed by a grid generator. Finer levels Li for i > 0 are constructed recursively from the coarser levels L i −1 . The tolerance for spatial grid refinement is set tolx = 10−2 . The Z 2 error estimator is called for every 3 time intervals during the primal solve. It adjusts the spatial grid by refining and coarsening, depending on the estimated spatial solution error of the elements. Accordingly, the control variable is interpolated based on the new grid construction. We used the same strategy to solve to adjoint problem with the spatial grid being adjusted according to the adjoint solution. In the Newton optimization algorithm, to evaluate the matrix–vector product in each iteration of the inner loop, see algorithm (1), the coarse grid dimension is used to up-date the control solution. Thus our Newton direction is an inexact one: it is determined from solving the Newton system on the coarse grid iteratively up to a certain tolerance tolNew . We set tolNew = 10−3 | ˆJ ( I e,k )|. Since | ˆJ ( I e,k )| → 0 in case of convergence to a critical point we can expect to obtain superlinear convergence of the overall numerical optimization strategy. Of course, this can be achieved only if function and gradient evaluations are sufficiently accurate. For our numerical experiments, we developed an optimization code based on the public domain software package DUNE [1]. 5. Numerical results In this section we present numerical results based on the adaptive and second order methods. We shall see that the proposed methodology is capable of dampening an excitation wave of the transmembrane potential by properly applying an extracellular potential, even if the control domain is relatively small. The numerical experiments were carried out using different static grid sizes and as well as the AMR technique. Here we consider 4 msec of simulation time. The computational domain Ω = [0, 1] × [0, 1] and various relevant subdomains are depicted in Fig. 1. The excitation domain and control domains are Ωexi and Ωcon = Ωcon1 ∪ Ωcon2 . Further Ω f 1 , Ω f 2 are neighborhoods of Ωcon1 , Ωcon2 and the observation domain is Ωobs = Ω\(Ω f 1 ∪ Ω f 2 ). In the computations equidistant time steps are used. For the spatial adaptive grid refinement, a uniform 60 × 60 triangular grid is considered as coarse grid and the subsequent multi level grids are constructed based on the primal or adjoint solution. In simulations the weight of the cost of the control is fixed to be α = 1 × 10−3 and the iterations were terminated when ∇ J k ∞ 10−3 (1 + | J k |), is satisfied or the difference between two successive optimization iterations of cost functional minimization value is less than 10−3 . If this condition is not met within the prescribed iterations, let us say 100 optimization iterations, then we terminate the simulations. The line search algorithm starts with initial step length α = 1 for the inexact Newton method and is reduced by a factor of 2 for subsequent rejected step sizes, see Nocedal and Wright [16]. For the computational set up, we considered the domains Ωexi = [0.498, 0.502] × [0.498, 0.504], Ωcon1 = [0.3, 0.4] × [0.45, 0.55], Ωcon2 = [0.6, 0.7] × [0.45, 0.55], Ω f 1 = [0.28, 0.42] × [0.43, 0.57] and Ω f 2 = [0.58, 0.72] × [0.43, 0.57]. Parameters used in the simulation The following parameter values [7] are used for our numerical experiments: σil = 3 × 10−3 −1 cm−1 , −1 cm−1 G = 1.5 S/cm2 , v th = 13 mV, v p = 100 mV, η1 = 4.4 S/cm2 , η2 = 0.012, η3 = 1.

10−4

σit = 3.1525 ×

60


Fig. 2. The norm of the gradient and minimum value of the cost functional are shown on left and right respectively for T = 4 msec of simulation time.

Fig. 3. The minimum value of

| V m |2 dx dt and

| I e |2 dx dt are on left and right respectively for T = 4 msec of simulation time.

Test case 1 First we discuss the results based on different static grid sizes and the AMR grid. We use a 60 × 60 triangular grid, consisting of 13,924 elements and 7081 nodal points, an 80 × 80 triangular grid, consisting of 24,964 elements and 12,641 nodal points, a 100 × 100 triangular grid, consisting of 40,000 elements and 20,201 nodal points, and AMR coarse grid which is a 60 × 60 triangular grid. The corresponding L 2 norms of the gradients of the cost functional and the minimum values of the cost functional themselves are depicted in Fig. 2. A logarithmic scale is considered for the presentation of L 2 norms of the gradients of the cost functional. As expected for a Newton method, the optimal solutions are obtained within a rather small number of iterations. For each one of therespective grids convergence was obtained within 8 or 9 iterations. The values of the two additive terms of the cost, 12 V m 2 dx dt and α2 I e 2 dx dt, are depicted in Fig. 3. Note that the two terms are of approximately the same order of magnitude. We observed that the control action has a strong impact at the beginning of iterations where the excitation of the wave front is dampened out and decreases with time, see Fig. 6. The total CPU time for solving the optimization problem, as well as the CPU time for all primal solves, all dual solves, and for all setup and evaluation times of the Hessian steps is given in the first 5 columns of Table 1. The last column contains the average number of CG-steps required for obtaining the inexact Newton algorithm. We note that the computational time for the gradient of the cost, which requires one primal and one dual solve per iteration, is much cheaper than updating the control variable on the basis of second order information. For all grids, updating the control variable on the basis of the second order system, needs approximately 18 times more CPU time than computing the first derivative. This is the expensive step in our calculations. However, as we experienced in our earlier work, minimization on the basis of only first order information requires at least several hundred iterations before comparable accuracy can be achieved. Let us also mention that additional Newton


61

Table 1 Computational time and an average inner CG iterations in Newton’s algorithm are presented for all grids. Grids

CPU time

Primal solves (s)

Dual solves (s)

Opt. solves (s)

Avg inner CG iters

60 × 60 80 × 80 100 × 100 AMR grid

58 min 51 sec 2 h 2 min 54 sec 3 h 13 min 14 sec 1 h 13 min 19 sec

93.64 197.03 323.89 598.52

87.24 180.46 292.27 551.26

3264.63 6813.89 10 759.4 3187.07

15.7 17.7 20 17

Table 2 Optimization iterations, norm of gradient of cost functional and order of convergence for the different grid constructions are presented using inexact Newton’s algorithm. 60 × 60

80 × 80

100 × 100

AMR grid

∇ J

∇ J i +1 ∇ J i

∇ J

∇ J i +1 ∇ J i

∇ J

∇ J i +1 ∇ J i

∇ J

∇ J i +1 ∇ J i

59.9128 27.9293 19.8697 3.40178 3.47448 1.60263 0.31174 0.02236 0.00050

– 0.4661 0.7114 0.1712 1.0213 0.4612 0.1945 0.0717 0.0226

62.5567 25.8370 18.8618 3.20784 2.62626 1.23709 0.22913 0.01267 0.00024

– 0.4130 0.7300 0.1700 0.8187 0.4710 0.1852 0.0553 0.0191

59.3120 18.9710 5.0035 2.64203 2.31805 1.95699 0.18502 0.00700 –

– 0.3199 0.2637 0.5280 0.8774 0.8442 0.0945 0.0378 –

66.5305 30.5083 5.8358 3.07498 0.67403 0.05105 0.00180 0.00010 –

– 0.4586 0.1913 0.5269 0.2192 0.0758 0.0353 0.0556 –

Fig. 4. Uncontrolled solution (V m ) at 0.3 msec, 2 msec and 4 msec of simulation time respectively.

steps, beyond those which are documented, allow to further decrease ∇ J which we could not observe with the nonlinear CG algorithm [14]. The AMR technique takes 598 seconds to solve the primal problem, out of which it needs 96.87 seconds for grid adaption. For the dual solve the grid adaption process takes 81.96 seconds. We can observe that AMR takes less overall CPU time and achieves results comparable to the 100 × 100 grid. Obviously the AMR technique shows a good improvement over the static grids for the current problem. Table 2 confirms superlinear convergence of the inexact Newton method for all grid constructions at the end of iterations. The line search algorithm takes small step lengths during the initial iterations and full step lengths after 3 or 4 iterations which leads to superlinear convergence. The 2D contour snap shots of the uncontrolled solution, and of the optimally controlled solutions are shown in Figs. 4, and 5 at times 0.3 msec, 2 msec and 4 msec for the 100 × 100 mesh. We can see that the uncontrolled wave front of the transmembrane voltage spreads almost over the entire domain during the time interval from 0 msec to 4 msec, see in Fig. 4. The corresponding controlled action potential is first slowly dampened at 0.3 msec and is almost completely damped at 4 msec. The control action has a tremendous effect on the action potential at the beginning to control and dampens the excitation wave It has much less effect at 4 msec time, see Fig. 6. The AMR grid of the uncontrolled solution can be seen in Fig. 7. The refinement/coarsening strategy follows the interface of the wave propagation. For the first optimization iteration the fine grid for the primal solve consists of 34,038 elements and 17,138 number of nodal points. At the last optimization iteration the fine grid consists of 21,700 number of elements and 10,969 number of nodes where the maximum level of refinement is 5. The AMR grids corresponding to the optimal state solution are shown in Fig. 8. We can note that the grid at time t = 4 msec is equivalent to the coarse grid, which we considered as the level-0 grid. This is consistent with the fact that there is no further wave propagation within the domain. Finally, the presented numerical results evidently show that the excitation wave propagation is dampened out by properly adjusting the extracellular potential using the optimization algorithm. Further, the AMR method shows a good computational improvement over the static grids.

62


Fig. 5. Controlled solution (( V m )opt ) at 0.3 msec, 2 msec and 4 msec of simulation time respectively.

Fig. 6. Controlled (I e ) at 0.3 msec, 2 msec and 4 msec of simulation time respectively.

Fig. 7. Uncontrolled solution (V m ) grid view at 0.3 msec, 2 msec and 4 msec of simulation time respectively for AMR technique.

Fig. 8. Controlled solution (V m ) grid view at 0.3 msec, 2 msec and 4 msec of simulation time respectively for AMR technique.


63

Fig. 9. The norm of the gradient and minimum value of the cost functional are shown on left and right respectively for T = 4 msec of simulation time.

Fig. 10. The minimum value of

| V m |2 dx dt and

| I e |2 dx dt are on left and right respectively for T = 4 msec of simulation time.

Test case 2 In this test case the placement of control domain is changed and isotropic conductivity tensors are considered. With respect to the control domains the observation domains are chosen in an asymmetric manner. More specifically, Ωcon1 = [0.3, 0.4] × [0.45, 0.55], Ωcon2 = [0.5, 0.6] × [0.53, 0.6], Ωobs1 = [0.25, 0.45] × [0.40, 0.60] and Ωobs2 = [0.45, 0.65] × [0.47, 0.65]. In this test case the weight of the cost of the control is reduced to α = 5 × 10−4 to achieve the dampening of exciting wave propagation. The L 2 norms of the gradient and the minimum value of the cost functional are depicted in Fig. 9 for different grid constructions. For all static grids 7 optimization iterations are required, while the AMR method takes 6 iterations. In this case the AMR method shows a clear improvement over the static grids in terms of fast converging to the optimal solution. The corresponding additive values of 12 V m 2 dx dt and α2 I e 2 dx dt are depicted in Fig. 10. We observed that the control term has an especially strong impact during the first minimization iterations, during which the excitation of the wave front is dampened out. This can be seen on the right-hand side of Fig. 10. 2D contour snap shots of the uncontrolled solution and the optimally controlled solution are shown in Figs. 11, and 12 at times 0.3 msec, 2 msec and 4 msec the 100 × 100 mesh. We can see that the uncontrolled wave front of the transmembrane voltage spreads uniformly in the x- and y-directions over the domain during the time interval from 0 msec to 4 msec, see in Fig. 11. The corresponding controlled action potential is shown in Fig. 13 at times 0.3 msec, 2 msec and 4 msec respectively. In contrast to the Test case (1), we can observe that the controlled action potential is dampened much more rapidly, which is due to the isotropic conductivity tensors, σil = σit = 3 × 10−3 and the fact that the control cost parameter is reduced by a factor of 2. Also observe that the control domain that is closer to the excitation domain has much more effect than the other one. We also carried out an experiment with the isotropic conductivity tensor of the form σil = σit = 3 × 10−4 . In this case the excitation wave is not dampened out by the chosen control configuration, due to the domination of the reaction

64


Fig. 11. Uncontrolled solution (V m ) at 0.3 msec, 2 msec and 4 msec of simulation time respectively.

Fig. 12. Controlled solution (( V m )opt ) at 0.3 msec, 2 msec and 4 msec of simulation time respectively.

Fig. 13. Controlled (I e ) at 0.3 msec, 2 msec and 4 msec of simulation time respectively.

part. We remark that for long time horizons this procedure is effective from the point of dampening the wave. For short time horizons, however, it is less successfull when compared to locally distributed (w.r.t. space) control action. Again we can observe super-linear convergence of the optimization algorithm. 6. Conclusions We discussed solution strategies based on the Newton method and adaptive grid refinement strategies to solve optimal control of the action potential in cardiac electrophysiology based on the monodomain model. The numerical results show the capability of dampening the excitation wave propagation by properly applying an extra cellular current as a control variable. The presented optimization algorithm exhibits super linear convergence of the inexact Newton method. Moreover the overall performance is more efficient when compared to first order methods, see [14]. The results motivate us to continue our investigations for the bidomain model. Adaptive time-stepping should be considered to further save computational time. We also need to strive for more insight with respect to longer time horizons, with more realistic geometries, and finer meshes.


65

Acknowledgements The authors gratefully acknowledge the Austrian Science Foundation (FWF) for financial support under SFB 032, “Mathematical Optimization and Applications in Biomedical Sciences”. Appendix A. Optimization algorithm Here is a brief outline of complete a Newton’s optimization algorithm, by utilizing the spatial grid adaptivity during the primal and dual solves, which is used to carry out numerical computations. Algorithm 1 Line search Newton-CG optimization algorithm. 1: primal variables: v, w. 2: dual variables: p, q. 3: repeat 4: set v (0) := v 0 , w (0) := w 0 and solve the primal problem to obtain v (t ), w (t ) by utilizing the spatial grid adaptation during the intermediate time steps. 5: solve the dual problem for p (t ), q(t ) using terminal conditions p ( T ) := 0, q( T ) := 0 by utilizing the spatial grid adaptation during the intermediate time steps. 6: update the gradient using the adjoint solution ˆJ = p + α I ek .

7: solve the system ˆJ δ I e = − ˆJ by linear conjugate gradient method. 8: set step length βk := 1.0 and compute optimal βk using backtracking method by checking the strong Wolfe conditions, see [16]. 9: update I ek+1 := I ek + βk δ I e using modified βk . 10: k ← k + 1. 11: until ∇ J k ∞ 10−3 (1 + | J k |)

References [1] P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, O. Sander, A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in dune, Computing 82 (2008) 121–138. [2] R. Becker, H. Kapp, R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim. 39 (2000) 113–132. [3] R. Becker, R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer. 10 (2001) 1–102. [4] Y. Bourgault, Y. Coudiére, C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology, Nonlinear Analysis: Real World Applications 10 (2009) 458–482. [5] E.M. Cherry, H.S. Greenside, C.S. Henriquez, A space–time adaptive method for simulating complex cardiac dynamics, Phys. Rev. Lett. 84 (2000) 1343– 1346. [6] L. Dede’, A. Quarteroni, Optimal control and numerical adaptivity for advection–diffusion equations, M2AN 39 (2005) 1019–1040. [7] P.C. Franzone, P. Deuflhard, B. Erdmann, J. Lang, L.F. Pavarino, Adaptivity in space and time for reaction–diffusion systems in electrocardiology, SIAM Journal on Numerical Analysis 28 (2006) 942–962. [8] C.S. Henriquez, Simulating the electrical behavior of cardiac tissue using the bidomain model, Crit. Rev. Biomed. Eng. 21 (1) (1993) 77. [9] M. Hinze, K. Kunisch, Second order methods for optimal control of time-dependent fluid flow, SIAM J. Control Optim. 40 (2001) 925–946. [10] M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE Constraints, Springer, 2008. [11] J. Lang, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems, Lecture Notes in Computational Science and Engineering, vol. 16, SpringerVerlag, Berlin, 2001. [12] P.K. Moore, An adaptive finite element method for parabolic differential systems: Some algorithmic considerations in solving in three space dimensions, SIAM J. Sci. Comput. 21 (1999) 1567–1586. [13] C. Nagaiah, K. Kunisch, G. Plank, Numerical solutions for optimal control of monodomain equations in cardiac electrophysiology, in: Proceedings of BFG-09, Recent Advances in Optimization and its Applications in Engineering, in press. [14] C. Nagaiah, K. Kunisch, G. Plank, Numerical solution for optimal control of the reaction–diffusion equations in cardiac electrophysiology, Comput. Optim. Appl. (2009) 1–30. [15] B.F. Nielsen, T.S. Ruud, G.T. Lines, A. Tveito, Optimal monodomain approximations of the bidomain equations, Appl. Math. Comput. 184 (2007) 276–290. [16] J. Nocedal, S.J. Wright, Numerical Optimization, 2nd ed., Springer Verlag, New York, 2006. [17] G. Plank, M. Liebmann, R.W. dos Santos, E. Vigmond, G. Haase, Algebraic multigrid preconditioner for the cardiac bidomain model, IEEE Trans. Biomed. Eng. 54 (4) (2007) 585–596. [18] R. Plonsey, Bioelectric sources arising in excitable fibers (ALZA lecture), Ann. Biomed. Eng. 16 (1988) 519–546. [19] M. Potse, B. Dube, J. Richer, A. Vinet, R. Gulrajani, A comparison of monodomain and bidomain reaction–diffusion models for action potential propagation in the human heart, IEEE Trans. Biomed. Eng. 53 (2006) 2425–2435. [20] J.M. Rogers, A.D. McCulloch, A collocation-Galerkin finite element model of cardiac action potential propagation, IEEE Trans. Biomed. Eng. 41 (1994) 743–757. [21] J.A. Trangenstein, C. Kim, Operator splitting and adaptive mesh refinement for the luo-rudy i model, J. Comput. Phys. 196 (2004) 645–679. [22] L. Tung, A bi-domain model for describing ischemic myocardial DC potentials, Ph.D. thesis, MIT, Cambridge, MA, 1978. [23] R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley & Teubner, 1996. [24] E.J. Vigmond, R. Weber dos Santos, A.J. Prassl, M. Deo, G. Plank, Solvers for the cardiac bidomain equations, Prog. Biophys. Mol. Biol. 96 (2008) 3–18. [25] R. Weber dos Santos, G. Plank, S. Bauer, E. Vigmond, Parallel multigrid preconditioner for the cardiac bidomain model, IEEE Trans. Biomed. Eng. 51 (2004) 1960–1968. [26] O.C. Zienkiewicz, J.Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Num. Meth. Eng. 24 (1987) 337– 357.




