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=
1 - e~Xh
(2.22)
and has the property (X,h) = h + 0(Xh2).
(2.23)
The linear harmonic oscillator is modeled by the following second-order ODE d?u
,
(2.24)
where a; is a real constant. The two linearly independent solutions are u(1>(i) = e*",
uW{t) = e-* wf .
(2.25)
Substitution of these functions into Eq. (2.13) gives uk
uk+i uk+2
e™hk A
+1
e^ e*"h = 0.
(2.26)
e-^Hk+2)
Evaluation of the determinant, followed by simplification of the algebra yields the following second-order difference equation uk+i - [2 cos(uih))uk + u*-i = 0.
(2.27)
Exact Schemes
9
Using the identity 2 cos(w/i) = 2 — 4 sin'
(f).
(2.28)
Eq. (2.27) can be transformed to Ujfe+l - 2Uk +Uk-1 4 ^ ; „ 22 / M
(^)sin (¥)
2 +WUt=0.
(2.29)
This is the exact finite difference scheme for the harmonic oscillator. Consider now the coupled, linear system of first-order ODE's with con stant coefficients [2] du — = au + bw, at
(2.30a)
dw , — = cu + dw, at
(2.30b)
where ad-bc^0,
uo = u(t0),
w0 = w(t0).
(2.30c)
The solution of Eqs. (2.30) is w0
„A,(t-i 0 ) cAa(t-*o) t
w{t) =
~('t^€)[{hP)Uo~Wo. / X2-a\
lYAi
-a\
(2.31)
„A,(t-M „A 2 (t-t 0 )
(2.32)
where 2Ai,2 = {a + d)± s/(a +
d)2-A(ad-bc).
(2.33)
10
Nonstandard Finite Difference Schemes
The exact finite difference scheme for Eqs. (2.30) is obtained by making the following substitutions in Eqs. (2.31) and (2.32) t0->tk=hk, uo-Mifc, w0^wk,
t-)tk+i =h(k+l), u(t)-*uk+i, w(t)-¥Wk+i-
(2.34)
The results of these substitutions are the expressions = auk + bwk,
4> wk+i. -
jwk
= cuk + du>k,
(2.35a)
(2.35b)
where lC
V- =
.
r
4> = —,
,
j — ■
(2.36a)
(2.36b)
Consider again the linear harmonic oscillator. In system form, Eq. (2.24) can be written du -dt=W'
dw , —- = — u) u. dt
(2.37)
Comparison with Eqs. (2.30) gives a = 0,
6=1,
c=
-LJ2,
d = 0,
(2.38)
and sin(o;/i)
ih — c.na(uih).
(2.39)
CJ
Thus, the exact finite difference scheme for Eq. (2.24), in the form of a system of two first-order equations, is u*+i ~ cos(w/i)ui
[
gin(n;ft)l
w
J
= wk,
(2.40a)
Exact Schemes
11
iDjfc+i — cos(u>h)wk o • = -u uk. 8in(ii;ft)l
[
w
(2.40b)
J
In a similar manner, the damped harmonic oscillator cPu
du
(2.41)
or du
dw
Tt=w>
-d7 =
(2.42)
-u-2ew>
in system form, has the following exact finite difference representations uk+1 - 2uk + ufc_i
2
+ 2e
Uk - i>uk~\
0
2
+
2
2(l-VOu* + (0 + V- -l)«
-uk-
2ewk,
(2.44b)
where te~th
e~th cos ( v ' l - e 2 /i)
= -f==
■ sin ( v / T ^ / i ) .
(2.45a)
(2.45b)
The general Logistic differential equation is given by the following firstorder, nonlinear equation du = Aiu - A2u2, ~dl
u(«o) = uo,
(2.46)
Nonstandard Finite Difference
12
Schemes
where (Ai, A2) are positive parameters. This equation can be solved by the method of separation of variables [15] to give the solution U(
(Ai - u 0 A 2 ) exp[-Ai(t - t0)} + A2u0 '
The exact scheme for Eq. (2.46) is gotten by carrying out the replacements to -» tk,
t-*tk+i,
u0-+uk,
u(t)^uk+i.
(2.48)
Making these substitutions and rearranging the resulting expression gives *+1
k
K=0
= Xiuk - X2Uk+iuk,
(2.49)
as the exact finite difference scheme for Eq. (2.46). Note that if Ai = - A and A2 = 0, Eq. (2.49) becomes the relation in Eq. (2.19). Likewise, for Ai = 0 and A2 = 1, the exact scheme for
is obtained Uk+l ~ Uk
h
= -Uk+\uk.
