A- and B-stability for Runge-Kutta methods-characterizations and equivalence

Numerische Mathematik

Numer. Math. 48, 383-389 (1986)

9 Springer-Verlag1986

A- and B-Stability for Runge-Kutta Methods Characterizations and Equivalence* E. Hairer Section de math6matiques 2-4, rue du Li~vre, Case postale 240, CH-1211 Gen6ve24, Switzerland

Summary. Using a special representation of Runge-Kutta methods (Wtransformation), simple characterizations of A-stability and B-stability have been obtained in [9, 8, 7]. In this article we will make this representation and their conclusions more transparent by considering the "exact RungeKutta method". Finally we demonstrate by a numerical example that for difficult problems B-stable methods are superior to methods which are "only" A-stable.

Subject Classifications: AMS(MOS): 65L20; CR: G.1.7.

1. Introduction For the numerical solution of the system

y'=f(x,y),

y(xo)=y o

(1.1)

we consider m-stage (implicit) Runge-Kutta methods m

g i = y o + h ~. aijf(Xo+Cjh, gj), j=l

i= 1..... m,

Yl = y o + h ~ blf(Xo+Cih, gi),

(1.2a)

(1.2b)

i=1

which are of order at least one. It is well-known that for stiff equations (1.1) the integration scheme must possess good stability properties. Let us shortly recall two of the most important stability concepts. The oldest one - A-stability - has been introduced in the classical paper of Dahlquist [4]. It is related to the scalar equation y ' = 2 y and requires the *

Talk, presented at the conference on the occasion of the 25th anniversary of the founding of

Numerische Mathematik, TU Munich, March 19-21, 1984

384

E. Hairer

boundedness of the numerical solution (for every stepsize h>0) whenever Re2=O

n>0

0

(2.3)

A- a n d B - S t a b i l i t y for R u n g e - K u t t a M e t h o d s

385

and by using the integration formulas (cf. [10], p. 211) x

SP,(s)ds=~.+IP,+I(x)-~,P._I(x),

n>l,

0 x

~Po(s)ds=~l Pl(X)+89

'

~,-

o

1 2l/4n 2 - 1

we arrive at (2.4a)

where the infinite matrix ~ .... t is given by

f~ ) 9

--~2.

The equation (2.1b) now simply reads Y(1)=y0 +fo.

(2.4b)

In the next section we try to find a similar representation for the "finite" Runge-Kutta method (1.2).

3. A new Interpretation of the W-Transformation

In the following we treat only R u n g e - K u t t a methods with positive weights b i and distinct c i and we again assume x 0 =0, h = 1. Instead of the functions (2.2) we consider - for a given R u n g e - K u t t a m e t h o d - the polynomials g(x) and f ( x ) (of degree m - l ) that interpolate the values gz and f(ci, gi) (i=1 .... ,rn), respectively. We then write them as m--1

m-1

g ( x ) = ~ ~,,p,(x),

/(x)=

n=O

~f,p,(x),

(3.1)

n=O

where the polynomials p,(x) (of degree n - 1) are o r t h o n o r m a l with respect to ( f , g)~ = ~ bJ(ci) g(ci).

(3.2)

i=1

Because of g(cz)=g i it follows from (3.1) that (gl, .-., g,,)T = W (~,~....

, gin-

1) T

(3.3)

386

E. Hairer

where the matrix W is given by

w=IP~

I .

(3.4)

\PO('Cm)"'" Pro-i(Cm)/ (This transformation W has been introduced in [9, 10] for the study of Bstable Runge-Kutta methods). If we insert (3.3) and an analoguous relation for f; into (1.2) we obtain the transformed Runge-Kutta method

( gOI

1 /~

/fO ~

-, ----'"o+ fo t:",0//§ Vo- ',)' yl=Yo +fo .

(3.5b)

Here the matrix ~r is given by

= W- 1AW- 89 e T,

(3.6)

where the entries of A are the coefficients aij of (1.2) and e 1=(1, 0. . . . . 0) T. A comparison of (3.5) with (2.4) makes it plausible that "the better b' approximates y .... t the higher is the order of the method". Indeed, the following result has been given in [9].

Theorem 1. Let the quadrature formula 0/.,.

Y~---

-(1 0

0 \

(b i >

0 and distinct ci) be of order p. If

k = [ ( p - 1)/23,

~karbitrary

-~k

~k ~

(3.7)

(erl~Ckel= 0 for p even),

then the Runge-Kutta method, whose coefficients bl, ci are given by the quadrature formula and aij by (3.6), is of order p. [] 4. A-Stability, B-Stability and their Equivalence We show in this section that the use of the transformed Runge-Kutta method (3.5) leads to simple characterizations of A- and B-stability in terms of the matrix ~#.

A-Stability For the test equation y ' = A y it holds f,=2g,,. After a short calculation, using the representation (3.5), we obtain YI R(2)Yo where =

A- and B-Stabilityfor Runge-KuttaMethods

l+89

R(z)= 1 - 89

'

387 0(z) =ze~(l - z y ) - ' e,.

(4.1)

A well-known property of the M6bius-transform implies that A-stability (i.e. condition (1.3)) is equivalent to Re0(z)