A dimensional split preconditioner for Stokes and linearized Navier–Stokes equations Michele Benzi a,∗,1 , Xue-Ping Guo b,2 a b

Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA Department of Mathematics, East China Normal University, Shanghai, 200062, PR China

a r t i c l e

i n f o

Article history: Received 23 March 2009 Received in revised form 2 August 2010 Accepted 9 August 2010 Available online 19 August 2010 Keywords: Saddle point problems Matrix splittings Iterative methods Preconditioning Stokes problem Oseen problem Stretched grids

a b s t r a c t In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier–Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixed-point iteration. The basic iteration is accelerated by a Krylov subspace method like restarted GMRES. The corresponding preconditioner requires at each iteration the solution of a set of discrete scalar elliptic equations, one for each component of the velocity field. Numerical experiments illustrating the convergence behavior for different finite element discretizations of Stokes and Oseen problems are included. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction We consider the solution of the incompressible Navier–Stokes equations governing the flow of viscous Newtonian fluids. For an open bounded domain Ω ⊂ Rd (d = 2, 3) with boundary ∂Ω , time interval [0, T ], and data f, g and u0 , the goal is to find a velocity field u = u(x, t ) and pressure field p = p (x, t ) such that

∂u − ν u + (u · ∇)u + ∇ p = f on Ω × (0, T ], ∂t div u = 0 on Ω × [0, T ],

(2)

u=g

(3)

on ∂Ω × [0, T ],

u(x, 0) = u0 (x)

on Ω,

(1)

(4)

where ν is the kinematic viscosity, is the vector Laplacian, ∇ is the gradient and div the divergence. Implicit time discretization and linearization of the Navier–Stokes system by Picard fixed-point iteration result in a sequence of (generalized) Oseen problems of the form

*

Corresponding author. E-mail addresses: [email protected] (M. Benzi), [email protected] (X.-P. Guo). 1 The work of this author was supported in part by the National Science Foundation Grants DMS-0511336 and by a grant by the University Research Committee at Emory University. 2 Part of this work was performed while visiting CERFACS, 42 Ave. Gaspard Coriolis, 31057 Toulouse Cedex, France. 0168-9274/$30.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2010.08.005

M. Benzi, X.-P. Guo / Applied Numerical Mathematics 61 (2011) 66–76

67

σ u − ν u + (v · ∇)u + ∇ p = f in Ω,

(5)

div u = 0 in Ω,

(6)

u = g on ∂Ω,

(7)

where v is a known velocity field from a previous iteration or time step (the ‘wind’) and σ is proportional to the reciprocal of the time step (σ = 0 for a steady problem). When v = 0 we have a (generalized) Stokes problem. Spatial discretization of the Stokes or Oseen problem using LBB-stable finite elements (cf. [18,19]) results in large, sparse linear systems in saddle point form:

BT 0

A −B

u p

=

f , −g

or Ax = b,

(8)

where now u and p represent the discrete velocity and pressure, respectively, A is the discretization of the diffusion, convection, and time-dependent terms, B T is the discrete gradient, B the (negative) discrete divergence, and f and g contain forcing and boundary terms. The efficient solution of (8) calls for rapidly convergent iterative methods. Much work has been done in developing efficient preconditioners for Krylov subspace methods applied to this problem; see, e.g., [4,5,8,14,15,17,18,22]. The ultimate goal is to develop robust solvers with optimal complexity. In particular, the rate of convergence should be independent of the mesh size h. For the Oseen problem, the rate of convergence should also depend only weakly on the kinematic viscosity ν (equivalently, on the Reynolds number Re = O (ν −1 )), although this goal is difficult to achieve in practice. 2. Dimensional splitting For simplicity, in this paper we limit ourselves to the 2D case. Extension to the 3D case is possible (see Section 7), but will not be described here. The system matrix A admits the following splitting:

⎡

A1

B 1T

0

A=⎣ 0 A2 −B1 −B2

⎤

⎡

A1

0

B 2T ⎦ = ⎣ 0 −B1 0

B 1T

⎤

⎡

0

0

0 ⎦ + ⎣0

0

0

0

0

A2

B 2T

0 −B2

⎤

⎦ = A1 + A2 .

(9)

0

Here each diagonal submatrix A i is a scalar discrete convection-diffusion-reaction operator:

Ai = σ M + ν L + Ni B 1T ,

B 2T

(i = 1, 2),

(10)

are discretizations of the partial derivatives ∂∂x , ∂∂y , respectively. Note that A 1 and A 2 act, respectively, on u (the

and x-component of the velocity field u) and on v (the y-component of u). Denoting by n1 , n2 and m the number of degrees of freedom of u, v and p, respectively, then A 1 ∈ Rn1 ×n1 , A 2 ∈ Rn2 ×n2 , B 1 ∈ Rm×n1 and B 2 ∈ Rm×n2 . Thus, A ∈ R(n+m)×(n+m) with n = n1 + n2 . In (10), M denotes the velocity mass matrix, L the discrete (negative) Laplacian, and N i the convective terms. Note that for the discrete Stokes problem the convective terms are absent (N i = 0) so that each A i is symmetric and positive definite. For the discrete Oseen problem A i = A iT , but A i + A iT is positive definite (i = 1, 2). As a consequence, A1 and A2 in (9) are nonsymmetric but positive semidefinite, in the sense that A1 + A1T and A2 + A2T are both symmetric positive semidefinite. In particular, A1 and A2 are singular. We refer to (9) as to a dimensional splitting, since A1 contains terms that correspond to the x-component of the solution and A2 contains terms that correspond to the y-component of the solution. Although this splitting is somewhat reminiscent of ADI (alternating direction implicit) methods [29], it is actually quite different since we do not split the operators A i into their constituent components. To distinguish it from ADI splitting, we refer to (9) as to dimensional splitting (or DS for short). We further mention that our approach is also different from previous ADI schemes for saddle point problems, such as those described in [11] and [13]. Let now α > 0 be a parameter, and denote by I the identity matrix of order n1 + n2 + m. Then A1 + α I and A2 + α I are both nonsingular, nonsymmetric, and positive definite. Consider the two splittings of A,

A = (A1 + α I ) − (α I − A2 ) and A = (A2 + α I ) − (α I − A1 ). Associated to these splittings is the alternating iteration 1

(A1 + α I )xk+ 2 = (α I − A2 )xk + b, k +1

(A2 + α I )x

k+ 12

= (α I − A1 )x k+ 12

(k = 0, 1, . . .). Eliminating x k +1

x

k

= Tα x + c,

+b

(11) (12)

from these, we can rewrite (11)–(12) as the stationary scheme

k = 0, 1 , . . .

where Tα is the iteration matrix

Tα = (A2 + α I )−1 (α I − A1 )(A1 + α I )−1 (α I − A2 )

(13)

68


and c is a certain vector. As in [9], one can show that there is a unique splitting A = Pα − Qα with Pα nonsingular such that the iteration matrix Tα is the matrix induced by that splitting, i.e., Tα = Pα−1 Qα = I − Pα−1 A. Furthermore, c = Pα−1 b. Matrices Pα and Qα are given by

Pα =

1 2α

(A1 + α I )(A2 + α I ),

Qα =

1 2α

(α I − A1 )(α I − A2 ).

(14)

We refer to iteration (11)–(12) as the DS iteration, and to Pα as the DS preconditioner. Besides the already mentioned resemblance to the Peaceman–Rachford ADI method, the DS iteration bears some resemblance to another alternating method, the Hermitian and skew-Hermitian splitting (HSS) iteration [2–4,6]. While the HSS method has proved quite successful in solving such problems as the generalized Stokes problem and the rotation form of the Navier–Stokes equations (see [6]), it is not well-suited for the standard (convection) form (1)–(2). This limitation of HSS was one of the main motivations for introducing the DS approach. 3. Convergence of the fixed-point iteration We now prove that, under standard assumptions on the saddle point problem (8), the alternating iteration (11)–(12) converges to the solution of (8) for any choice of α > 0 and for all initial guesses. First we state two auxiliary results. The first one is classical, and is known as Kellogg’s Lemma; see [21]. In the following, a (not necessarily Hermitian) matrix A ∈ Cn×n is said to be positive definite (semidefinite) if the Hermitian matrix A + A ∗ is positive definite (resp., semidefinite) in the usual sense. Lemma 1. Let A ∈ Cn×n be positive semidefinite. Then

(α I n + A )−1 (α I n − A ) 1 2

for all α > 0. Furthermore, if A is positive definite then

(α I n + A )−1 (α I n − A ) < 1 2

for all α > 0. Lemma 2. Assume that the (1, 1) block A in (8) has positive definite symmetric part and that B has full row rank. Then the following are equivalent: (i) The matrix

⎡ A 1 Cα := ⎣ 0

1

T

B B A α2 1 2 2 A2

−B1

−B2

⎤

B 1T + α12 B 1T B 2 B 2T ⎦ B 2T

(15)

0

has no purely imaginary eigenvalues. (ii) The spectral radius of the iteration matrix Tα in (13) is strictly less than unity. Proof. First we note that under the assumptions made on A and B, the matrix A in (8) is nonsingular; see [4, Lemma 1.1]. Let λ be an eigenvalue of the iteration matrix Tα = I − Pα−1 A. Then λ = 1 − μ where μ is a generalized eigenvalue of the matrix pencil (A, Pα ); that is, there exists a vector x = 0 such that Ax = μPα x. Expanding the right-hand side, we get

Ax =

μ 2α

(A1 A2 + α A + α 2 I )x.

Collecting terms in A, we rewrite this as

1−

1 2

μ Ax =

μα 2

1 I + 2 A1 A2 x .

(16)

α

Since both A and Pα are nonsingular, it must be μ = 0. Also, it must be 1 − (I + α12 A1 A2 )x = 0 has a nonzero solution, but this is impossible since

⎡

G := I +

1

α2

I n1

A1 A2 = ⎣ 0

is clearly nonsingular. Hence,

0

− α12 B 1T B 2 I n2 0

0

⎤

0 ⎦ Im

μ = 2 and we can set (as in [25, p. 378])

μα 2α 2θ θ := , from which μ = 2 − = . 2−μ θ +α θ +α

1 2

μ = 0 for otherwise (16) implies that


69

Hence, the generalized eigenproblem (16) can be reformulated as

Ax = θ G x , that is,

⎡

A1

0

⎣ 0 A2 −B1 −B2

⎡ ⎤ I n1 u T ⎦ v ⎣ =θ 0 B2 B 1T

p

0

⎡ G − 1 Ax = θ x ,

0

I n2

0

or

⎤ u 0 ⎦ v ,

− α12 B 1T B 2 0 1

I n1

T

B B α2 1 2

where G −1 = ⎣ 0

0

⎤

0 ⎦.

I n2

0

p

Im

0

Im

This eigenproblem is precisely Cα x = θ x, where Cα = G −1 A is the matrix in (15). Note that Cα is necessarily nonsingular. Recall now that the eigenvalues of Pα−1 A are of the form μ = 2θ/(θ + α ). It must be |1 − μ| 1, since λ = 1 − μ is an eigenvalue of the iteration matrix

Tα = I − Pα−1 A = (α I + A2 )−1 (α I − A1 )(α I + A1 )−1 (α I − A2 ), which is similar to

α = (α I − A1 )(α I + A1 )−1 (α I − A2 )(α I + A2 )−1 , T hence

(Tα ), the spectral radius of Tα , satisfies

(Tα ) = (T α ) (α I − A1 )(α I + A1 )−1 2 (α I − A2 )(α I + A2 )−1 2 1.

(17)

The last inequality is an immediate consequence of Kellogg’s Lemma. Denoting by (θ) and (θ) the real and imaginary parts of θ , respectively, we claim that (Tα ) < 1 if and only if (θ) = 0 for every eigenvalue θ of Cα ; equivalently, (Tα ) = 1 if and only if there exists at least one θ with (θ) = 0. Indeed, we have

|1 − μ| = 1

⇔

2θ =1 − 1 θ + α

⇔

θ − α θ + α = 1

⇔

|θ − α | = |θ + α |.

The last equality can be rewritten as ((θ) − α )2 + (θ)2 = ((θ) + α )2 + (θ)2 , or

(θ) − α

Therefore,

2

2 = (θ) + α

⇔

4α (θ) = 0

⇔

(θ) = 0.

(Tα ) = 1 if and only if Cα has at least one purely imaginary eigenvalue. The proof is complete. 2

We are now in a position to prove the following convergence result. Theorem 3. Under the assumptions of Lemma 2, the iteration (11)–(12) is unconditionally convergent; that is, (Tα ) < 1 for all α > 0. Proof. By Lemma 2, it suffices to show that for all α > 0, the matrix Cα in (15) has no purely imaginary eigenvalues. We will argue by contradiction. Recall that Cα is nonsingular. Thus, let θ = 0 be an eigenvalue of Cα corresponding to an eigenvector x = [u ; v ; p ], where u ∈ Cn1 , v ∈ Cn2 and p ∈ Cm are not all equal to zero. Expanding Cα x = θ x we obtain

A1u +

1

α2

1

B 1T B 2 A 2 v + B 1T p +

α2

B 1T B 2 B 2T p = θ u ,

(18)

A 2 v + B 2T p = θ v ,

(19)

− B 1 u − B 2 v = θ p.

(20)

Assuming that the eigenvector x has been normalized so that x 2 = 1, we have

θ = x ∗ Cα x ,

θ¯ = x∗ CαT x and (θ) =

Therefore, letting H 1 = ( A 1 +

A 1T )/2

(θ) = u ∗ H 1 u + v ∗ H 2 v +

θ + θ¯ 2

and H 2 = ( A 2 +

1 2α

2

1 = x∗ Cα + CαT x. 2

A 2T )/2,

we find after some easy algebraic manipulations

u ∗ B 1T B 2 A 2 v + B 2T p + v ∗ A 2T + p ∗ B 2 B 2T B 1 u .

70


First we observe that at least one between u and v must be nonzero, for otherwise (20), together with the fact that θ = 0, implies p = 0 and thus x = 0, a contradiction. Since H 1 and H 2 are symmetric positive definite we have that u ∗ H 1 u + v ∗ H 2 v > 0, therefore

(θ) = 0

u ∗ B 1T B 2 A 2 v + B 2T p + v ∗ A 2T + p ∗ B 2 B 2T B 1 u < 0,

⇒

(21)

showing that if θ is purely imaginary, it must necessarily be u = 0. Next, consider the case where v = 0. Then Eqs. (18)–(20) reduce to

A 1 u + B 1T p = θ u ,

(22)

B 2T

p = 0,

(23)

− B 1 u = θ p,

(24)

that is, Ax = θ x where x = [u ; 0; p ] = 0. Hence, θ is an eigenvalue of A; but then (θ) > 0 by virtue of Lemma 1.1 in [4]. So it must be v = 0. Furthermore, if p = 0, then Eq. (19) becomes A 2 v = θ v, hence (θ) > 0 since A 2 is positive definite (has positive definite symmetric part) by assumption, and therefore all its eigenvalues have positive real part. Thus, it must be u = 0, v = 0 and p = 0. Now, using (19) we rewrite the necessary condition in (21) in the form

¯ B 2 v )∗ B 1 u < 0. θ u ∗ B 1T B 2 v + θ¯ v ∗ B 2T B 1 u = θ( B 1 u )∗ B 2 v + θ(

(25)

Now, from (20) we obtain B 2 v = − B 1 u − θ p which substituted into (25) yields

θ u ∗ B 1T (− B 1 u − θ p ) + θ¯ −u ∗ B 1T − θ¯ p ∗ B 1 u < 0,

or, equivalently,

−θ u ∗ B 1T B 1 u − θ 2 u ∗ B 1T p − θ¯ u ∗ B 1T B 1 u − θ¯ 2 p ∗ B 1 u < 0. Now, if (θ) = 0 then θ = iξ for some ξ ∈ R, ξ = 0, where i = ∗ T

2 ∗

−θ u B 1 B 1 u − θ u

B 1T

p − θ¯ u B 1 B 1 u − θ p B 1 u = ξ ∗ T

¯2 ∗

√

2

−1. After simplification, we find

u ∗ B 1T p + p ∗ B 1 u ,

therefore condition (21) becomes

u ∗ B 1T p + p ∗ B 1 u < 0 and since u ∗ B 1T p + p ∗ B 1 u = 2(u ∗ B 1T p ), we conclude that

(θ) = 0

u ∗ B 1T p < 0.

⇒

Likewise, from (20) we obtain B 1 u = − B 2 v − θ p; substituting this into (25) and going through the same algebraic operations as before, we also find that

(θ) = 0

v ∗ B 2T p < 0.

⇒

Therefore,

( B 1 u + B 2 v )∗ p = u ∗ B 1T + v ∗ B 2T p < 0,

but together with (20) this implies ((−θ p )∗ p ) < 0, or (iξ p 22 ) < 0, which is clearly absurd since iξ p 22 is imaginary. This proves that Cα cannot have purely imaginary eigenvalues. 2 The restriction in Theorem 3 that A have positive definite symmetric part is not essential. If A + A T is only positive semidefinite (and singular), the alternating iteration (11)–(12) is still well defined. Moreover, the spectral radius of the iteration matrix cannot exceed 1. Indeed, the iteration matrix Tα still satisfies (Tα ) 1, and if the symmetric part of A and B have no null vectors in common, the coefficient matrix A in (8) is still nonsingular; see again Lemma 1.1 in [4]. Hence, 1 is not an eigenvalue of Tα = I − Pα−1 A. However, it may happen that (Tα ) = 1 for some choices of α > 0. A simple example is given by

⎡

I 0 A=⎣0 0

⎤

0 I ⎦.

0 −I 0

Note that this matrix is nonsingular. It is easy to see that for α = 1, the iteration matrix Tα has only three distinct eigenvalues: λ = 0, λ = i and λ = −i. Hence, the spectral radius is 1. Nevertheless, a simple modification of the basic algorithm yields a convergent iteration. To this end, recall that (Tα ) 1 for all α > 0; see (17). Let γ ∈ (0, 1) be a parameter, then the matrix (1 − γ )I + γ Tα has spectral radius less than 1 for all α > 0. Indeed, the eigenvalues of (1 − γ )I + γ Tα are of the form 1 − γ + γ λ, where λ is an eigenvalue of Tα . It is easy to see that since |λ| 1 and λ = 1, all the quantities


71

1 − γ + γ λ have magnitude strictly less than 1. In practice, however, this modification is seldom used. In the next section we discuss Krylov subspace acceleration, which is much more effective and is applicable whether or not (Tα ) < 1. Nevertheless, knowing that (Tα ) < 1 is useful because it implies that the spectrum of the preconditioned matrix lies entirely in the right half-plane, a desirable property for Krylov subspace acceleration. Moreover, the smaller is (Tα ), the more clustered the spectrum of the preconditioned matrix is around 1. 4. Krylov subspace acceleration The basic method (11)–(12) although unconditionally convergent, is not competitive as a solver for problem (8), mainly due to the fact that convergence is generally slow. Fortunately, the rate of convergence can be greatly improved by Krylov subspace acceleration. In other words, Pα can be used as a preconditioner for GMRES [24] or any other nonsymmetric Krylov method. It should be noted that when Pα is used as a preconditioner, the pre-factor 21α in (14) is irrelevant and can be neglected. In this paper we use the restarted GMRES algorithm with restart parameter m. Preconditioning is applied on the right. The rate of convergence of nonsymmetric Krylov iterations (like GMRES) preconditioned by Pα depends on the particular choice of α . Finding the value of α that optimizes the rate of convergence appears to be a difficult problem in general. Indeed, in practice the convergence rate depends to a large extent on the size, shape, and location of the entire spectrum of the preconditioned matrix Pα−1 A, and not just on the spectral radius of Tα = I − Pα−1 A. (The rate of convergence may also be affected by the conditioning of the eigenbasis of the preconditioned matrix, but this is usually difficult to estimate; see, e.g., [26, p. 17].) Numerical experiments (see below) suggest that the value (or values) α∗ of α for which the number of preconditioned iterations is minimized is a rather small number (0 < α∗ 1). Moreover, the convergence rate is not overly sensitive to small relative changes in α . 5. Implementation aspects For the proposed approach to be successful, it is imperative that the action of the DS preconditioner be computed efficiently within each GMRES iteration. Written out explicitly, system (11) reads

A 1 + α I n1 0 −B1

B 1T 0 α I n3

0 α I n2 0

⎡

⎤

⎡

⎤

α uk + f 1 u k+ 2 ⎣ v k+ 12 ⎦ = ⎣ (α I n2 − A 2 ) v k − B 2T pk + f 2 ⎦ , 1 B 2 v k + α pk − g p k+ 2 1

(26)

while system (12) becomes

α I n1

0 A 2 + α I n2 −B2

0 0

0 B 2T α I n3

u k +1 v k +1 p k +1

⎡

=⎣

1

1

(α I n1 − A 1 )uk+ 2 − B 1T pk+ 2 + f 1

αv B1u

k+ 12

k+ 12

+ f2

+ αp

k+ 12

⎤ ⎦.