(2.51)
An interesting example to consider is the first-order, nonlinear differen tial equation
§--...
cu*
Using the change of variable z = u3, this equation becomes §
= -2,'.
(2.53)
The corresponding exact finite difference scheme is Zk+1
Zk
h
=-2zk+izk.
(2.54)
In terms of u*, Eq. (2.54) becomes 1l2 - U2 "*+l "* _
,,2
,,2
/o r r \
Exact
Schemes
13
or
= - L 2 u *r„) u*+iu*-
(256)
The structure of this scheme is not one that would be written down by most persons familiar only with standard numerical methods. The last ODE example to be studied is the following second-order, linear equation cPu
du
,
Its general solution consists of linear combinations of the two functions u (1) (t) = 1,
u(2>(t) = ext.
(2.58)
The exact finite difference scheme is obtained from the expression uk uk+1 uk+2
1 e \hk A/, +1 1 e (* > = 0, 1 eAh(*+2>
(2.59)
which, after a little algebraic exertion, can be written as uk+i - 2uk + uk-i
(^i)»
_ . fuk-uk-i\
-H-T-J-
.
(2 60,
.
•
The situation regarding exact finite difference schemes is more prob lematic for PDE's. As for the case of ODE's, the question as to whether exact schemes can be constructed for PDE's is very dependent upon the existence of known solutions to the PDE's of interest. Another difficulty is the problem of defining precisely what is to be understood as the general solution for a given PDE [16]. The following presentation shows how to construct exact schemes for a certain class of PDE's, namely, first-order, linear advection equations. The 1-dim unidirectional wave equation for u — u(x, t) ut + ux= 0,
u(x, 0) = f(x),
(2.61)
has the solution u{x,t) = f{x-t),
(2.62)
Nonstandard Finite Difference Schemes
14
where f(z) is assumed to have a first-derivative in z. Since the linear partial difference equation
t4 + 1 =«£,-!,
(2-63)
has as its general solution an arbitrary function of (m — k) [14], i.e., ukm = F(m - k),
(2.64)
it follows that if Ax = At, then Eq. (2.61) has the exact scheme given by Eq. (2.63) where xm = {Ax)m,
tk = {At)k.
(2.65)
With its restriction, Eq. (2.63) can be rewritten to the form 7I*~M
71*
71*
— 71*
,
-w^-%4^0'
Ax=At
>
(2 66)
-
where <j)(z) is arbitrary, except that it must satisfy the condition 4>{z) =z + 0(z2).
(2.67)
An example of a PDE having linear advection and a nonlinear reaction term is ut + ux = u(l - u),
u(x,0) = f{x),
(2.68)
where f(z) and its first derivative is assumed to exist. The nonlinear trans formation «(«,*) = ^
,
(2-69)
gives the linear, inhomogeneous equation wt + wx = 1 - w,
(2.70)
which can be solved to give [17] w(x, t) = g(x - t)e~l + 1,
(2.71)
where g(z) is an arbitrary function having a first-derivative. This function can be determined from the initial condition in Eq. (2.68), i.e.,
9(I)=i
75r-
(272)
Exact Schemes
15
Using this result with Eqs. (2.69) and (2.71), the solution to Eq. (2.68) is
^Vta^w-f Solving for f(x -t)
(2 73)
'
gives /(
*-')=l-(l-e-'M«,0-
( 7 )
Making the substitutions x ->xm = (Ax)m, f(x -t)-¥
t -ttk
= (At)*,
u(x, t) -»• u „ ,
(2.75a)
f[h(m - k)] = fa, with At = Ax = h,
(2.75b)
and using the fact that fa satisfies Eq. (2.63), it follows that p-h(k+l),,k+l e
p-hk e
"m
1 _ [1 _ c -k(*+D] u *+i
u
k m-l
1 - [1 - c-**] < _ , '
(2.76)
After some algebraic manipulations this equation becomes
Ax = At = h.