(27)

−g 1

Both systems (26) and (27) are highly reducible. Indeed, the second equation in (26) immediately yields v k+ 2 = α1 [(α I n2 − A 2 ) v k − B 2T pk + f 2 ] together with the reduced system

A 1 + α I n1 −B1

B 1T α I n3

1

u k+ 2 1 p k+ 2

=

α uk + f 1 . B 2 v k + α pk − g

(28) 1

1

Likewise, system (27) is equivalent to uk+1 = α1 [(α I n1 − A 1 )uk+ 2 − B 1T pk+ 2 + f 1 ] together with the reduced system

A 2 + α I n2 −B2

B 2T α I n3

v k +1 p k +1

=

1

α v k+ 2 + f 2 . 1 k+ 12 B1u + α p k+ 2 − g

(29)

Both systems (28) and (29) can be further reduced. Let ck = α uk + f 1 and dk = B 2 v k + α pk − g. For (28), the second equation yields 1

p k+ 2 =

1 1 k d + B 1 u k+ 2

(30)

α

which, substituted in the first one, yields

A 1 + α I n1 +

1

α

1

B 1T B 1 uk+ 2 = ck −

1

α

B 1T dk .

1

(31) 1

1

1

1

Once this equation has been solved for uk+ 2 , we recover pk+ 2 from (30). Similarly, let ck+ 2 = α v k+ 2 + f 2 and dk+ 2 = B1u

k+ 12

+ αp

k+ 12

− g, then the second equation of (29) yields

72


p k +1 =

1 k+ 1 d 2 + B 2 v k +1

(32)

α

which, substituted in the first one, yields

A 2 + α I n2 +

1

α

1

B 2T B 2 v k+1 = ck+ 2 −

1

α

1

B 2T dk+ 2 .

(33)

Once this equation has been solved, the new value pk+1 of the pressure can be obtained from (32). Hence, the bulk of the work in applying the preconditioner is in the solution of the two reduced systems (31) and (33). Each of these is a discrete analogue of a scalar, second-order, elliptic, anisotropic convection–diffusion–reaction equation. The anisotropy is in a sense artificial, since it depends on the size of the algorithmic parameter α : the smaller α is, the stronger the anisotropy in the diffusion terms. Note that for the Stokes and generalized Stokes problems, the convection terms are missing in (31) and (33) and the coefficient matrices are symmetric and positive definite. Some remarks on the solution of these two subsystems are given in the next section. On the basis of the foregoing discussion, it is clear that the DS approach can be regarded as a dimensionally segregated method, i.e., a method where the values of the velocity components u and v (or u, v and w in 3D) are updated separately through a decoupling process; the new value of the pressure p is obtained at very low cost from the new velocity values. We conclude this section with a discussion of diagonal scaling. In the alternating iteration (11)–(12), it is possible to replace the (n + m)-by-(n + m) identity matrix I with an arbitrary symmetric positive definite matrix D , leading to a preconditioner of the form

α = P

1 2α

(A1 + α D)(A2 + α D).

ˆ where

xˆ = b, It is easy to check that this is equivalent to applying the original DS preconditioner Pα to the linear system A

:= D− 12 AD− 12 , xˆ = D− 12 x, and bˆ = D− 12 b. A natural choice for finite element problems is to take D to be a block A

diagonal matrix with the velocity and pressure mass matrices on the main diagonal. In order to reduce the cost of applying the preconditioner, the mass matrices can be lumped or replaced by their main diagonals. In this paper we form D from the diagonals of the mass matrices. We found that this scaling, which is used in all numerical experiments in the next section, is quite effective in improving the convergence rate of DS preconditioning, especially for problems on stretched grids. Incidentally, it was noted in [4] that diagonal scaling is very beneficial for the HSS preconditioner as well. 6. Numerical experiments In this section we report the results of numerical experiments on linear systems from Stokes and Oseen models of incompressible flow. We consider Q2–Q1 finite element discretizations of two standard model problems: the leaky-lid driven cavity problem, and the backward facing step problem (see [18]). For the driven cavity problem, we consider both uniform and stretched grids of increasing size. All test problems were generated under Matlab using the IFISS software package [16] (see also [18]). We use the DS preconditioner in conjunction with restarted GMRES with m = 30 as the restart. In all cases the initial guess was the zero vector, and the stopping criterion was a reduction of at least six orders of magnitude of the initial residual norm. We discuss experiments for both steady and unsteady cases. 6.1. Steady problems Application of the DS preconditioner requires, at each iteration, the solution of the linear systems (31) and (33), where the general form of A i is given in (10). For the steady Stokes problems, σ = 0 and N i = 0; hence, the coefficient matrices in these two systems are symmetric positive definite. The systems can be solved very efficiently with a sparse Cholesky factorization with an approximate minimum degree (AMD) ordering; see [1,12]. The factorization is computed once and for all at the outset, and only forward and backward triangular solves need to be performed at each GMRES iteration. For the steady Oseen problem (σ = 0, N i = 0 in (10)) the two systems (31) and (33) are nonsymmetric, although structurally symmetric. We compute sparse LU factorizations [12] using again an AMD reordering. These direct methods are much faster than iterative methods in the case of 2D problems; in the solution of large 3D problems iterative methods will have to be used instead, necessitating the use of a flexible Krylov method (like flexible GMRES [23]) for the outer iteration. The first set of experiments is aimed at assessing the performance of the DS preconditioner on steady Stokes and Oseen problems, in particular to investigate the dependence on the discretization parameter h and on the viscosity ν . We begin with the leaky-lid driven cavity problem. In Table 1 we show iteration counts for DS-preconditioned GMRES (30) applied to the steady Stokes problem on a sequence of uniform grids. We report results for the optimal choice of α , determined experimentally. We see that the iteration count is independent of mesh size. Our tests show that for the Stokes problem, small changes in the value of α do not have a dramatic effect on the number of iterations. In Table 2 we report iteration counts for the steady Oseen problem on a sequence of uniform grids and for different values of ν , using optimal or near-optimal values of α . We found that for small values of the viscosity, scaling (as described


Table 1 Preconditioned GMRES iterations for Stokes problem with the optimal

73

α.

Grid

Its

αopt

16 × 16 32 × 32 64 × 64 128 × 128

11 12 12 10

0.006 0.001 0.0006 0.0002

Table 2 Preconditioned GMRES iterations for Oseen problem for different values of ν , optimal α (lid driven cavity, uniform grids). Grid

ν = 0.1

ν = 0.01

ν = 0.001

16 × 16 32 × 32 64 × 64 128 × 128

14 15 15 14

19 20 19 18

44 57 50 39

Table 3 Preconditioned GMRES iterations for Oseen problem for different values of ν , optimal α (backward facing step, uniform grids). Grid

ν = 0.1

ν = 0.01

ν = 0.005

16 × 48 32 × 96 64 × 192 128 × 384

19 19 20 19

21 22 24 25

22 27 30 32

Table 4 Preconditioned GMRES iterations for Stokes problem with the optimal driven cavity, stretched grids).

α (lid

Grid

Its

αopt

16 × 16 32 × 32 64 × 64 128 × 128

9 9 9 9

0 .2 0 .3 0 .3 0 .4

Table 5 Preconditioned GMRES iterations for Oseen problem for different values of ν , optimal α (lid driven cavity, stretched grids). Grid

ν = 0.1

ν = 0.01

ν = 0.001

16 × 16 32 × 32 64 × 64 128 × 128

14 14 14 14

30 37 39 39

137 166 177 189

in the previous section) dramatically improves the rate of convergence of preconditioned GMRES iteration. For example, for ν = 0.001 the preconditioned iteration without scaling requires over 200 iterations on the fine grid. One can clearly see again that DS preconditioning results in h-independent convergence rates. There is a mild dependence of the rate of convergence on the viscosity. Note, however, that the convergence rate on the finest grid remains excellent. Results for the backward facing step problem (Oseen only) are presented in Table 3. Note that for this problem, the number of cells is different in the horizontal and vertical directions. Here the smallest value of the viscosity we consider is ν = 0.005, since the flow is unsteady for ν ≈ 0.001. The experiments show a fairly robust convergence behavior with respect to both h and ν . Next, we present some results using stretched grids, using the default stretch factors provided by IFISS. These are 1.2712 for the 16 × 16 grid, 1.1669 for the 32 × 32 grid, 1.0977 for the 64 × 64 grid, and 1.056 for the 128 × 128 grid. The stretching is done in both the horizontal and vertical directions starting at the center of the domain, resulting in rather fine grids near the boundaries. In practice, stretched grids of this kind are often used to resolve boundary layers, if present. However, using stretched grids typically results in linear systems that are considerably more difficult to solve with iterative methods. Tables 4–5 report results on numerical experiments for the driven cavity Stokes and Oseen problems (respectively) discretized on a sequence of stretched grids. Clearly, the results for the Stokes problem are very good—indeed, even better than on the uniform grids. It is important to mention that without the diagonal scaling (using the diagonals of the mass matrices), the

74


Table 6 Preconditioned GMRES iterations for generalized Stokes driven cavity problem (ν = 1, σ = h−1 ) with the optimal α . Grid

Its

αopt

16 × 16 32 × 32 64 × 64 128 × 128

11 12 11 12

0.006 0.001 0.0005 0.0001

Table 7 Preconditioned GMRES iterations for generalized Oseen driven cavity problem (σ = h−1 ) for different values of ν , optimal α . Grid

ν = 0.1

ν = 0.01

ν = 0.001

16 × 16 32 × 32 64 × 64 128 × 128

12 14 16 19

13 15 18 22

13 15 19 23

convergence behavior is much worse; in particular, the number of iterations increases as the mesh is refined. Also note that the optimal value of α is much larger now than for the uniform grid case. Table 5 shows that mesh independent convergence is observed also for the steady Oseen problem, although there is a noticeable dependence on the viscosity. We note that without scaling with the mass matrices, the rate of convergence deteriorates very rapidly for decreasing mesh size and viscosity. Hence, diagonal scaling helps mitigate the negative impact of stretched meshes and removes the dependence on mesh size; however, it does not appear to be enough to achieve robustness for very small values of ν . In our experience, this kind of degradation in solver performance is observed also with other preconditioners when stretched grids are used. 6.2. Unsteady problems Next, we report on analogous experiments involving the generalized Stokes problem (with ν = 1) and the generalized Oseen problem (for several values of ν ). A sequence of linear systems of this type needs to be solved when the timedependent Stokes or Navier–Stokes equations are integrated numerically using implicit time-stepping schemes. Now the matrices A 1 and A 2 in (31)–(33) are of the form (10) where σ = h−1 and M is the velocity mass matrix; also, N i = 0 for generalized Stokes and N i = 0 for generalized Oseen. For brevity, we only consider the driven cavity problem on square meshes. As one can see from Table 6, for the generalized Stokes problem the results are virtually the same as those obtained in the steady case. The behavior, however, is somewhat different for the generalized Oseen problem. Indeed, we can see from the results in Table 7 that the rate of convergence for DS-preconditioned GMRES(20) is essentially independent of viscosity, while showing a mild dependence on h. 6.3. Choosing α As with all parameter dependent preconditioners, some guidelines need to be provided for the choice of α . The analytic determination of the value of α which results in the fastest convergence of the preconditioned GMRES iteration appears to be quite difficult, especially in the case of the Oseen problem. Our numerical experiments indicate that for problems posed on uniform meshes, α should be taken very small; for the Q2–Q1 discretization used in our experiments, the best value of α is often of the order of 10−3 or even smaller. Moreover, the best value of α gets smaller as the mesh is refined. Note that taking too small a value of α can lead to excessive ill-conditioning in the subsystems (31)–(33) to be solved at each GMRES iteration; in our experiments, however, this was never a problem. For problems posed on stretched meshes, on the other hand, our experiments show that with proper diagonal scaling the optimal α is often of the order of 10−1 . A possible rule of thumb, applicable in case of uniform meshes, is to tie α to the discretization parameter h. Taking α ≈ h2 was found to give pretty good results in most cases, at least for h small enough. In the case of a uniform mesh, the rate of convergence of DS-preconditioned GMRES does not appear to be overly sensitive to the choice of α , in the sense that small relative changes in the size of α do not usually cause the number of iterations to change too drastically. In Fig. 1 we show the total number of GMRES and GMRES(30) iterations for the solution of the steady Oseen problem (with ν = 0.01) as a function of α , for two choices of h. Fig. 2 displays the corresponding data for the unsteady case. From these plots, it appears that slightly overestimating the parameter α does not lead to drastic changes in the number of iterations, especially in the steady case. Underestimating the optimal α can be more harmful, but this is easy to avoid. One way to do this is to find a (near-)optimal value of α on a coarse grid, and then use the same value of α on finer grids. Since the optimal α tends to decrease as the grid id refined, this strategy will generally overestimate the optimal α . This strategy, again, assumes uniform meshes are used.


(a) 16 × 16 grid

(b) 64 × 64 grid

Fig. 1. Number of GMRES iterations vs. the value of

(a) 16 × 16 grid Fig. 2. Number of GMRES iterations vs. the value of

75

α , steady Oseen problem, ν = 0.01. Uniform mesh.

(b) 64 × 64 grid

α , unsteady Oseen problem, ν = 0.01. Uniform mesh.

7. Conclusions and future work In this paper we have introduced a “dimensional splitting” approach for the solution of saddle point systems in which the (1, 1) block can be partitioned into a two-by-two block diagonal structure. Saddle point systems of this kind arise in a number of applications: in this paper we focused on linear systems arising from the discretization of two-dimensional incompressible flow problems. We have established the convergence of the fixed-point iteration, and investigated experimentally its use as a preconditioner for restarted GMRES on a set of Stokes and Oseen problems (both steady and unsteady) discretized by Q2–Q1 finite elements on both uniform meshes and stretched meshes. The numerical experiments indicate that for uniform meshes, the preconditioner results in fast convergence for a wide range of mesh sizes and Reynolds numbers. Diagonal scaling was found to be essential to achieve robustness for small values of the viscosity. For stretched meshes, on the other hand, we found that diagonal scaling (using the diagonals of the velocity and pressure mas matrices) is absolutely necessary to retain mesh-independent convergence for steady Stokes and Oseen problems. On stretched meshes, however, the performance of the preconditioner deteriorates for small values of the viscosity. This problem occurs with other preconditioners as well. Future work should include further analysis of the preconditioned iteration, including using Local Fourier Analysis [28,3] for estimating the optimal value of the relaxation parameter α , and extension to the 3D case. We observe here that the basic alternating iteration (11)–(12) is of Peaceman–Rachford type and cannot be directly extended to the case of three splittings. Extension to the 3D case requires the alternating iteration to be of Douglas–Rachford type; see [27, pp. 244–245]. From the viewpoint of implementation, the 3D case necessitates using (inexact) inner iterative solves for the subproblems that occur in the application of the preconditioner. The effect of inexact solves on the performance of the DS preconditioner needs to be investigated. We mention that a promising approach for solving systems of the type (31)–(33) was recently described

76


in [10]. Another possibility would be to use the scalable algebraic multilevel solvers for scalar elliptic PDEs provided in the state-of-the-art Trilinos software package [20]. Finally, we mention that the method presented in this paper forms the starting point for a new preconditioning scheme, called Relaxed Dimensional Factorization, which is currently under development; see [7]. Acknowledgements We would like to thank Zhen Wang and Qiang Niu for useful comments and for help with some of the numerical experiments. We are also indebted to Daniel Szyld and two anonymous referees for constructive criticism that resulted in several improvements. References [1] P.R. Amestoy, T.A. Davis, I.S. Duff, An approximate minimum degree ordering algorithm, SIAM J. Matrix Anal. Appl. 17 (1996) 886–905. [2] Z.Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003) 603–626. [3] M. Benzi, M.J. 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Iterative adaptive RBF methods for detection of edges in two-dimensional functions Jae-Hun Jung a,∗ , Sigal Gottlieb b , Saeja Oh Kim b a b

Department of Mathematics, The State University of New York at Buffalo, Buffalo, NY 14260-2900, United States Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA 02747-2300, United States

a r t i c l e

i n f o

Article history: Received 23 November 2009 Received in revised form 30 June 2010 Accepted 9 August 2010 Available online 20 August 2010 Keywords: Two-dimensional edge detection Gibbs phenomenon Radial basis function methods Adaptive methods

a b s t r a c t Radial basis functions have gained popularity for many applications including numerical solution of partial differential equations, image processing, and machine learning. For these applications it is useful to have an algorithm which detects edges or sharp gradients and is based on the underlying basis functions. In our previous research, we proposed an iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities in one-dimensional problems. The iterative edge detection method is based on the observation that the absolute values of the expansion coefficients of multiquadric radial basis function approximation grow exponentially in the presence of a local jump discontinuity with fixed shape parameters but grow only linearly with vanishing shape parameters. The different growth rate allows us to accurately detect edges in the radial basis function approximation. In this work, we extend the one-dimensional iterative edge detection method to two-dimensional problems. We consider two approaches: the dimension-by-dimension technique and the global extension approach. In both cases, we use a rescaling method to avoid ill-conditioning of the interpolation matrix. The global extension approach is less efficient than the dimension-by-dimension approach, but is applicable to truly scattered two-dimensional points, whereas the dimensionby-dimension approach requires tensor product grids. Numerical examples using both approaches demonstrate that the two-dimensional iterative adaptive radial basis function method yields accurate results. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction Radial basis function (RBF) methods have been actively investigated for a broad range of application areas, including machine learning and neural networks [15], image processing and reconstruction [3,13], and numerical approximations of partial differential equations (PDEs) [10,12]. RBF methods work on a wide variety of meshes including scattered points, which makes them attractive for problems on complex geometries and problems featuring a large degree of localization which benefits from irregularly distributed points. The grid points used for the RBF method are called centers. Centers can be distributed arbitrarily based on the given geometry, without any constraints from the method itself. For this reason RBF methods are referred to meshless methods. Furthermore, the RBF approximation yields fast convergence when the function to be approximated is smooth enough [1,2]. However, when the function is non-smooth, the fast rate of convergence deteriorates significantly unless the centers or the shape parameters of the RBFs are adapted appropriately. Such an adaptive approach [4] is necessary to obtain a stable RBF approximation. To apply the adaptive method, it is important to be able to

*

Corresponding author. E-mail addresses: [email protected] (J.-H. Jung), [email protected] (S. Gottlieb), [email protected] (S.O. Kim).


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J.-H. Jung et al. / Applied Numerical Mathematics 61 (2011) 77–91

identify the location of the discontinuity or sharp gradient in the RBF approximation. That is, one needs an accurate RBF edge detection algorithm. Several edge detection algorithms for high order methods have been developed. Examples include the Fourier conjugate sum and the enhanced conjugate sum edge detection methods developed by Gelb and Tadmor [6]. These methods are based on the Fourier method and have been applied successfully to many practical problems. These methods are particularly useful for problems which have Fourier data as an inherent part of the method. For problems which feature RBFs as their basis functions, it is more efficient to exploit the properties of the RBFs to determine the edges. The numerical approximation of discontinuous problem with the RBF method is highly oscillatory near the local jump discontinuities, a feature known as the Gibbs phenomenon [5,8]. Interestingly, the Gibbs phenomenon of the RBF method can be used to provide an accurate and efficient edge detection algorithm [9]. In [9], an iterative adaptive RBF method was proposed for one-dimensional edge detection. It was shown that the proposed method performs well with multiple local jump discontinuities. As explained in the following sections, the iterative adaptive RBF method exploits the different growth/decay rates of the expansion coefficients. At or near the discontinuity, the absolute values of the expansion coefficients attain local maxima and grow exponentially with N, the total number of center points, if the shape parameter is fixed for all N and it is greater than a certain value. The magnitudes of the expansion coefficients also decay exponentially away from the local discontinuity. This exponential decay rate changes dramatically into a linear one when the shape parameters at or near the discontinuity are set to vanish. The success of the iterative adaptive method is due to the finding that the growth rate of the expansion coefficients can change significantly when the shape parameters vary adaptively. This dramatic change in the growth rate makes it easy to determine whether the region considered contains a discontinuity or not. In such a way, the iterative RBF method can serve as an efficient edge detection method for RBF approximations. In this paper, we extend the 1D iterative adaptive RBF method to the 2D problems, using two approaches: a slice-byslice approach and a global extension approach. The slice-by-slice approach finds edges in the x-direction first with a given y value, and then find edges in y-direction with a given x value. The final edge map is the union of these two edge sets. The advantage of this approach is that it is simple to implement and computationally inexpensive. However, it requires tensor product grids, so that the points may be scattered in one dimension but not in the other. On a truly scattered 2D grid, we require a more sophisticated approach, the global extension method, in which we use the global expansion coefficients of the 2D RBF basis, and follow the same edge detection approach as the 1D method. The maximum absolute values of the expansion coefficients are used to define the edge maps. This approach can be applied to scattered grids, and yields more rigorous results than the slice-by-slice approach because it considers the variations of the 2D function both in x- and y-directions simultaneously. However, this approach is computationally costly. This paper presents preliminary numerical results which verify that the 2D iterative adaptive RBF method, using both the slice-by-slice and the global extension approaches, yields accurate results on the 2D data. The paper has the following structure. In Section 2, a brief explanation of the RBF approximation is given. In Section 3, both the global extension method and the slice-by-slice method are explained. In Section 4, various numerical examples are given. In Section 5, a brief summary will be presented and future research directions are discussed. 2. 1D iterative adaptive RBF methods 2.1. Radial basis function approximation Radial basis functions are defined using a set of points, known as centers, and a set of corresponding shape parameters. Consider a center set X composed of the center points xi such that X = {xi | xi ∈ [−1, 1], i = 1, . . . , N }, and let f denote the set of function values at the centers, f = { f i | f i = f (xi ) ∈ R, xi ∈ X } for the real-valued function f (x). In this work, we use the multiquadric (MQ) RBFs, defined by

φ j (x) =

(x − x j )2 + 2j ,

x ∈ [−1, 1], x j ∈ X

where the shape parameters j may be determined a priori for the approximation or can be adaptively determined depending on the regularity of the given function f (x) [11]. In the special case that the shape parameter vanishes (i = 0) for all the given centers, the corresponding RBF basis function becomes the piecewise linear function |x − xi | and its first derivative has a jump at x = xi . Although the strength of RBF methods is their ability to handle irregular grid points, in the following 2 . discussion we assume for simplicity that we have uniform centers xi = −1 + (i − 1)x for ∀i = 1, . . . , N where x = N − 1 The MQ RBF approximation s f , X (x) to the function f (x) is given by the linear combination of MQ RBFs [1,7]:

s f , X (x) =

N

λ j (x − x j )2 + j 2 ,

(1)

j =1

where λ j are the expansion coefficients, which are obtained by the interpolation condition s f , X (xi ) = f (xi ), ∀xi ∈ X , that is,

λ = M−1 f,

(2)


79

where the interpolation matrix M has its elements M i j = (xi − x j )2 + 2j , f = ( f 1 , . . . , f N ) T and λ = (λ1 , . . . , λ N ) T . Unfortunately, it is well known that the above linear system, Eq. (2) is ill-posed for a large N. The RBF approximation in Eq. (1) allows a simple computation of the derivative sf , X (x), given by the linear combination of the derivatives of RBF basis functions. At each center x = xi , the derivative is given by

sf , X (xi ) =

N

λ j (xi − x j )/ (xi − x j )2 + j 2 .

(3)

j =1

Here note that if j = 0 the derivative is not defined at xi = x j . For the adaptive method used in the following discussion to be well posed, we will let sf , X (xi ) = 0 at xi = x j when j = 0. Let D denote the derivative matrix D i j =

(xi − x j )/ (xi − x j )2 + j 2 with non-zero j , then the derivative of s f , X (x) at center points is given by s = Dλ,

where s

(4)

= (s

(x ), . . . , sf , X (xN ))T . We note that one has to exercise caution when using the RBF derivative approximation. f ,X 1

In general, this operation is not exact in the sense that m + 1 times RBF derivative operations for the polynomial of degree m does not vanish because RBFs are not polynomials. In fact, if f (x) is a constant function, say f (x) = 1, we expect its derivative to be zero. However, the matrices D and M are non-singular, so that setting DM−1 f = 0 yields f (x) = 0, which is a contradiction. 2.2. 1D iterative adaptive RBF methods The iterative adaptive RBF edge detection method uses the difference in the rate of growth or decay of the RBF coefficients. To understand this method, we use three properties of RBF approximations which were observed numerically and discussed in [9]: Property 1 (Linear growth). Given the sign function f (x) = sgn(x) with x ∈ [−1, 1], and the set of centers X = {xi | xi = 2 −1 + (i − 1)x, i = 1, . . . , N } where x = N − and N is even, the MQ expansion coefficients are given by 1

λi =

⎧ 0 ⎪ ⎨

N −1 2 ⎪ ⎩ N −1 − 2

i ∈ I \{ N /2, N /2 + 1}, i = N /2 ,

(5)

i = N /2 + 1 ,

if the shape parameters are given by (proved in [9]).

i = > 0, i ∈ I \{ N /2, N /2 + 1} and i = 0, i ∈ { N /2, N /2 + 1}, where I = {1, . . . , N }

Property 2 (Exponential growth). With the same conditions used in Property 1, except a uniform shape parameter i = > 0, ∀i ∈ I , for some it was observed in [9] that the maximum absolute value of the expansion coefficients grows exponentially with the number of centers,

max |λi | ≈ q N ,

(6)

i∈I

where the constant q is q > 1 and independent of N.

N

Property 3 (Multiple edges). Let f (x) be f (x) = i =s 1 αi (x) H c i (x) with c i ∈ Ω = (−1, 1). Here H c i (x) is the Heaviside function whose jump exists at x = c i . αi (x) are analytic functions in Ω , and are permitted to be constant functions. f (x) possibly contains maximum N s discontinuities due to the singular feature of the Heaviside functions H c i . Properties 1 and 2 yield Property 3 that the local maxima of |λi | occur at centers where c i exist (see [9]). These three properties are used to define the iterative adaptive RBF edge detection method, which is based on the growth/decay properties of the expansion coefficients λi . A coefficient λi whose absolute value is larger than a certain tolerance level suggests a discontinuity near xi . The 1D iterative method defines a concentration map C as

C = C i C i = |λi |, xi ∈ X , i ∈ I .

(7)

The edge exists near the center point where the value of the element of C is large. The higher the contrast between the largest value of the element of C and the others the easier to determine the edge location. In fact, the concentration map itself contains the edge as the function. To improve the method and obtain sharper edges, we can incorporate derivative information as well [9] to define concentration map C for sf , X (xi ) = 0:

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Table 1 Pseudo MATLAB code for 1D RBF edge detection with = 0.1 and η = 12 . function [edge] = EdgeMap ( f , M, D) N = length( f ); = 0.1 · ones( N , 1); edge = zeros( N , 1); λ previous = zeros( N , 1); η = 12 ; Residue = 1; while Residue 10−10 λ = M −1 f ; f = D · λ; C = | f ||λ|/ max(| f ||λ|); for i = 1 : N if C (i ) η edge (i ) = 1; (i ) = 0; end end Update M and D; Residue = λ − λprevious ; λ previous = λ; end

C = C i C i = λi sf , X (xi ) , xi ∈ X , i ∈ I , where C i is equivalent to C i = |λi

j

(8)

D i j λ j |. We then define the normalized concentration map Cˆ as

Cˆ = Cˆ i Cˆ i = C i / max C i , i ∈ I . i∈I

(9)

Here note that ∀C i = 0 iff f (x) = 0 at every xi for Eq. (7). If f (x) = 0 at x = xi ∈ X , the problem is trivial and the edge detection method is not applied. For Eq. (9), we need to show that if not every f (xi ) vanishes, there exists at least one nonvanishing element of C i for Eq. (8). Then the concentration map is well defined. Otherwise the problem becomes trivial. The concentration map reflects the smoothness of the function and indicates the possible location of the local jump. We define the edge set S such that

S = { S i | S i = xi for Cˆ i η, xi ∈ X , i ∈ I },

(10)

where 0 < η < 1 is the tolerance level given a priori. Since the expansion coefficients decay exponentially away from the local jump, the value η = 12 is a reasonable value for the tolerance level. Iterative adaptive RBF methods: The iterative method first detects the edge set S based on the expansion coefficients λi with the fixed shape parameters i (Property 2). Then the adaptive method is applied at the center points in S. According to the Property 1, the expansion coefficients at the centers xi corresponding to S i then become algebraically small. If new expansion coefficients are computed after the adaptation of the shape parameters, the maximum of the expansion coefficients occurs at the second jump discontinuity. By repeating this procedure, all the possible jumps are detected (Property 3). A description of this algorithm is given in Table 1. In Table 1, a pseudo MATLAB code is provided. The pseudo code generates a new function edge at centers xi whose values are either 0 or 1. The value of 1 indicates that the possible edges are located on or near the corresponding center points. 3. Iterative adaptive RBF method for 2D edges To extend the 1D RBF iterative edge detection method to the 2D problems we can adopt two possible approaches. The first approach is the global extension method. The global extension method relies on the 2D RBF basis functions, and uses the 2D global expansion coefficients. The second approach is the dimension-by-dimension or slice-by-slice application of the 1D RBF edge detection method. The slice-by-slice method treats each dimension separately and the final edges are obtained by summing up the edges obtained from each dimension. The slice-by-slice method requires more strict 2D grid system or 2D reconstruction from the arbitrary grid system to define slice, but the global method does not require such complication on any grid system. The global method, however, suffers from ill-conditioning due to the large size of the interpolation matrix and requires us to separate the problem into multiple sub-domain problem for the search of the edges in each sub-domain. The second approach reduces this complication, but can only be applied on tensor product grids. The two approaches are described below.


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3.1. 2D global extension method The 2D global approach uses the global two-dimensional MQ RBFs based on a distribution of points {xi } and { y j } in 2D, and basis functions

φi , j (x, y ) =

(x − xi )2 + ( y − y j )2 + i2j .

The 2D MQ RBF approximation s f , Z (x, y ) is given by

s f , Z (x, y ) =

Ny Nx

λi j (x − xi )2 + ( y − y j )2 + i2j ,

(11)

i =1 j =1

where Z is the set of the 2D centers, λi j the expansion coefficients, i j the shape parameters, N x and N y numbers of centers in x- and y-directions, respectively. Suppose that the real-valued function f (x, y ) are defined at the 2D center points (xi , y j ). Let the expansion coefficient vector be

Λ = (λ11 , λ12 , . . . , λ N x N y −1 , λ N x N y )T , the function vector

T

f = f (x1 , y 1 ), f (x1 , y 2 ), . . . , f (x N x , y N y −1 ), f (x N x , y N y )

,

and the interpolation matrix M

M (i j )(kl) =

(xi − xk )2 + ( y j − yl )2 + kl ,

(12)

where (i j ) = (i − 1) N y + j and (kl) = (k − 1) N y + l with i , k = 1, . . . , N x and j , l = 1, . . . , N y . With these notations the 2D expansion coefficients are found by

Λ = M−1 f.

(13)

The size of the interpolation matrix M is ( N x N y ) × ( N x N y ). The complexity of the direct method for the interpolation is about O ( N x3 N 3y ). If N x = N y = N, then it becomes O ( N 6 ). This makes the 2D global extension method costly. For this reason, a domain decomposition method should be considered to implement the global approach if N x and N y are large [8]. Similar ideas have been proposed such as the RBF domain decomposition techniques for elliptic PDEs found in [16]. To understand the domain decomposition technique, consider the 2D MQ RBFs, where the center set Z as in Eq. (11) is denoted by

Z = (xi , y j ) xi ∈ [−1, 1], y j ∈ [−1, 1], i ∈ I x , j ∈ I y

with I x = {i | i = 1, . . . , N x , x ∈ Z∗ } and I y = { j | j = 1, . . . , N y , y ∈ Z∗ }. For the domain decomposition, the given do-

L

main Ω is split into L subdomains Ωi , i = 1, . . . , L where Ω = i =1 Ωi , and for some i = j , i , j = 1, . . . , L, Ωi ∩ Ω j can be a non-empty set, that is, the overlapping subdomains are allowed. We also decompose a given function f (x, y ) into L f i (x, y ), (x, y ) ∈ Ωi where f (x, y ) = i =1 f i (x, y ), (x, y ) ∈ Ωi . Once this domain decomposition is performed, it is as though the method is performed on smaller domains. This approach has the advantage of creating smaller systems that are more efficient to solve and better conditioned. Whether using one domain or multiple smaller domains, the next step is to apply the procedure similar to the 1D method. We assume that the behavior of the 2D expansion coefficients in Eq. (13) is similar to that of the 1D coefficients. However, we observed that in the global 2D case the use of the derivative (or, more accurately, the gradient) does not improve the concentration map, and is a costly procedure which makes the 2D edge detection less efficient. Thus, we return to the original definition of the concentration map which relies only on the expansion coefficients. Next, the critical step is to make the shape parameters vanish at or near the local discontinuity. Table 2 shows the pseudo MATLAB code for 2D RBF method. The overall algorithm is the same as the 1D. The input variable is a 2D array f . The 2D array is converted into a 1D string. Once the 2D array is converted into a 1D string, the same 1D algorithm is applied. In the subroutine EdgeMap, the construction procedure for M should be modified to correspond to the 1D string made of f . It is interesting to note that with the vanishing shape parameter, the 2D RBF also has a jump discontinuity in its derivative both in x- and y-directions as in the 1D case. Fig. 1 shows the 2D RBF function with the vanishing shape parameter (left) and non-zero shape parameter (right). Only the difference is that the 1D RBF with vanishing shape parameter is a piecewise linear function, but the 2D RBF with vanishing shape parameter, i j = 0 is not linear unless either direction is fixed as shown in Fig. 1.

82


Fig. 1. The 2D MQ RBF. Left: With the vanishing shape parameter. Right: With the non-zero shape parameter,

= 0.1.

Table 2 A pseudo MATLAB code for the 2D RBF global edge detection. Input a 2D array f ; Convert f into a 1D string g; Construct the corresponding M; edge = zeros( N x N y , 1); [edge ] = EdgeMap(g, M); Convert edge into a 2D array;

3.2. Edge detection by slice-by-slice The global extension approach in the previous section is the true 2D generalization of the 1D method. However, it is a costly approach, requiring the inversion of a much larger matrix. The slice-by-slice approach is a straightforward extension of the 1D RBF edge detection method in a dimension-by-dimension approach, with the edge map determined by the union of the edge maps obtained from the x- and y-directions. This approach is straightforward, less costly, and easy to implement, as long as the 2D points are given on a tensor product grid. The numerical examples in the next section also verify that this approach yields accurate results. Each array has an arbitrary domain size in the x- and y-directions, and the major concern is the ill-conditioning problem of the interpolation matrix. One approach that resolves this issue is the rescaling of the domain. 3.2.1. Ill-conditioning and rescaling domain intervals To combat the exponential growth with N of the condition number κ (M) of the MQ RBF interpolation matrix M, first we rescale the domain interval for the interpolation matrix. For example, when dealing with the 2D images, the domain intervals of x- and y-directions can be arbitrarily chosen for the reconstruction. If we use a fixed domain interval for any size of array, e.g. x = [−1, 1] and y = [−1, 1] and a fixed shape parameter i (e.g. i = 0.1 such as that used for the 1D iterative edge detection method), the interpolation matrix becomes highly ill-posed as N becomes large. This ill-conditioning can be reduced by simply stretching the domain, which suggests that the domain size should be determined for the edge detection as a function of the given number of grid points in each direction, and the initial domain interval. Numerical studies allow us to investigate the optimal size of the domain for a given number of center points. Consider a simple sign function, f (x) = sgn(x) in x-dimension. We seek the best reconstruction domain Ω ∗ = [−xc , xc ] such that with the given even N and X the best edge set S is accurately detected as the best edge set S ∗ = {x N /2 , x N /2+1 } for the be the minimum domain size needed for the best domain Ω ∗ for the best edge set S ∗ . For uniform distribution. Let xmin c the numerical experiments, the shape parameter is fixed for every value of xc and N. Figs. 2 and 3 show the numerical experiments on the relation between xc and the best edge set. The numerical computation, the MATLAB “\” operation is used with machine accuracy of about 10−16 . Fig. 2 shows the edge sets versus N with different xc . From left to right, top to bottom, xc = 1.0, 1.12, 1.24, 1.36, 1.48, 1.6, respectively. The figure shows that the maximum number of N that allows the best edge set increases if xc increases. For xc = 1.0, the maximum N for the best edge set is N ∼ 200. For N 200, many wrong edges are detected due to the ill-posedness of the interpolation matrix M. Fig. 3 shows the instability of the interpolation matrix on xc –N plane. The left figure shows the stable and unstable regions. The stable region is where the best edge set S is detected and is given in blue color while the unstable region is where false edges are detected. It is interesting that there are blue spots even in the unstable region. The right figure shows the contour plot of the total number of elements of the edge set. As the figure shows, the number of elements of the detected edge set increases rapidly after a certain value of N for a fixed xc . Fig. 4 shows the histogram of the total number of elements of the detected edge set, #( S ), versus N for several values of xc = 1, 1.303, 1.4545, 1.6 in logarithmic scale based on base 2. In the figure, the value of 1 in the y-axis means that two edge elements are found. Note that the best edge set is composed of only two elements. The figure shows that for large values of N there are more than two edge elements are found due to the ill-posedness of the interpolation matrix after a certain value of N, say, N ∗ . Also note that some large values of N yield the best edge map although N > N ∗ . This is confirmed by the blue stable dots in the unstable region clearly shown in the left figure of Fig. 3.


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Fig. 2. Edge sets detected with the MQ RBF iterative method versus N for different domain intervals xc , xc = 1.0, 1.12, 1.24, 1.36. 1.48, 1.6. The uniform shape parameters are used = 0.1 for every N and xc .

Fig. 3. Left: Stability of the iterative method for fixed shape parameters = 0.1 on the xc –N plane. Stable region (blue area) is where the best edge set is obtained. Right: The contour plot of the total number of elements of the edge set. (For interpretation of the reference to color in this figure legend, the reader is referred to the web version of this article.)

Based on these numerical experiments, we can roughly estimate the relation between xmin and N in the range of at least c N ∈ [100, 340] as

xmin ≈ c

6 1000

N − 0.2.

(14)

Using these values, we can rescale the domain to enhance stability. 3.2.2. 1D domain decomposition methods Beside the rescaling method, one can also use the domain decomposition method for large N in one direction when the slice-by-slice method is applied. For example, we know that Ω ∗ = [−1, 1] for N = 128, according to our 1D numerical experiments Eq. (14). If we have N > 128, then we decompose the domain into smaller subdomains, either overlapping or non-overlapping. The final edge map is the union of the edge maps from each subdomain.

84


Fig. 4. Number of elements in the edge set S versus N for different xc = 1.0, 1.303, 1.4545, 1.6 in logarithmic scale based on base 2.

Fig. 5. Broadening effect. Left: The original square image with 40 × 40 resolution. Right: Edges detected during the whole iteration.

4. Numerical results 4.1. Global extension method In this section, we provide several simple numerical examples of the 2D iterative global edge detection method. For this method, we assume that N x = N y and the interpolation matrix has a size of N 2 × N 2 . The complexity is, at worst, about O( N 6 ). The 2D global edge detection process is different from the 1D case. For the 1D case, the edges are the sum of edges detected at each iteration step. For the 2D case, however, our numerical results show that the final edges should be those found at the last iteration step. The numerical results show that if all the edges from each iteration step are collected as the final edge set, the final edge has the broadening effect. That is, the edge set is not sharp but contains broad neighborhoods of real edges. Fig. 5 shows such broadening effect. For the numerical example, a square box inside 40 × 40 arrays is used. The left figure shows the original square image and the right figure shows all the edges detected from every iteration step. For the edge criteria, we use

1

|λi j |/ max |λi j | η = . ij

2

(15)


85

Fig. 6. Final edge map on 60 × 60 array. Left: The original square image. Right: The edge map detected at the last iteration step.

Fig. 7. Final edge map. Left: The original circle image. Right: The edge map detected at the last iteration step.

To stop the iteration we use the condition that if the difference between the current expansion coefficients and those from the previous iteration step is small enough in a certain norm the iteration stops. That is, the iteration stops if

i +1 λ − λi t ,

where λi +1 and λi are the coefficient vectors at the i + 1 and i iteration steps and t is the tolerance level, for which t = 10−8 was used for the numerical experiments. Here · denotes the vector norm and we use the vector 2-norm · 2 for the numerical experiments. In the right figure all the edges detected at each step are plotted. As a result, the broadening effect is clearly seen in the edge map. Figs. 6 and 7 show the edges detected at the final iteration step for the square box image (Fig. 6) and the circle image (Fig. 7). In each, the left figure shows the original image and the right figure shows the edge map with the RBF method. For the numerical experiments, the same condition was used, Eq. (15). Compared with Fig. 5, we observe that the sharper image is obtained when the edge map is constructed based on the edges found at the last iteration step only. To explain the broadening effect, we show the edge map at each intermediate iteration step in Figs. 8 and 9 for the square and circle images, respectively. As shown in the figures, the 2D global RBF edge detection method yields interesting intermediate patterns. Unlike to the 1D case with which each iteration step does not depend on the previous iteration, each iteration of the 2D method depends on the previous edge detection result. For the 1D case, once the edges are detected, the corresponding shape parameters are set to vanish and the image is smoothed at and near these edges. Once smoothed, the detected edges are no longer edges and the next iteration only detects next edges. In this way, the smoothing does not affect the next edge detection iteration. This is because the 1D case has only one direction of gradient. For the 2D case, however, after the shape parameters are set to vanish, the smoothed image may cause a new edge. The reconstructed image is smoothed near the detected edges but possibly in a particular direction only resulting in creating a new edge in the reconstruction. Thus the next iteration is affected by the wrong edges generated in the previous iteration. For example, in Fig. 6, the 2D method detects edges at the 4 corners of the square at the first iteration. We observed that the edges at the next iteration center around those 4 edges from the first iteration. In Fig. 7, we also observe that the edges at the second iteration are found along the alignment of edges from the first iteration. Thus we know that the 2D global method finds and generates wrong edges at each iteration and the collection of all these false edges yields the broadening effect. Despite the wrong edge detection during the intermediate iteration steps, we also observe that interestingly edges detected at the last iteration are close to the real edges.

86


Fig. 8. Edge map for the square image at each iteration with the first iteration through the last iteration from left top to right bottom.

The 2D global method captures edges satisfactorily at the last iteration and fine features in the test images, but is costly and requires additional techniques to enhance its efficiency. The full development and testing of this approach is left to future work. 4.2. Dimension by dimension method In this section, we provide numerical examples of the slice-by-slice approach. First, we repeat the simple image used in the previous section, a square of size 10 × 10 pixels on a 40 × 40 pixel domain. Although this is a simple image, the problem itself has low resolution (the image itself is very small) and so is challenging from the standpoint of edge detection. Fig. 10 shows the edge detections applied to the x- and y-slices and the final figure which is the union of the two edge maps from x- and y-directions, from left to right, respectively. The method easily and efficiently identifies the edges in each slice, and this procedure produces a reliable edge image. We also note that if a discontinuity does not exist inside the sliced domain the domain boundaries are included in the edge map. The boundary edges are excluded in the edge map otherwise. The slice-by-slice approach is inexpensive and can handle larger data sets efficiently, which allows us to consider more difficult problems without applying the 2D domain decomposition techniques. For the second numerical example, we consider the Shepp–Logan phantom image. The Shepp–Logan phantom image is the simplified version of the human brain phantom composed of ten ellipses. Each ellipse has different major and minor axes with different rotation angle and is filled by a single gray level. That is, the 2D Shepp–Logan phantom image is the 2D piecewise constant function. To generate the Shepp–Logan phantom image, we use the MATLAB intrinsic command

P = phantom(‘Modified Shepp–Logan’, N ), where P is the two-dimensional square array with the dimension of N × N. For the numerical example, N = 256 is used. We use two different domain intervals xc , i.e. xc = 1 and xc = 3. Fig. 11 shows the Shepp–Logan phantom image with N = 256. The image was created using the MATLAB command imagesc. Fig. 12 shows edges detected by the iterative RBF method. The figure shows the edges in x- and y-directions (left and middle columns, respectively) and the addition of edges (right column). The top row figure shows the results with xc = 1 and the bottom with xc = 3. As shown in the figure, the RBF


87

Fig. 9. Edge map for the circle image at each iteration with the first iteration through the last iteration from left top to right bottom.

Fig. 10. The x-slice edges detected (left), y-slice edges detected (middle), and the union of the two (right).

method detects many false edges including the real ones due to round-off errors for xc = 1. We adopt the formula derived in Eq. (14) in the previous section, which suggests that xc > 1.34. We used xc = 3 and the bottom figures in Fig. 12 show the results. The RBF method with xc = 3 properly detects the real edges without false edges found in the results with xc = 1.

88


Fig. 11. Shepp–Logan phantom image for N = 256.

Fig. 12. Edges detected by the iterative RBF method in x-direction, y-direction and their addition from left to right, respectively. Top: Edges with xc = 1. Bottom: Edges with xc = 3.

The following numerical examples take simple sample images of a flower and a maple leaf with N = 125. These images have very clear naturally defined edges, and the edge detection algorithm easily identifies the edges, thus recreating the images quite clearly (Figs. 13 and 14). In our final examples, we use standard images taken from the USC Signal and Image Processing Institute data base [14]. The first two images are a picture of a clock and an aerial photo of an airplane, both with N = 256. The edge detection method, with a rescaling of xc = 5, reliably captures these images, as seen in Figs. 15 and 16. Finally, we study a more complicated image, known as a resolution test (N = 256). The edge detection method successfully identifies these edges as well (Fig. 17). The performance of the RBF edge detection method is better for the resolution test image than the clock and aerial images. 5. Conclusions In this paper, we extended the previously developed 1D iterative adaptive MQ RBF edge detection method to 2D problems based on the global extension and the dimension-by-dimension approaches. In the dimension-by-dimension approach, we obtain the edge map by the collection of edges in each direction. To obtain the best edge map and improve the conditioning of the linear system, we rescale the domain interval in each direction. Several numerical examples are given to demonstrate that the dimension-by-dimension multiquadrics iterative edge detection approach yields accurate edge maps. Future work will focus on the development of RBF edge detection


Fig. 13. Left: Flower image. Right: Edge map.

Fig. 14. Left: Maple image. Right: Edge map.

Fig. 15. Left: Clock image. Right: Edge map of the image.

89

90


Fig. 16. Left: Aerial photograph of a plane on a runway. Right: Edge map of the image.

Fig. 17. Left: Resolution test image. Right: Edge map of the image.

methods based on other RBF bases, including Gaussians and Wendland functions which are localized, thus leading to a sparse interpolation matrices which are more efficient to implement. While the dimension-by-dimension approach works well for problems with a tensor product grid, for truly scattered images a global method will be necessary. For this reason, we explore a global extension method based on 2D MQ RBFs. We studied this method for images with small number of pixels and demonstrated that the global extension method performs well. For this method, we observe that there is a broadening effect in the final edge maps if the 1D method is directly extended to the 2D method. Unlike the 1D method, the 2D global extension method needs to use the final profile of the expansion coefficients obtained in the final iteration to avoid the broadening effect. We also observe that for the 2D global extension method, there exist interesting intermediate edge patterns during the iterations. These patterns are made due to the successive detection and elimination of edges at each iteration step. If the domain is large and contains many centers, a domain decomposition approach together with rescaling will need to be considered. A direct implementation of the global extension method yields a computational complexity of O ( N 6 ) for N × N array data if applied in a direct manner. Thus the domain decomposition technique becomes critical for the efficiency of the 2D global method. This will comprise our future work. The motivation of this work is the need for efficient edge detection in the RBF approximations of PDEs. The RBF approximations possibly contain local jump discontinuities such as shocks, or even sharp gradients which appear as discontinuities when the grid is not sufficiently refined. For PDEs, the continuity of the derivative(s) in the solution is also important for enhancing the accuracy and stability of the RBF approximations. In this paper we only focused on the continuity of the given function. In our future work, we will also further develop the RBF method for the detection of jumps in the derivative(s) of the given function or solutions of PDEs.


91

Acknowledgement This work was supported by the National Science Foundation under the grant DMS 0608844. References [1] M.D. Buhmann, Radial Basis Functions: Theory and Implementations, first ed., Cambridge Univ. Press, Cambridge, 2003. [2] M.D. Buhmann, N. Dyn, Spectral convergence of multiquadric interpolation, Proc. Edinb. Math. Soc. 36 (1993) 319–333. [3] J. Carr, R. Beatson, J. Cherrie, T. Mitchell, W. Fright, B. McCallum, T. Evans, Reconstruction and representation of 3D objects with radial basis functions, in: SIGGRAPH, 2001, pp. 67–76. [4] T.A. Driscoll, A.R.H. Heryudono, Adaptive residual subsampling methods for radial basis function interpolation and collocation problems, Comput. Math. Appl. 53 (2007) 927–939. [5] B. Fornberg, N. Flyer, The Gibbs phenomenon for radial basis functions, in: A. Jerri (Ed.), Advances in the Gibbs Phenomenon with Detailed Introduction, in: The Gibbs Phenomenon in Various Representations and Applications, Sampling Publishing, Potsdam, NY, 2008, pp. 201–224. [6] A. Gelb, E. Tadmor, Detection of edges in spectral data II: Nonlinear enhancement, SIAM J. Numer. Anal. 38 (2000) 1389–1408. [7] R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 176 (1971) 1905–1915. [8] J.-H. Jung, A note on the Gibbs phenomenon with multiquadric radial basis functions, Appl. Numer. Math. 57 (2007) 213–229. [9] J.-H. Jung, V. Durante, An iteratively adaptive multiquadric radial basis function method for the detection of local jump discontinuities, Appl. Numer. Math. 59 (2009) 1449–1466. [10] E.J. Kansa, Muliquadrics—A scattered data approximation scheme with applications to computational fluid dynamics: II. Solutions to parabolic, hyperbolic, and elliptic partial differential equations, Comput. Math. Appl. 19 (1990) 147–161. [11] E.J. Kansa, R.E. Carlson, Improved accuracy of multiquadric interpolation using variable shape parameters, Comput. Math. Appl. 24 (1992) 99–120. [12] E. Larsson, B. Fornberg, A numerical study of some radial basis function based solution methods for elliptic PDEs, Comput. Math. Appl. 46 (2003) 891–902. [13] Y. Ohtake, A. Belyaev, H.-P. Seidel, 3D scattered data approximation with adaptive compactly supported radial basis functions, in: Proceeding of International Conference on Shape Modeling and Applications, Genova, Italy, 2004, IEEE Computer Society, 2004, pp. 31–39. [14] USC Signal and Image Processing Institute Data base, http://sipi.usc.edu/database/database.cgi?volume=misc. [15] P.V. Yee, S. Haykin, Regularized Radial Basis Function Networks: Theory and Applications, John Wiley & Sons, Inc., New York, 2001. [16] X. Zhou, Y.C. Hon, J. Li, Overlapping domain decomposition method by radial basis functions, Appl. Numer. Math. 44 (2003) 241–255.




Numerical solutions of a two-phase membrane problem F. Bozorgnia Faculty of Sciences, Persian Gulf University, Boushehr 75168, Iran

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 21 April 2009 Received in revised form 13 August 2010 Accepted 13 August 2010 Available online 19 August 2010 Keywords: Free boundary problems Two-phase membrane Finite element method Error estimate Regularization

In this paper different numerical methods for a two-phase free boundary problem are discussed. In the first method a novel iterative scheme for the two-phase membrane is considered. We study the regularization method and give an a posteriori error estimate which is needed for the implementation of the regularization method. Moreover, an efficient algorithm based on the finite element method is presented. It is shown that the sequence constructed by the algorithm is monotone and converges to the solution of the given free boundary problem. These methods can be applied for the one-phase obstacle problem as well. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction A free boundary problem is a partial differential equation where the equation changes qualitatively across a level set of the solution u of the equation. A general form of elliptic free boundary problems with the Laplacian can be written as

u = f (x, u , ∇ u ) in Ω,

(1)

where the right-hand side term is supposed to be piecewise continuous, having jumps at some values of the arguments u and ∇ u and where Dirichlet boundary conditions are considered. Suppose that λ± : Ω → R are positive and Lipschitz continuous functions and Ω is a bounded open subset of Rn with smooth boundary. Assume further that g ∈ W 1,2 (Ω) ∩ L ∞ (Ω) and g changes sign on ∂Ω . Let K = { v ∈ W 1,2 (Ω): v − g ∈ 1, 2 W 0 (Ω)}. The functional

I (v ) =

1 2

|∇ v | + λ max( v , 0) − λ min( v , 0) dx, 2

+

−

(2)

Ω

is convex, coercive on K and weakly lower semi-continuous, hence it attains its infimum in K . The Euler–Lagrange equation corresponding to the functional (2) is (see [17])

u = λ+ χ{u >0} − λ− χ{u 0 ∪ ∂ x ∈ Ω : u (x) < 0 ∩ Ω, is called the free boundary. Properties of the solution, regularity of the solution, Hausdorff dimension and the regularity of free boundary have been derived in [14–17]. In [2] a perturbation of coefficients is considered by the author. Note that if we let λ− = 0 and let g be non-negative on the boundary we obtain the one-phase obstacle problem. There are numerous papers about numerical solutions of elliptic free boundary problems, variational inequalities and one-phase obstacle problem, see for instance [1,5,8,10–13]. In the next section we present numerical methods for (3) and then we discuss the regularization method and an error estimate which is needed for the implementation of the regularization method. Finally, an algorithm based on the finite element method is described. 2. Numerical approximation of the two-phase membrane problem Let ψ ∈ C 02 (Ω), and consider the following equation:

u = λ+ χ{u >ψ} − λ− χ{u 0} − (λ− + ψ)χ{ w 0}, ⎪ 1 ⎨ u 1 = λ u 1 = g1 on ∂Ω ∩ ∂{u 1 > 0}, ⎪ ⎩ u1 = 0 on ∂{u 1 > 0} \ ∂Ω,

and

⎧ − in {u > 0}, ⎪ 2 ⎨ u 2 = λ u 2 = g2 on ∂Ω ∩ ∂{u 2 > 0}, ⎪ ⎩ u2 = 0 on ∂{u 2 > 0} \ ∂Ω.

Since u is also a solution to (3),

(u 1 − u 2 ) = λ+ χ{u 1 >0} − λ− χ{u 2 >0} .

(5)

Eq. (5) shows that u 1 − u 2 ∈ C 1,1 (see [14,15]). Therefore, on the free boundary we have

∇ u 1 = −∇ u 2 . Now for a given uniform mesh on Ω ⊂ R2 , let x = y = h, and use standard finite difference methods for Eq. (5), to obtain

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F. Bozorgnia / Applied Numerical Mathematics 61 (2011) 92–107

1 h2

u 1 (xi −1 , y j ) + u 1 (xi +1 , y j ) − 4u 1 (xi , y j ) + u 1 (xi , y j −1 ) + u 1 (xi , y j +1 )

−

1 h2

u 2 (xi −1 , y j ) + u 2 (xi +1 , y j ) − 4u 2 (xi , y j ) + u 2 (xi , y j −1 ) + u 2 (xi , y j +1 )

= λ+ χ{u 1 (xi , y j )>0} − λ− χ{u 2 (xi , y j )>0} .

(6)

If u 1 (xi , y j ) and u 2 (xi , y j ) are obtained from (6) and the conditions

u 1 (xi , y j ) · u 2 (xi , y j ) = 0 and

u 1 ( x i , y j ) 0,

u 2 ( x i , y j ) 0,

are imposed, then the iterative method for u 1 and u 2 will be as follows.

• Initialization:

( 0)

u 1 ( xi , y j ) =

0

if (xi , y j ) is an interior point,

g 1 (xi , y j ) if (xi , y j ) is a boundary point,

( 0)

u 2 ( xi , y j ) =

0

if (xi , y j ) is an interior point,

g 2 (xi , y j ) if (xi , y j ) is a boundary point.

• Step k + 1, k 0: Let u 1 (xi , y j ) and u 2 (xi , y j ) denote the average of u 1 and u 2 , respectively, for all neighbors of the points (xi , y j ). Then we iterate over all interior points by setting

uk1+1 (xi ,

y j ) = max

−λ+ h2 4

k

k

+ u 1 (xi , y j ) − u 2 (xi , y j ), 0 ,

and

uk2+1 (xi , y j ) = max

−λ− h2 4

+ u 2 k (xi , y j ) − u 1 k (xi , y j ), 0 .

2.2. Regularization method A major difficulty in solving problem (2) numerically is that the terms max(u , 0) and min(u , 0) are non-differentiable. In the case of the one-phase obstacle problem, the method of regularization is discussed in [8,9]. The idea of the regularization method is to approximate the non-differentiable terms by a sequence of differentiable terms. Convergence is obtained when the parameter ε tends to 0. One should note that when ε tends to 0 the conditioning of a regularized problem deteriorates. On the other hand, to obtain a more accurate approximation we have to choose ε small but when ε is too small the numerical solution of regularized problem cannot be computed accurately. Therefore one needs to know about a posteriori error estimates to compute error bounds of the approximated solution. If the estimated error is within the given error tolerance, the solution of the regularized problem is accepted as the exact solution but if the estimated error is large, then we need to use a smaller parameter ε . We consider a family of regularized equations

u ε = λ+ χ ε u ε − λ− χ ε −u ε , in Ω, uε = g

on ∂Ω,

where χ ε (t ) is a non-decreasing, smooth approximation of the Heaviside function such that

χ ε (t ) =

1 t ε, 0 t −ε .

A solution u ε can be obtained by minimizing the following functional I ε on K

I ε (v ) =

1 2

|∇ v |2 + λ+ φ ε ( v ) − λ− φ ε (− v ) dx,

Ω

t

where φ ε (t ) = −∞ χ ε (s) ds. By the maximum principle of Alexandrov (see [7]),

ε u

L ∞ (Ω)

g L ∞ (Ω) + C (n, Ω) max λ+ L ∞ , λ− L ∞ .


95

Consequently, applying the interior L p -estimates, we will get

ε u

W 2, p ( K )

C p , K , Ω, λ+ , λ− , g for any K Ω and 1 < p < ∞.

Thus the family {u ε } is uniformly bounded in W 2, p ( K ), and there exists a subsequence denoted in the same notation such that 2, p

uε u

weakly in W loc (Ω). 2, p

We have u ∈ W loc (Ω) for any 1 < p < ∞. In addition, 1, p

uε → u

strongly in W loc (Ω). 1, 2

1, 2

Since u ε − g ∈ W 0 (Ω) and W 0 (Ω) is a closed subspace of the Hilbert space W 1,2 (Ω), one gets 1,2

u − g ∈ W 0 (Ω). Applying Fatou’s lemma and the dominated convergence theorem,

I (u ) lim inf I ε u ε lim inf I ε ( v ) = I ( v ), 1, 2

for any v such that v − g ∈ W 0 (Ω). Thus u is the minimizer of the functional (2). Since the minimizer is unique, the 1, p whole sequence {u ε } converges to u in W loc (Ω). By the Sobolev embedding theorem, we have

1,α

uε → u

in C loc (Ω) as ε → 0.

Consequently, the locally uniform convergence implies that

u =

λ+

a.e. in {u > 0},

−λ−

a.e. in {u < 0}.

Since u ∈ W 2,2 (Ω) then u = 0 a.e. in the set {x ∈ Ω : u (x) = 0} ∩ {x ∈ Ω : ∇ u (x) = 0}. Furthermore, the set {x ∈ Ω : ∇ u (x) = 0} ∩ {x ∈ Ω : u (x) = 0} is a C 1 -surface and thus has measure zero, hence

u = 0 a.e. in {u = 0}. Therefore u satisfies (3). Different estimates for the solution of the regularized problem can be obtained. We consider approximating problems

u ε = f ε u ε in Ω, uε = g

on ∂Ω,

where f ε are smooth non-decreasing functions such that

f ε (t ) =

λ+

t ε,

−λ− t −ε .

By the maximum principle we have

inf min{ g , 0} u ε sup max{ g , 0}. ∂Ω

∂Ω

ε1 and ε2 if we choose ε = max{ε1 , ε2 }, then we obtain λ+ λ+ λ+ s − 2ε ε2 , s ε2 + 2ε , s ε1 , ε ε 2 1 f ( s − 2ε ) = = f ( s ) = − − − −λ −λ −λ s − 2ε −ε2 s −ε2 + 2ε s −ε1 .

Also for given

Therefore, u ε1 − 2ε is a subsolution of

ε2 -problem. Hence,

u ε1 − 2 ε f ε2 u ε1 − 2 ε . The monotonicity of f and the maximum principle imply

u ε1 − 2 ε u ε2 .

96


Changing the roles of u ε1 and u ε2 we obtain

u ε2 − 2 ε u ε1 . Hence,

−2ε u ε1 − u ε2 2ε . Therefore the estimate

supu ε1 − u ε2 2 max{ε1 , ε2 },

(7)

Ω

follows.1 Note that we have

u ε1 − u ε2 = f ε1 u ε1 − f ε2 u ε2 . Multiplying by (u ε1 − u ε2 ) and integrating by parts gives

ε ∇ u 1 − u ε2 2 dx =

Ω

ε u 2 − u ε1 f ε1 u ε1 − f ε2 u ε2 dx.

Ω

From the estimate (7) it follows that

ε ∇ u 1 − u ε2 22

L (Ω)

C · max{ε1 , ε2 } max λ+ , λ− .

In numerical experiments, the non-differential terms max(u , 0) and min(u , 0) can be approximated by many different differentiable sequences. In functional (2) set

max(u , 0) =

u + |u |

min(u , 0) =

and

2

u − |u | 2

.

The term |u | can be regularized by the following sequence: 1. φ1ε (u ) =

√

u2 + ε2 .

| u | ε +1 2. φ2ε (u ) = ε+1 . ⎧

3. φ3ε (u ) =

u ε,

u ⎪ ⎨

1 u2 ( 2 ε

+ ε ) |u | ε , ⎪ ⎩ −u u −ε . ⎧ u − ε /2 u ε , ⎪ ⎨ u2 4. φ4ε (u ) = |u | ε , ⎪ ⎩ 2ε ε −u − 2 u −ε . In order to obtain a posteriori error estimates of the approximate solution, we use the duality method by conjugate functions [6,9,12]. Let V and Q be two normed spaces, V ∗ and Q ∗ their dual spaces and let ·,· denote the duality pairing. Assume that there exists a continuous linear operator A from V to Q , A ∈ L ( V , Q ). The adjoint A∗ ∈ L ( Q ∗ , V ∗ ) of the operator A is defined through the relation

A∗ q∗ , v = q∗ , Av

∀ v ∈ V , q∗ ∈ Q ∗ .

Let J be a function from V × Q to R = R ∪ {−∞, +∞}. Consider the minimization problem

inf J ( v , Av ).

(8)

v ∈V

Define its dual by

sup − J ∗ A∗ q∗ , −q∗ ,

(9)

q∗ ∈ Q ∗

where the convex conjugate function of J is given by

J ∗ v ∗ , q∗ =

sup

v ∈ V , q∈ Q

v , v ∗ + q, q∗ − J ( v , q) ,

v ∗ ∈ V , q∗ ∈ Q ∗ .

The relation between (8) and (9) is stated in the following theorem (see [6]).

1

The estimate (2.4) is communicated to us by Arshak Petrosyan.


97

Theorem 2.1. Assume that V is a reflexive Banach space and Q is a normed vector space, and let A ∈ L ( V , Q ). Let J : V × Q → R be proper l.s.c. strictly convex such that 1. There exists v 0 ∈ V , such that J ( v 0 , Av 0 ) < ∞ and q → J ( v 0 , q) is continuous at Av 0 . 2. J ( v , Av ) → +∞, as v → ∞, v ∈ V . Then problem (8) has a solution u ∈ V also problem (9) has a solution q∗ ∈ Q ∗ , and

J (u , Au ) = − J ∗ A ∗ q∗ , −q∗ .

(10)

In the case that the function J is of a separated form, i.e.,

J ( v , q) = F ( v ) + G (q)

v ∈ V, q ∈ Q,

then the conjugate of J is

J ∗ v ∗ , q∗ = F ∗ v ∗ + G ∗ q∗ , where F ∗ and G ∗ are the conjugate functions of F and G, respectively. To calculate the conjugate function when the functional is defined by an integral, we use the following theorems which can be found in [6,9]. Theorem 2.2. Assume h : Ω × Rn → R is a Carathéodory function with h ∈ L 1 (Ω) and suppose

G (q) =

h x, q(x) dx. Ω

Then the conjugate function of G is

G ∗ q∗ =

h∗ x, q∗ (x) dx

∀q∗ ∈ Q ∗ ,

Ω

where

h∗ (x, y ) = sup y · ξ − h(x, ξ ) . ξ ∈Rn

Let u ∈ V be a solution of the minimization problem (8) and q∗ be defined as in (10). For any v ∈ V , define the energy difference

E D (u , v ) = J ( v , Av ) − J (u , Au ). Theorem 2.3. Suppose the assumptions in Theorem 2.1 are satisfied. Then

∀ v ∈ V , q∗ ∈ Q ∗ . E D (u , v ) J ( v , Av ) + J ∗ A ∗ q∗ , −q∗ Note that by the relations max(u , 0) =

I (v ) =

1 2

|∇ v |2 +

u +|u | 2

and min(u , 0) =

u −|u | , 2

we can rewrite functional (2) as

1 + 1 λ + λ− | v | + λ+ − λ− v dx. 2 2

Ω

We introduce the following

A v = (∇ v , v ),

F (v ) =

1

Ω 2 (λ

∞

∗

V ∗ = H 1 (Ω) ,

V = H 1 (Ω),

J ( v , Av ) = F ( v ) + G ( A v ), +

n

Q = Q ∗ = L 2 (Ω)

− λ− ) v dx v ∈ g + H 01 (Ω), otherwise,

× L 2 (Ω),

98


1

G (q) =

2

|q1 |2 +

1 + λ + λ− |q2 | dx, 2

Ω

where q = (q1 , q2 ), q1 ∈ ( L 2 (Ω))n , q2 ∈ L 2 (Ω). We use Theorem 2.3 to find a bound for u − u ε . We have

∗

F

∗ ∗

A q

=

+ − sup v ∈ H 1 (Ω) Ω (q∗1 · ∇ v + vq∗2 − ( λ −λ ) v ) dx 2 g

∞

= where

otherwise

Ω

(q∗

· ∇g +

1

gq∗

2

−(

λ+ −λ− 2

) g ) dx

∞

∗

if q∗

∈

Q ,

otherwise,

∗

v ∈ g + H 01 ,

Q = q ∈Q :

∗

∗

q1 · ∇ v + vq2 −

λ + − λ−

v dx = 0 ∀ v ∈

2

H 01

,

Ω

and

G

∗

∗ q

= sup q∈ Q

1 −q1 · q1 − q2 q2 − |q1 | − λ+ + λ− |q2 | dx = 2 2 ∗

1

∗

2

Ω

1 ∗ 2 Ω 2 |q1 | dx

∞

if |q∗2 | 12 (λ+ + λ− ), otherwise.

Denoting the constraint set

∗

∗

∗

Qc = q ∈ Q :

∗

∗

q1 · ∇ v + vq2 −

λ + − λ− 2

∗ 1 + − λ + λ a.e. in Ω , v dx = 0, q2 2

Ω

where the integral condition should hold for all v ∈ H 01 , the term J ∗ (A∗ q∗ , −q∗ ) can be written as

J

∗

∗ ∗

A q , −q

∗

1 ∗ 2 Ω ( 2 |q1 |

=

+ q∗1 · ∇ g + gq∗2 − ( λ

+ −λ−

2

) g ) dx q∗ ∈ Q c∗ ,

∞

otherwise.

Now consider the energy difference

ED u , u ε =

+ + − − u ε − |u | + λ − λ ∇ u ε 2 − 1 |∇ u |2 + λ + λ u ε − u dx.

1 2

2

2

2

Ω

We show that the following inequality holds:

1 ε ∇ u − u 22 ED u , u ε . L (Ω) 2 This is equivalent to show

|∇ u |2 + λ+ max(u , 0) − λ− min(u , 0) dx

Ω

∇ u · ∇ u ε + λ+ max u ε , 0 − λ− min u ε , 0 dx.

Ω

The difference between two integrals is

∇ u · ∇ u − u ε + λ+ max(u , 0) − λ− min(u , 0) − λ+ max u ε , 0 + λ− min u ε , 0 dx

Ω

=

u u ε − u + λ+ max(u , 0) − λ− min(u , 0) − λ+ max u ε , 0 + λ− min u ε , 0 dx

Ω

=

λ+ χ{u >0} − λ− χ{u 0} − χ{u ε >0} ) + λ− u ε (χ{u ε 0} ) + λ− u ε (χ{u ε F i− , ( j ,1) ( j) I2 = i ∈ I 2 : ( AU )i < F i+ , ( j ,2) j ( j ,0) ( j ,1) , I2 = I2 \ I2 ∪ I2 ( j ,0) ( j) ( j) I3 = i ∈ I 3 : ui 0 . I1

Define

⎧ j +1 ( j ,0) ( j ,0) ⎪ ⎪ ⎨ I1 = I1 ∪ I2 , j +1 ( j ,1) ( j ,2) ( j ,0) I2 = I1 ∪ I2 ∪ I3 , ⎪ ⎪ ⎩ j +1 j +1 j +1 I 3 = I \ ( I 1 ∪ I 2 ).

(27)

The algorithm stops when I n2 = I n2+1 and this condition is equivalent to

∀i ∈ I 2 : ( AU )i F i− and ( AU )i F i+ . Remark. Assume B r ⊆ supp (φi ). If we multiply u = −λ− by a basis function φi and integrate by parts we have

−

∇ u · ∇φi dx + Br

∂{u 0} − 8χ{u 0 to I 2 . Similarly we add half of the points on I 3 such that u i < 0 to I 2 . Fig. 7 illustrates the iterations in modified method. Example 4.5. Consider

u = 2χ{u >0} − χ{u 1 are generally hard to solve and are still under active research. The theory of IAEs appeared from early attempts by Gear in the 1990 that determined the difficulties of these equations. He introduced the “index reduction procedure” for IAEs system in [8] similar to that in [9] for DAEs in which if the process is terminated, then the index is determined. This means that under suitable conditions, there is a solution for the resulting regular system of integral equations. Since then, several authors have investigated the existence, uniqueness and numerical analysis of IAEs systems. Bulatov [3] in 1997, gave the existence and uniqueness results of solution for IAEs systems with convolutions kernels and defined the index in analogy to Gear’s approach. (See [4] for further details.) Kauthen [16] in 2000, applied the polynomial spline collocation method for a semi-explicit IAEs with index-1 and established global convergence as well as local superconvergence. Furthermore, Brunner [2] defined the index-1 tractable for a semi-explicit form of IAEs and investigated the existence of a unique solution for these types of systems.

*

Corresponding author. E-mail address: [email protected] (M. Hadizadeh).


132

M. Hadizadeh et al. / Applied Numerical Mathematics 61 (2011) 131–148

In this paper, we study the numerical solvability of a mixed system of Volterra integral equations of the first and second kind. More precisely, we consider the semi-explicit system of integral algebraic equations

y (t ) = f (t ) + (ν11 y )(t ) + (ν12 z)(t ),

(1.1)

0 = g (t ) + (ν21 y )(t ), where the linear Volterra integral operators

νkl are given by:

t (νkl ϕ )(t ) =

K kl (t , s)ϕ (s) ds,

t ∈ I = [0, T ]

(k, l = 1, 2)

0

and y : I → Rd1 , z : I → Rd2 , K 11 (·,·) ∈ L (Rd1 ), K 12 (·,·) ∈ L (Rd2 , Rd1 ), K 21 (·,·) ∈ L (Rd1 , Rd2 ) and L (·,·) is the linear transformation space. Here, the word “algebraic” assumes a wider meaning, in that it refers to the “non-differential” constraints forming part of the system in analogy to DAEs. In order to give an application of these models, consider the following heat equation with initial and mixed boundary conditions which represents a boundary reaction in diffusion of chemicals where α (t )u x represents the diffusive transport of materials to the boundary. Following [5, p. 79], for continuously differentiable function f and for continuous functions g , h, α , β , and γ , the solution of the problem

⎧ ut = u xx , 0 < x < 1, 0 < t , ⎪ ⎪ ⎪ ⎨ u (x, 0) = f (x), 0 < x < 1, ⎪ ut (0, t ) + α (t )u x (0, t ) + β(t )u (0, t ) = g (t ), ⎪ ⎪ ⎩ u x (1, t ) + γ (t )u (1, t ) = h(t ), 0 < t ,

(1.2)

0 < t,

has the representation

1 u (x, t ) =

θ(x − ξ, t ) − θ(x + ξ, t ) f (ξ ) dξ − 2

0

t

τ ∂θ (x, t − τ ) f (0) + φ1 (η) dη dτ ∂x

0

t +2

0

τ ∂θ (x − 1, t − τ ) f (1) + φ2 (η) dη dτ , ∂x

0

(1.3)

0

if and only if φ1 and φ2 are continuous functions that satisfy in the coupled system of Volterra integral equations

⎧ t t ⎪ ⎪ ⎪ ⎪ g (t ) = φ1 (t ) + 2α (t ) θ(ξ, t ) f (ξ ) dξ − 2α (t ) θ(0, t − τ )φ1 (τ ) dτ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ t t ⎪ ⎪ ⎪ ⎪ ⎪ + 2α (t ) θ(−1, t − τ )φ2 (τ ) dτ + β(t ) f (0) + β(t ) φ1 (τ ) dτ , ⎪ ⎪ ⎨ 0

0

t t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪h(t ) = 2 θ(1 + ξ, t ) f (ξ ) dξ − 2 θ(1, t − τ )φ1 (τ ) dτ ⎪ ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ t t ⎪ ⎪ ⎪ ⎪ + 2 θ(0, t − τ )φ2 (τ ) dτ + γ (t ) f (1) + γ (t ) φ2 (τ ) dτ , ⎪ ⎪ ⎩ 0

0

where θ(x, t ) is a well-known Theta function which is given in [5]. However, in some special cases (e.g. system of the form (1.1) as follows:

(1.4)

γ (t ) =

−2

t 0

θ(0,t −τ )φ2 (τ ) dτ t , u (1, t ) = 0), it is possible to reduce the system (1.4) to a 0 φ2 (τ ) dτ

f (1)+


133

⎧ t t ⎪ ⎪ ⎪ ⎪ ⎪ g ( t ) = φ ( t ) + 2 α ( t ) θ(ξ, t ) f (ξ ) d ξ − 2 α ( t ) θ(0, t − τ )φ1 (τ ) dτ 1 ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ t t ⎨ + 2α (t ) θ(−1, t − τ )φ2 (τ ) dτ + β(t ) f (0) + β(t ) φ1 (τ ) dτ , ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ t t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪h(t ) = 2 θ(1 + ξ, t ) f (ξ ) dξ − 2 θ(1, t − τ )φ1 (τ ) dτ . ⎪ ⎩ 0

0

The outline of this paper is as follows: Section 2 is devoted to the introduce of IAEs and its preliminary concepts. In Section 3, we investigate the existence of a unique solution for the semi-explicit IAEs system (1.1). The numerical analysis and the error estimation of the Jacobi collocation method in weighted L 2 norm are given in Sections 4 and 5. Finally, in Section 6 some numerical experiments are reported which confirm the theoretical results of the paper. 2. Preliminaries A general integral algebraic equations system takes the form:

t A (t ) X (t ) = G (t ) +

K t , s, X (s) ds,

(2.1)

0

where X : I → Rd , G : I → Rd , K : I × I × Rd → Rd are continuous and A (t ) ∈ L (Rd ) is singular matrix with continuous entries (rank(A) 1, det( A ) = 0). Note that since A is singular and the system has a structure similar to DAEs of index greater than zero, there is no guarantee that the above system is solvable. (See e.g. [8,9].) The semi-explicit linear version of (2.1) is as follows:

y (t ) = f (t ) + (ν11 y )(t ) + (ν12 z)(t ), 0 = g (t ) + (ν21 y )(t ) + (ν22 z)(t ),

(2.2)

where the Volterra integral operators νkl and the matrix kernels K kl (k, l = 1, 2), are defined as (1.1). The notion of index is crucial for the behavior of solutions of IAEs and it must contain the properties of feasible numerical solutions. There exist several different definitions of index for IAEs in [2,4,8] which are often closely related. These definitions are conceptually based on the “index reduction procedure” i.e. a differentiation process of the algebraic constraints which yields a system of regular VIEs and the “tractability index” that is the algebraic constraints which are locally solvable for the algebraic components of the IAEs solution. The following definition regarding to index-1 tractable for the semi-explicit system (2.2) was given by Brunner which is analogous to that defined for DAEs by März [17]: Definition 1. (See [2].) The semi-explicit IAEs (2.2), is said to be index-1 tractable if the first-kind VIE:

t K 22 (t , s) w (s) ds = h(t ),

t ∈ I,

0

is uniquely solvable in C ( I ), whenever h ∈ C 1 ( I ) and h(0) = 0. This means that IAEs system (2.2) is index-1 tractable if its second equation (e.g. the linear first kind VIE) is uniquely solvable for z ∈ C ( I ) as an algebraic component. Following [2], assume that the functions f , g , K i j (i , j = 1, 2) are sufficiently smooth and for all t ∈ I , |det K 22 (t , t )| k0 > 0, where k0 is a positive constant and g (0) = 0. Using the index reduction procedure [8] and Theorem (2.1.8) in [2, p. 64], we can show that the linear VIE possesses a unique solution on I and so the solvability and regularity of its solution have been obtained. 3. IAEs of index-2 As a consequence of the previous section, it is possible to define the index-2 for the semi-explicit IAEs system (1.1) based on minimal regularity conditions and investigate the existence of a unique solution. Considering the system (1.1) and differentiating from its second equation with respect to t, we obtain:

t

0 = g (t ) + K 21 (t , t ) y (t ) + 0

∂ K 21 (t , s) y (s) ds. ∂t

134


Now, by replacing y (t ) from the first equation of (1.1):

t 0 = g 1 (t ) +

t ∂ K 21 (t , s) K 21 (t , t ) K 11 (t , s) + y (s) ds + K 21 (t , t ) K 12 (t , s) z(s) ds, ∂t

0

(3.1)

0

g (t ) + K 21 (t , t ) f (t ).

where g 1 (t ) = Obviously, (3.1) together with the first equation of (1.1) are the same as the index-1 semi-explicit IAEs system (2.2). In accordance with the terminology introduced by Gear [8], we will refer to the system (1.1) as a semi-explicit IAEs of index-2. However it has to be pointed out that this reduction to IAEs of index-1 (2.2) is not practical from a numerical point of view. We can focus again on the tractability index-2 for the IAEs (1.1), in other view point. The IAEs (1.1) is called index-2 tractable, if the algebraic constraints are locally solvable for the algebraic components z i.e. by replacing y (t ) from the first equation of (1.1) into its second equation we have

t s 0 = g (t ) + f 1 (t ) +

K 21 (t , s) K 12 (s, x) z(x) dx ds,

(3.2)

0 0

where

t f 1 (t ) =

t s K 21 (t , s) f (s) ds +

0

K 21 (t , s) K 11 (s, x) y (x) dx ds. 0 0

Exchanging the two integrals in Eq. (3.2) with respect to s, then x, we obtain:

t 0 = g (t ) + f 1 (t ) +

K (t , x) z(x) dx,

(3.3)

0

t

where K (t , x) = x K 21 (t , s) K 12 (s, x) ds. One of our result in this section is given in the following definition which gives the tractability index-2 for the semiexplicit IAEs system (1.1) using (3.3) and the extension of Definition 1: Definition 2. The semi-explicit IAEs (1.1), is said to be index-2 tractable if the first-kind VIE:

t K (t , x) w (x) dx = h(t ),

t ∈ I,

0

is uniquely solvable in C ( I ) whenever h ∈ C 1 ( I ) and h(0) = 0. Now, we will make use of the result of Brunner [2], which gives the existence and uniqueness results for IAEs (1.1). So, ∂ K 21 (t ,s) are sufficiently smooth the VIE (3.1) is uniquely solvable for z ∈ C ( I ), if the given functions f , g , K i j (i , j = 1, 2) and ∂t and

g 1 (0) = 0,

and

det K 21 (t , t ) K 12 (t , t ) k0 > 0,

∀t ∈ I

where k0 is a positive constant. The following theorem gives the relevant conditions concerning K 1l (l = 1, 2), K 21 , f and g for the investigation of the unique solution of IAEs (1.1) analogous to Theorem (8.1.5) in [2, p. 472]: Theorem 1. Let ν 0 and assume that 1. K 1l ∈ C ν ( D ) for l = 1, 2 and D = I × I , 2. K 21 ∈ C ν +1 ( D ) and |det( K 21 (t , t ) K 12 (t , t ))| k0 > 0, 3. f ∈ C ν ( D ), g ∈ C ν +1 ( D ) and g 1 (0) = 0, then the IAEs (1.1) possesses a unique solution y , z ∈ C ν ( I ). It is also possible to obtain the desired information on the regularity of the solutions in accordance to Theorem (8.1.6) of [2, p. 473]:


Theorem 2. Assume that the hypotheses given in Theorem 1 hold with unique solution of the IAEs (1.1) is given by the representation

t y (t ) = f (t ) +

ν 0, then for K 21 (t , t ) K 11 (t , s) +

135

∂ K 21 (t ,s) ∂t

= 0 on D, the

t R 11 (t , s) f (s) ds + k12 (t ) g 1 (t ) +

0

Q 12 (t , s) g 1 (s) ds, 0

z(t ) = k21 (t ) g 1 (t ) − k−1 g 1 (t ) +

t Q 22 (t , s) g 1 (s) ds, 0

where

k21 = −

R 22 (t , t )

∂ Q 22 = ∂s

k

k12 = −

,

R 22 (t , s)

k

2R 11 (t , t ) − R 22 (t , t ) k

,

∂ 2R 11 (t , s) − R 22 (t , s) , Q 12 = ∂s k

,

and R 11 (t , s) denotes the resolvent kernels associated with the kernel K 11 (t , s) in (1.1). Proof. The proof is mainly based on the proof of the corresponding theorem in [2] and we refrain from going into details. It is sufficient we take k = K 21 (t , t ) K 12 (t , t ) and H 22 (t , s) = −k−1 ∂∂t ( K 21 (t , t ) K 12 (t , s)). Also, R 22 (t , s) is the resolvent kernels associated with the kernel H 22 (t , s). Under these conditions, the results of the theorem will be obtained. 2 For considering the stability issue of the problem, we use the results of Gear [8], who has defined the index of IAEs by considering the effect of perturbations of the equations on the solutions and obtained the relationship between the differential and the perturbation index in which they are identical. Theorem 2 reveals us the stability issue in the sense of perturbation index as follows: The IAEs system (1.1) can be written as compact form

BY = G, where Y = ( y , z) T , G = ( f , g ) T and B =

I −ν11 −ν12 −ν21 0

(3.4)

.

The corresponding perturbed system may be considered as

BY˜ = G + δ, with δ = (δ1 , δ2 ) T . Differentiating the second equation of the perturbed system with respect to t and substituting y˜ from the first equation, yields

B1 Y˜ = G1 + δ , t I −ν −ν where B1 = −V11 −V12 , Vi = 0 Ki (t , s) ds, i = 1, 2, with 1

2

K1 (t , s) = K 21 (t , t ) K 11 (t , s) + and

∂ K 21 (t , s) , ∂t

T

δ = δ1 , K 21 (t , t )δ1 + δ2 ,

K2 (t , s) = K 21 (t , t ) K 12 (t , s)

G1 = ( f , g 1 ) T

with g 1 (t ) = g (t ) + K 21 (t , t ) f (t ).

Also, differentiating the second equation of (3.5) with respect to t, gives

B2 Y˜ = G2 + δ , where

B2 =

Vi =

t 0

I − ν11

−ν12

−V1

− K 21 (t , t ) K 12 (t , t ) − V2

∂ Ki (t , s) ds, ∂t

G2 = ( f , g 2 ) T ,

, T

δ = δ1 , K 21 (t , t )δ1 + K 21 (t , t )δ1 + δ2 ,

(3.5)

136


and

∂ K 21 (t , t ) g 2 (t ) = g 1 (t ) + K 21 (t , t ) K 11 (t , t ) + y˜ (t ). ∂t

Furthermore, for the original system (3.4) we have

B2 Y = G2 . 1 Applying Theorem 2, we can deduce that the operator B2 is a linear, bounded and bijective such that B− 2 exists and is bounded. Then it follows from the above relations

1 , Y − Y˜ C B− 2 δ1 + δ1 + δ2 where C is a constant independent of t. This yields the stability estimate of the IAEs system (1.1) in the sense of the perturbation index. 4. The Jacobi collocation scheme This section is devoted to applying the Jacobi collocation method to numerically solve the IAEs system of index-2. To do so, we consider a collocation method including the matrix–vector multiplication representation of the equations. Let P N (Λ) be the space of all polynomials with degree not exceeding N on Λ, where Λ stands for the open interval (−1, 1) and w α ,β (x) = (1 − x)α (1 + x)β , for α , β > −1 denotes a weight function in the usual sense. It is well known, the set α ,β 2 2 of Jacobi polynomials { J N }∞ N =0 forms a complete L w α ,β (Λ) orthogonal system, where L w α ,β (Λ) is the space of functions f : [−1, 1] → R with f 22

L α ,β w

< ∞, and

1 f 2L 2 α ,β

= f , f L2

w α ,β

w

| f (x)|2 w α ,β (x) dx.

= −1

So, any function u (x) in the space L 2w α,β (Λ) admits the expansion u (x) =

u˜ N =

1

1

∞

Ñ N =0 u

α ,β

JN

with

α ,β

u (ξ ) J N (ξ ) w α ,β (ξ ) dξ.

α ,β J 2

L 2 α ,β −1 w

k

Now, let H m denotes the Sobolev space of all functions u (x) on Λ such that u (x) and all its weak derivatives up to w α ,β

order m are in L 2w α,β (Λ), with the norm and the semi-norm as

u (x)2 m

H α ,β w

= (Λ)

|u | H m, N

w α ,β

(Λ)

2 m k ∂ u ( x ) 2 ∂ xk m

=

,

L α ,β (Λ) w

k =0

( j ) 2 u 2

j =min(m, N +1)

1/2

L α ,β (Λ) w

(4.1)

.

Let Π N be the orthogonal projective operator from L 2w α,β (Λ) on to P N (Λ). It means that for any function ϕ in L 2w α,β (Λ), Π N ϕ belongs to P N (Λ) and satisfies

1 (ϕ − Π N ϕ )(ξ )ψ N (ξ ) w α ,β (ξ ) dξ = 0.

∀ψ N ∈ P N (Λ), −1

α ,β

For any u (x) ∈ C [−1, 1], we can define the projection I N Jacobi polynomials (see e.g. [19]) α ,β

I N u (x) =

N i =0

α ,β

u˜ i J i

(x) =

N

u (xi ) L i (x),

i =0

where the Lagrange polynomials L i (x) take the form

as a Lagrange interpolating polynomial associated with the


L i (x) = w i

137

N 1 α ,β α ,β J (xi ) J k (x) (i = 0, 1, . . . , N ) γ˜k k k =0

and the collocation points {xi }iN=0 represent the Jacobi–Gauss quadrature points used to compute the discrete expansion coefficients u˜ k , N 1

u˜ k =

γ˜k

α ,β

u ( xi ) J k

( xi ) w i ,

(4.2)

i =0

where γ˜k and w i were given in [11, p. 231]. Let us now turn our attention toward the application of Jacobi collocation method for the following linear semi-explicit IAEs system:

t A (t ) X (t ) = G (t ) +

t ∈ [0, T ]

K (t , s) X (s) ds,

(4.3)

0

where X : [0, T ] → Rd , G : [0, T ] → Rd , K ∈ L (Rd ), are continuous and A (t ) = diag(Id1 , Od2 ) ∈ L (Rd ) is a singular block matrix and X (t ) = {xi (t )}di=1 , K (t , s) = {ki j (t , s)}di, j =1 . To use the theory of orthogonal Jacobi polynomials, we consider the change of variables:

s=

T 2

(η + 1),

−1 η τ ,

t=

T

(τ + 1),

2

−1 τ 1

(4.4)

to rewrite the problem (4.3) as follows

A (τ ) X (τ ) = G (τ ) +

τ

K (τ , η) X (η) dη,

τ ∈ [−1, 1]

(4.5)

−1

where

T G (τ ) = G (τ + 1) ,

T X (τ ) = X (τ + 1) ,

2

T T T K (τ , η) = K (τ + 1), (η + 1) .

2

2

2

2

Approximating K (τm , η) with Jacobi polynomials gives

d K N (τm , η) = kˆ i j (τm , η) i , j =1 =

N

d (kˆ i j )mk J k (η) α ,β

k =0

i , j =1

d

= (kˆ ij )m i , j =1 ⊗ VW , α ,β

where V is a lower triangular coefficient matrix of Jacobi polynomials with { J i

W = 1, η, . . . , η N

(kˆ i j )mk =

T

(kˆ ij )m = (kˆ i j )m0 , . . . , (kˆ i j )mN ,

, 1

1 α ,β

J k 22

kˆ i j (τm , ξ ) J k

α ,β

(ξ ) w α ,β (ξ ) dξ

(4.6)

(η)}iN=0 = VW , and

(k = 0, 1, . . . , N )

L α ,β −1 w

and ⊗ is a Kronecker product. Consequently, we seek a solution α ,β

IN

X (η) = X N (η) =

N k =0

α ,β

where V = JV with J = { γ1˜ w k J j j

d α ,β

=

(˜xi )k J k (η) i =1

N

d (ˆxi )(τk ) Lk (η)

k =0

i =1

= {ˆxi }di=1 ⊗ V W ,

(4.7)

(τk )}kN, j =0 , xˆ i = ((ˆxi )(τ0 ), (ˆxi )(τ1 ), . . . , (ˆxi )(τ N )), {τk }kN=0 are the Gauss–Jacobi collocation

α ,β

points (i.e. the zeros of J N +1 ), the coefficients {(˜xi )k }kN=0 are also given by (4.2) and { L k }kN=0 are the interpolating Lagrange polynomials. N Inserting the collocation points {τm }m =0 in (4.5), we obtain the collocation equation as

A (τm ) X (τm ) = G (τm ) +

τm

−1

K N (τm , η) X N (η) dη (m = 0, 1, . . . , N )

(4.8)

138


and using (4.6), (4.7) and (4.8), we get

A (τ

xi }di=1 m ){ˆ

τm

= G (τm ) +

(kˆ ij )m

d i , j =1

⊗ VW {ˆxi }di=1 ⊗ V W dη (m = 0, 1, . . . , N ).

(4.9)

−1

In this position, we give the following lemma, that will become instrumental in establishing the matrix vector multipliK N (τm , η) X N (η) in (4.8): cation representation of the product Lemma 1. Let P (η) =

N

i =0

N

p i η i and Q (η) =

j =0 q j

η j be two given polynomials, then

P (η) Q (η) = P(Q ⊗ M)W, where P = ( p 0 , p 1 , . . . , p N ), Q = (q0 , q1 , . . . , q N ), W = (1, η, η2 , . . . , η2N ) T , and M is a block sparse matrix of the form M = (k)

(MN )kN=0 with k

0 ... 0 1

j −k

⎞ ... 0 0 ... 0 . ⎜ ⎟ ⎜ 0 . . . 0 0 1 . . . .. 0 . . . 0 ⎟ (k) ⎟ Mj = ⎜ . ⎜. . . . .⎟ .. ⎝ .. . . . .. .. . 0 .. . . . .. ⎠ 0 . . . 0 0 . . . 0 1 0 . . . 0 ( N +1)( N +1+ j ) ⎛

0

N +1

Proof. Firstly, we will show that

ηm

N

(m)

p i ηi = PMm W m ,

i =0

where W m = (1, η, η2 , . . . , η N +m ) T . The validity of the equation for m = 0 is obvious. Now we proceed by induction. So, we assume the validity of the equation for m = k and transit to m = k + 1 as follows

η k +1

N

p i η i = ηk

i =0

N T (k) (k) ( p i η)ηi = { p i η}iN=0 Mk W k = PMk η, η2 , . . . , ηk+ N +1 . i =0

With some simple manipulations we conclude (k)

Mk

η , η 2 , . . . , η k + N +1

T

(k+1)

= Mk+1

1, η , . . . , η k + N +1

T

From these relations, we can write

η k +1

N i =0

(k+1)

p i ηi = PMk+1 W k+1 ,

and this proves the identity for m = k + 1. Now, consider the multiplication

P (η) Q (η) =

N

qj

j =0

N

pi η

i

ηj =

i =0

N

( j)

q j PM j W j .

j =0

(k)

Due to the structure of M = (M N )kN=0 , we have ( j)

( j)

MN W N = M j W j

( j = 0, 1, . . . , N ).

This leads to the following relation

P (η) Q (η) = P

N j =0

( j)

q j MN

W N = P(Q ⊗ M)W.

2

(k+1)

= Mk+1 W k+1 .


139

Algorithm 1. The construction of the Jacobi collocation method including the matrix–vector multiplication representation of the equations α ,β

(η), (i = 0, 1, . . . , N ) and compute the nonsingular co-

Step 1. Choose N, form the orthogonal bases J i efficient matrices V and V as

α ,β Ji

(η )

N i =0

= VW ,

and

V = JV α ,β

(τk )}kN, j =0 and W = (1, η, . . . , η N ) T . Compute the approximations of K (τm , η) and X (η) using τ Jacobi polynomials from (4.6) and (4.7). Compute the matrix M from Lemma 1 and take Wm = −m1 W dη with W = (1, η, η2 , . . . , η2N ) T . Solve the linear system (4.11) and obtain the entries of the vector solution {ˆxi }di=1 . Set X N (η) = {ˆxi }di=1 ⊗ (V W ).

where J = { γ1˜ w k J j j

Step 2. Step 3. Step 4. Step 5.

By the mentioned lemma and relations (4.6) and (4.7), the matrix–vector multiplication representation of the product

K N (τm , η) X N (η) can be obtained as follows:

K N (τm , η) X N (η) = = = =

d

d kˆ i j (τm , η)ˆx j (η)

j =1

i =1

d

j =1

(kˆ ij )m VW xˆ j V W

d

N

j =1

l =0

(kˆ ij )m V l η

l

d i =1

N

xˆ j V

d

l

η

l

l =0

i =1

d

d

(kˆ ij )m V xˆ j V ⊗ M W j =1

(4.10)

, i =1

(l)

where M = (M N )lN=0 is a block matrix which is defined in Lemma 1. Inserting (4.10) in (4.9), we obtain

A (τm ){ˆxi }di=1 = G (τm ) +

τm

j =1

= G (τm ) +

d

(kˆ ij )m V xˆ j V ⊗ M W dη −1

d

(kˆ ij )m V xˆ j V ⊗ M Wm j =1

d i =1

d (m = 0, 1, . . . , N )

(4.11)

i =1

τ

where Wm = −m1 W dη . Finally, we end up with a system of equations whose solution gives the unknown vectors {ˆxi }di=1 . The Algorithm 1 summarizes the proposed Jacobi collocation method. 5. Convergence analysis In this section, we will try to provide an error analysis which theoretically justifies the spectral rate of convergence of the proposed method. In order to describe the key ideas without having to resort to complex notation involving Kronecker products of matrices and vectors, we will consider the index-2 IAEs system (4.3) with d1 = d2 = 1. Our strategy is mainly based on the ideas in [20], together with the following auxiliary lemmas from [6] and [7]: Lemma 2. (See [6].) Let φ ∈ P N where P N denote the space of all polynomials of degree N, then for any integer r 1 and 0 p ∞, there exists a positive constant C independent of N such that

(r ) φ

p

L α ,β (−1,1) w

C N 2r φL p

w α ,β

(−1,1) .

140


N

α ,β

α ,β

Lemma 3. (See [6].) Assume that u ∈ H m (Λ) and P N u = k=0 uˆ k J k is the truncated orthogonal Jacobi series of u and I N u is the interpolation of u at any of the three families of Jacobi Gauss points (Gauss or Gauss–Radau or Gauss–Lobatto). Then for Λ = (−1, 1), the following estimates hold:

u − P N u L 2 u − I α ,β u N

C N −m |u | H m,N (Λ) ,

(Λ)

w α ,β

C N −m |u | H m,N

L 2 α ,β (Λ)

w α ,β

w

α ,β u − I u N

L 2 α ,β (Λ) w

(5.1)

,

(Λ)

C N 1−m |u | H m,N

w α ,β

(Λ)

(5.2)

.

We are now ready to prove the following main theorem which gives the convergence properties of the presented scheme: A (τ ) = diag(1, 0), G (τ ) = ( ˆf (τ ), gˆ (τ )) T and Theorem 3. Consider the IAEs of index-2 (4.3) and its transformed system (4.5) with T T ˆ ˆ ˆ ˆ X (t ) = ( y , z) . Assume that the hypotheses given in Theorem 1 hold with ν 0. If X N = ( y N , z N ) is an approximate solution of the proposed Jacobi collocation scheme with the Gauss–Jacobi collocation points, then the Jacobi collocation approximation errors yˆ − yˆ N and zˆ − zˆ N for m 1 satisfy:

yˆ − yˆ N L 2

w α ,β

(Λ)

5

O( N 2 +γ −m ),

ˆz − zˆ N L 2

w α ,β

(Λ)

O( N 2−m log N ), −1 < α , β − 12 , otherwise

O( N 4−2m (log N )2 ), −1 < α , β − 12 , O( N 5+2γ −2m ),

otherwise

where yˆ , zˆ ∈ H m (Λ), Λ = (−1, 1) and γ = max(α , β). Proof. The strategy we shall follow is to determine the error estimations for yˆ − yˆ N and then ˆz − zˆ N . The spectral collocation solution X N was considered as an interpolating polynomial of degree N on the interval (−1, 1). It is defined through the collocation equation (4.11)

yˆ (τi ) = fˆ (τi ) + D 1 (τi ) + D 2 (τi )

( i = 1, 2, . . . , N )

(5.3)

where

D 1 (τi ) = (kˆ 11 )m V yˆ V ⊗ M Wm ,

D 2 (τi ) = (kˆ 12 )m V zˆ V ⊗ M Wm .

We rewrite Eq. (5.3) in the form:

yˆ (τi ) = fˆ (τi ) +

τi

K 11 (τi , η) yˆ N (η) dη +

−1

τi

K 12 (τi , η)ˆz N (η) dη + q1 (τi ) + q2 (τi ),

(5.4)

−1

such that

τi q1 (τi ) = D 1 (τi ) −

K 11 (τi , η) yˆ N (η) dη,

−1

τi q2 (τi ) = D 2 (τi ) −

K 12 (τi , η)ˆz N (η) dη,

−1

and yˆ N , zˆ N are the approximate solutions of yˆ , zˆ which is approximated by k-th Lagrange interpolating polynomials L k : α ,β

IN

yˆ (τ ) = yˆ N (τ ) =

n

yˆ k L k (τ ),

α ,β

IN

k =0

where yˆ k = yˆ (τk ) and zˆ k = zˆ (τk ). It follows that

zˆ (τ ) = zˆ N (τ ) =

n k =0

zˆ k L k (τ ),


yˆ (τi ) = fˆ (τi ) +

τi

K 11 (τi , η)e (η) dη +

−1

τi +

τi

141

K 11 (τi , η) yˆ (η) dη

−1

K 12 (τi , η)ε (η) dη +

−1

τi

K 12 (τi , η)ˆz(η) dη + q1 (τi ) + q2 (τi ),

(5.5)

−1

where e (s) = yˆ N (s) − yˆ (s) and ε (s) = zˆ N (s) − zˆ (s). Multiplying the j-th equation of (5.5) by L j (τ ) and summing up over j from 0 to N, we get α ,β ˆ

α ,β f (τ ) + I N

yˆ N (τ ) = I N

τ

K 11 (τ , η) yˆ (η) dη

−1

α ,β

α ,β α ,β q1 (τ ) + I N q2 (τ ) + I N

+ IN

α ,β

τ

+ IN

τ

K 11 (τ , η)e (η) dη

−1

α ,β K 12 (τ , η)ˆz(η) dη + I N

−1

τ

K 12 (τ , η)ε (η) dη .

(5.6)

−1

From the first equation of (4.5), we may write α ,β

τ

IN

K 11 (τ , η) yˆ (η) dη +

−1

τ

α ,β K 12 (τ , η)ˆz(η) dη = I N yˆ (τ ) − fˆ (τ ) ,

−1

and replacing the above relation in (5.6), we get

τ 0=

K 11 (τ , η)e (η) dη +

−1

τ

α ,β α ,β K 12 (τ , η)ε (η) dη + I N q1 (τ ) + I N q2 (τ ) + P 1 + P 2 ,

(5.7)

−1

where α ,β

τ

P1 = IN

−1

α ,β

τ

P2 = IN

τ

K 11 (τ , η)e (η) dη −

K 11 (τ , η)e (η) dη,

−1

K 12 (τ , η)ε (η) dη −

−1

τ

K 12 (τ , η)ε (η) dη.

−1

Differentiating from (5.7) respect to

τ , yields:

− K 11 (τ , τ )e (τ ) − K 12 (τ , τ )ε (τ ) =

τ τ ∂ K 11 (τ , η) ∂ K 12 (τ , η) e (η) dη + ε(η) dη ∂τ ∂τ

−1

α ,β

+ IN

−1

α ,β q1 (τ ) + I N q2 (τ ) + P 1 + P 2 .

(5.8)

In order to investigate the convergence of the second equation of (4.5), we apply the vector–matrix multiplication representation form (4.11) for this equation. So, we have

0 = gˆ (τi ) + D 3 (τi ) (i = 1, 2, . . . , N )

(5.9)

where D 3 (τi ) can be written as:

D 3 (τi ) = (kˆ 21 )m V yˆ V ⊗ M Wm . Using a similar procedure as outlined in the first part, Eq. (5.9) give rise

τ 0= −1

α ,β K 21 (τ , η)e (η) dη + I N q3 (τ ) + P 3 ,

(5.10)

142


where P 3 and q3 (τi ) are similarly defined as: α ,β

τ

K 21 (τ , η)e (η) dη −

P3 = IN

−1

τ

K 21 (τ , η)e (η) dη,

−1

τi

K 21 (τi , η) yˆ N (η) dη.

q3 (τi ) = D 3 (τi ) − −1

Also, differentiating from (5.10) respect to

− K 21 (τ , τ )e (τ ) =

τ , gives:

τ ∂ K 21 (τ , η) α ,β e (η) dη + I N q3 (τ ) + P 3 . ∂τ

(5.11)

−1

Now, let E(τ ) = (e (τ ), ε (τ )) T , then Eqs. (5.8) and (5.11), can be rewritten in matrix notation

τ HE(τ ) =

$ K(τ , η)E(η) dη + D,

(5.12)

−1

with

H=

and

− K 11 (τ , τ ) − K 12 (τ , τ ) K 21 (τ , τ ) 0 −

α ,β IN

D=

α ,β

(q1 (τ )) + I N α ,β

IN

,

$ K(τ , η) =

∂ K 11 (τ ,η) ∂τ ∂ K 21 (τ ,η) ∂τ

∂ K 12 (τ ,η) ∂τ

,

0

(q2 (τ )) + P 1 + P 2

.

(q3 (τ )) + P 3

Due to the assumptions of Theorem 1, we have

K 21 (t , t ) K 12 (t , t ) > 0,

∀t ∈ I ,

this shows that H is an invertible matrix with the inverse of the form:

H

−1

=

−1 − K 21 (τ , η) . −1 −1 −1 K 21 (τ , η) K 12 (τ , η)( K 11 (τ , η) K 21 (τ , η)) − 0

Multiplying (5.12) by H−1 and using the Gronwall’s inequality (see e.g. Lemma 3.4 from [20]), we have

EL 2

w α ,β

(Λ)

C FL 2

w α ,β

(5.13)

(Λ) ,

where F = H−1 D. Then it follows from Lemma 2:

α ,β I q1 (τ ) N

L 2 α ,β (Λ) w

α ,β C N 2 I N q1 (τ ) L 2

w α ,β

(Λ)

(5.14)

,

indeed

α ,β I q1 (τ ) N

L 2 α ,β w

n q1 (τi ) max L i (τ ). max (Λ) 0 i N

τ ∈(Λ)

(5.15)

i =0

Applying the Cauchy–Schwarz inequality (see e.g. [6]), we can write:

τi ( |q1 (τi )| = K 11 ) N (τi , η) − K 11 (τi , η) yˆ N (η) dη −1

( K 11 ) N (τi , η) − K 11 (τi , η) L 2

w α ,β

and using (5.1) from Lemma 3, we have

(Λ)

yˆ N L 2

w α ,β

(Λ) ,


q1 (τi ) C N −m K 11 (τi , η)

m, N

H α ,β (Λ) w

yˆ N L 2

w α ,β

143

(5.16)

(Λ) .

In this position, we will make use of the result of Chen and Tang [7] and also [18], which gives the Lebesgue constant for the Lagrange interpolating polynomials associated with the nodes of the Jacobi polynomials. Actually, as stated in Lemma 3.4 of [7], the following relation for (5.15) holds n L i (η) = max

η∈[−1,1]

1

O( N γ + 2 ),

i =0

so, we have

α ,β I q1 (τ ) N

L 2 α ,β (Λ) w

−1 < α , β − 12 ,

O(log N ),

γ = max(α , β), otherwise

C N −m log N Θ11 , CN

1 2 +γ −m

Θ11 ,

−1 < α , β − 12 , otherwise

where Θ11 = max0i N | K 11 (τi , η)| H m,N (Λ) yˆ N L 2 (Λ) and w α ,β w α ,β Now, by using this relation for (5.14), we have

α ,β I q1 (τ ) N

L 2 α ,β w

(Λ)

and similarly

α ,β I q2 (τ ) N

α ,β I q3 (τ ) N

L 2 α ,β (Λ) w

L 2 α ,β (Λ) w

5

otherwise

C N 2−m log N Θ12 , −1 < α , β − 12 , 5

C N 2−m log N Θ11 , −1 < α , β − 12 , C N 2 +γ −m Θ11 ,

γ = max(α , β).

C N 2 +γ −m Θ12 , CN

2−m

otherwise

log N Θ21 , −1 < α , β − 12 ,

5

C N 2 +γ −m Θ21 ,

otherwise

where

Θ12 = max K 12 (τi , η) H m,N (Λ) ˆz N L 2 (Λ) , 0 i N w α ,β w α ,β Θ21 = max K 21 (τi , η) m,N yˆ N 2 . L α ,β (Λ) w

H α ,β (Λ) w

0 i N

Indeed, using (5.2) and (4.1) for m = 1 and then applying Hardy’s inequality (see e.g. Lemma 3.7 from [7]), we can write

P 1

τ ∂( K 11 (τ , η)) C K ( τ , τ ) e ( τ ) + e ( η ) d η 2 11 L α ,β (Λ) ∂τ w −1

L 2 α ,β (Λ) w

τ ∂( K 11 (τ , η)) C K 11 (τ , τ ) L 2 (Λ) e (τ ) L 2 (Λ) + e (η) dη ∂τ w α ,β w α ,β

C K 11 (τ , τ )

L 2 α ,β (Λ) w

C K 11 (τ , τ ) L 2

w

α ,β (Λ)

−1

C N −m | yˆ | H m,N (Λ) w α ,β

+ C e (τ )

+ C C N −m | yˆ | H m,N w

α ,β (Λ)

L 2 α ,β (Λ) w

L 2 α ,β (Λ) w

,

thus

P

1 L2 (Λ) w α ,β

C N −m K 11 (τ , τ ) L 2

w

α ,β (Λ)

+ C | yˆ | H m,N

w α ,β

(Λ)

C N −m | yˆ | H m,N

w α ,β

(Λ)

.

Similarly

P

2 L2 (Λ) w α ,β

P

3 L2 (Λ) w α ,β

C N −m K 12 (τ , τ ) L 2 (Λ) + C |ˆz| H m,N (Λ) C N −m |ˆz| H m,N (Λ) , w α ,β w α ,β w α ,β

C N −m K 21 (τ , τ ) 2 + C | yˆ | m,N C N −m | yˆ | m,N . L α ,β (Λ) w

H α ,β (Λ) w

The above estimates together with (5.13), yield

e L 2

w α ,β

ˆ − yˆ N L 2 (Λ) = y

w α ,β

(Λ) C N

−m

| yˆ | H m,N

w α ,β

+ (Λ)

H α ,β (Λ) w

C N 2−m log N Θ21 , −1 < α , β − 12 , 5

C N 2 +γ −m Θ21 ,

otherwise

(5.17)

144


Table 1 L 2w α,β errors for Example 1. N

x1 − u L 2

x2 − v L 2

x3 − w L 2

4 6 8 10 12 14

1.79 × 10−4

9.86 × 10−5

3.72 × 10−3 1.30 × 10−4 4.61 × 10−6 1.59 × 10−7 5.38 × 10−9 1.74 × 10−10

w α ,β

w α ,β

3.12 × 10−6 7.66 × 10−8 1.99 × 10−9 5.33 × 10−11 1.37 × 10−12

1.84 × 10−6 4.67 × 10−8 1.46 × 10−9 3.01 × 10−11 7.73 × 10−13

w α ,β

Also Θ11 can be written as:

Θ11 = max | K 11 (τi , η)| H m,N 0 i N

w α ,β

(Λ)

yˆ N L 2

w α ,β

(Λ)

K ( yˆ L 2ω (Λ) + e L 2ω (Λ) ),

where K = max0i N | K 11 (τi , η)| H m,N (Λ) . w α ,β Finally, the above estimates together with (5.13) and (5.17) give

ε L 2

w α ,β

(Λ)

= ˆz − zˆ N L 2 (Λ) C N −m | yˆ | H m,N (Λ) + |ˆz| H m,N (Λ) w α ,β w α ,β w α ,β ⎧ 2−m log N ( K ( yˆ L 2 (Λ) + e L 2 (Λ) ) + Θ12 + Θ21 ), −1 < α , β − 12 , ⎨CN w α ,β w α ,β + ⎩ C N 52 +γ −m ( K ( yˆ 2 otherwise L (Λ) + e L 2 (Λ) ) + Θ12 + Θ21 ), w α ,β

w α ,β

which leads to the estimate stated of the theorem, provided that N is sufficiently large and C is a constant independent of N. 2 6. Numerical results and discussions To incorporate our numerical approach, two semi-explicit IAEs system of index-2 together an applied problem are considered. These problems are solved using the proposed Jacobi collocation method for α = 12 and β = 13 based on matrix–vector

multiplication representation of equations. To examine the accuracy of the results, L 2w α,β errors are employed to assess the efficiency of the method. All the calculations were supported by the software Matematica® .

Example 1. Consider the following semi-explicit linear IAEs system of index-2

t A (t ) X (t ) = g (t ) +

K (t , s) X (s) ds, 0

where

A (t ) =

1 0 0 1 0 0

0 0 0

⎛

T

X (t ) = x1 (t ), x2 (t ), x3 (t )

,

3− s 2− s

2(2 − s)

⎞

⎟ ⎠, −1 1 s + 2 s2 − 4 0

T

g (t ) = 1, 2et − 1, −1 + et −t 2 + t + 1 . ⎜

K (t , s) = ⎝

,

3−2s 2− s −1 s −2

The exact solutions of the system are:

x1 (t ) = x2 (t ) = et ,

x3 (t ) = −

et 2−t

.

Let u , v , w be the approximation of the exact solutions x1 , x2 , x3 , respectively, that is given by (4.7). The discrete method described in Section 4 has been implemented for the problem and the L 2w α,β errors for different values of N = 4, 6, 8, . . . have been reported in Table 1. Graphs of the error functions and error behaviors for several values of N are also given in Figs. 1 and 2. We observe that the approximate solution of the equation represents the expected convergence behavior as described in Theorem 3. Example 2. Let us consider the IAEs system of index-2:

t A (t ) X (t ) = g (t ) +

K (t , s) X (s) ds, 0


Fig. 1. Error functions of the Jacobi collocation approximations of orders N = 6, 10, 12 and 14 for

α=

145

1 2

and β =

1 3

in Example 1.

Fig. 2. L 2w α,β errors versus the number of collocation points for Example 1.

with

A (t ) =

1 0 0 0

K (t , s) =

, T

X (t ) = x1 (t ), x2 (t )

,

et +s (s + 1)2 s+t +2 0

T

g (t ) = f 1 (t ), f 2 (t )

,

,

and

f 1 (t ) = sin t −

1

et 1 + et (− cos t + sin t ) −

2 f 2 (t ) = −(2 + t ) + 2(1 + t ) cos t − sin t .

1 4

−2 + 2(1 + t ) cos 2t + 1 + 4t + 2t 2 sin 2t ,

The exact solutions of the system are:

x1 (t ) = sin t ,

x2 (t ) = cos 2t .

We assume u and v are the Jacobi collocation approximation of the exact solutions x1 and x2 respectively, which are defined by (4.7).

146


Table 2 L 2w α,β errors for Example 2. N

x1 − u L 2

x2 − v L 2

4 6 8 10 12 14

2.41 × 10−5

5.29 × 10−4 4.44 × 10−6 1.86 × 10−8 4.85 × 10−11 1.97 × 10−13 1.45 × 10−13

w α ,β

4.25 × 10−8 4.18 × 10−11 2.39 × 10−14 4.05 × 10−15 3.94 × 10−15

w α ,β

Fig. 3. Error functions of the Jacobi collocation approximations of orders N = 6, 8, 10 and 14 for

α=

1 2

and β =

1 3

in Example 2.

Fig. 4. L 2w α,β errors versus the number of collocation points for Example 2.

The computational results have been reported in Table 2. Figs. 3 and 4 show the graphs of the Jacobi collocation error functions with α = 12 and β = 13 . It is observed that the errors decay exponentially.


147

Table 3 The absolute errors of Example 3 for different values of N. N

|u (0.95, 0.05) − uˆ (0.95, 0.05)|

|u (1, 0.05) − uˆ (1, 0.05)|

2 3 4 5

1.69 × 10−3 1.59 × 10−3 1.49 × 10−3 1.48 × 10−3

5.53 × 10−6 3.47 × 10−6 2.54 × 10−6 1.98 × 10−6

Example 3. As an applied test problem, consider the controlled heat equation (1.2) with f (x) = 1, h(t ) =

−1

α (t ) = 1, β(t ) = 0,

4t √1 , and g (t ) = e (1+32t ) . √ πt

2

πt 2

The exact solution of the equation is:

u (x, t ) = erf

√ .

1−x 2 t

It can be shown that, under the assumption

γ (t ) =

−49

13

6

e 4t (7−12e 4t +5e t )(9+π 2 t ) , 5 7 9π t (− erf( √ )+2 erf( √3 )−erf( √ )) 2 t

t

the system (1.4) can be reduced to a

2 t

system of the form (1.1) as follows

⎧ −1 t t ⎪ ⎪ e 4t (1 + 2t ) ⎪ ⎪ = φ1 (t ) − 2 θ(0, t − τ )φ1 (τ ) dτ + 2 θ(−1, t − τ )φ2 (τ ) dτ , ⎪ ⎪ ⎪ 2√π t 32 ⎨ 0

0

⎪ t ⎪ ⎪ 1 ⎪ ⎪ √ = −2 θ(1, t − τ )φ1 (τ ) dτ , ⎪ ⎪ ⎩ πt 0

√ 3 √ 3 2 where θ(0, t − τ ) √ 1 (1 + π3 t ), θ(−1, t − τ ) 34 t π 2 and θ(1, t − τ ) 12 t π 2 . 4π t

Let φˆ 1 (t ) and φˆ 2 (t ) be the approximation of the exact solutions φ1 (t ) and φ2 (t ) that are given by (4.7). Now, by replacing φˆ 1 (t ) and φˆ 2 (t ) in (1.3), the approximate solution of the problem (1.2) can be obtained as

1 uˆ (x, t ) =

ˆ x − ξ, t ) − θ( ˆ x + ξ, t ) dξ − 2 θ(

0

t +2

∂ θˆ ( x − 1, t − τ ) 1 + ∂x

0

t 0

τ

τ ∂ θˆ (x, t − τ ) 1 + φˆ 1 (η) dη dτ ∂x

φˆ 2 (η) dη dτ ,

0

(6.1)

0

ˆ x, t ) is the truncated series of the Theta function where θ(

ˆ x, t ) = √ θ(

1 4π t

3 m=−3

exp −

(x + 2m)2 4t

.

The approximate solution of uˆ (0.95, 0.05) and uˆ (1, 0.05) are special interest and computed using the Euler product integration method for (6.1). The differences between the numerical and analytical solutions for different values of N, are listed in Table 3. 7. Conclusion Here, we elaborated a spectral collocation method based on Jacobi orthogonal polynomials to obtain numerical solution of a class of IAEs system with index-2. We will achieve the goal by using the matrix–vector representation and the solution of the related system of equations. The strategy has been derived using some variable transformations to change the equation into an other IAEs defined on the standard interval [−1, 1], so the Jacobi orthogonal polynomial theory can be applied conveniently and the obtained index-2 system is solved directly without using the index reduction procedure which causes the simplification of the method. The spectral rate of convergence for the proposed method established in weighted L 2 norm. With the availability of this methodology, it will now be possible to investigate the approximate solution of other classes of IAEs systems. Although our convergence theory does not cover the nonlinear case, but it contains some complications and restrictions for establishing a convergent result similar to Theorem 3 which will be the subject of our future work.

148


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Applied Numerical Mathematics 61 (January 2011)

Applied Numerical Mathematics 61 (February 2011)

2DArtist Issue 061 January 2011 issue 61

Numerical Mathematics (Texts in Applied Mathematics)

Numerical Linear Algebra (Texts in Applied Mathematics)

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Applied Numerical Mathematics 61 (January 2011)

Applied Numerical Mathematics 61 (February 2011)

2DArtist Issue 061 January 2011 issue 61

Numerical Mathematics (Texts in Applied Mathematics)

Numerical Linear Algebra (Texts in Applied Mathematics)

PSM3 - January 2011

Computerworld - 24 January 2011

Consumer Reports January 2011

Popular science - January 2011

HAPPI January 2011

Coatings World January 2011

Ceramic Industry January 2011

BBC Focus – January 2011

Scientific American - January 2011

PC World January 2011

Food Engineering January 2011

Pollution Engineering January 2011

Vogue India January 2011

World Trade January 2011

Prepared Foods January 2011

Billboard Magazine - January 2011

GameInsider - January-February 2011

FX Magazine January 2011

Appliance Design January 2011

topCar Magazine - January 2011

Offshore Magazine - January 2011

Money Today January 2011

Candy Industry January 2011

Circuit Cellar January 2011

Process Heating January 2011

Quality January 2011

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