(2.77b)
Note that Eq. (2.77) is linear in u£+ l . Solving for u^1"1 gives
^1+(;r,w.-
(-a,
Thus, Eqs. (2.77) provide an exact finite difference scheme for Eq. (2.68). The form given in Eq. (2.78) shows that this scheme is explicit. Note that for both the linear and nonlinear equations, i.e., Eqs. (2.61) and (2.68), a functional relation exists between the space and time step-sizes: Ax = At. The 2-dim versions of Eqs. (2.61) and (2.68) are ut + aux + buy = 0,
(2.79)
ut + aux +lmy = u{\ - u),
(2.80)
Nonstandard Finite Difference
16
Schemes
where a and b are constants. Following the same procedures as for Eqs. (2.61) and (2.68), the Eqs. (2.79) and (2.80) have exact finite difference schemes given by the respective expressions [18] *+i _ * m,n "m,n
u
At
+
",myn
(£)
u
+ UyJ
"m-l,n +
' "m,n—1
m-l,n +uro,n
u
rn-l,n-l
^m-l.n-l +
(2.81)
u
m,n-l
= 0,
and *+i _ * m,n "m,n
u
+a
0i (At) x
+b
k m,n ~
u
m-l,n
u
k n»-l,n
02 (Ax)
"m-l.n-1
03 (Ay)
= « £ . - l , n - l ( l - 0 . (2-82)
where 0x (At) = eAt - 1,
02(Ax) = a (e A */ a - l ) ,
03(Ay)=6(eA«'/6-l),
(2.83)
and the following relations hold among the step-sizes Ax = aAt,
Ay = 6At.
(2.84)
The following notation is used x m = (Ax)m,
y„ = (Ay)n,
u
m,n
tk = (At)fc,
(2.85)
=u{xm,yn,tk).
These two exact schemes are both linear in u ^ and can be rewritten to the respective forms <J=t&_i,„_i,
( 2 - 86 )
and u fc+1 = u
m,n
J
-At. +
m-l,n-l
(l-e-^)
/2\
dd>
d2d>
,
,
,
where k is a real parameter. Define u(r, t) and w(r) to be u(r, t) = r(r, t),
w(r) = rtp(r).
(2.90)
Substitution of these expressions, respectively, into Eqs. (2.88) and (2.89) gives d2u
d2u
(Pw
l2
„
tnM.
The solutions to these equations are u(r,t)=f(r-t)+g{r
+ t),
(2.92)
w(r) = ^cos(fcr) + Bsin(fcr),
(2.93)
where f(z) and g{z) are arbitrary functions having first- and second-order derivatives, and A and B are arbitrary constants. The function w(r) sat isfies the linear harmonic oscillator differential equation whose exact finite difference scheme has been found; see Eqs. (2.24) and (2.29). Consequently, for r m = (Ar)ro and wm = w(rm), wm+1-2wm+wm.1
(£)sin2(Mr)
2
+kw\ \rmJ \ 2Ar ) (At)2 (2.98) A direct calculation shows the unidirectional wave equation having spherical symmetry du u du
d
-
^
( 2 ")
has the following exact difference scheme [21] , , n + l _ ,.n m
m A
At
.,n
.,n _ ,,n
+ ^2=i + rm
m
m 1 A
Ar
-
= 0,
(2.100) v
'
where At = Ar. A detailed examination and analysis of the various exact finite differ ence schemes of this section lead to several important conclusions. First, discrete models for the derivative require more complicated structures for the functional dependence on the step-size than those given by conventional methods. In general, a substitution like the following is required at
and <j> depend on the step-size, h = At, and satisfy the conditions V>(/i) = l + 0(/i 2 ),
(2.102)
+
a(h)]-mu(t) (h)
where ij>(h) and (h) satisfy Eqs. (2.101) and (2.103), and a(h) has the property (j{h) = h + 0{h2).
(2.105)
The function <j>(h) is called the "denominator function." The paper by Mickens and Smith [22] investigated certain of its properties and how they in fluenced the solution behaviors of finite difference schemes. Second, a major characteristic of exact schemes is the discrete modeling of nonlinear terms by nonlocal representations on the computational grid. For the Logistic equation, see Eq. (2.46), the nonlinear term u 2 was replaced in the exact finite difference scheme by u2->uk+1uk.
(2.106)
This is in contrast to standard methods which use a local representation, i.e., u2-+u\.
(2.107)
In a similar fashion, for the linear advection equation with a nonlinear reaction term, see Eq. (2.68), the u2 was discretely modeled as tia->t&_i«m+l.
(2-108)
Note that this term is nonlocal in both the discrete space and time variables. Third, the exact schemes for the various PDE's studied always gave explicit functional relations between the various time and space step-sizes. For the wave-type PDE's considered, a linear relation occurs, i.e., At oc Ax. 1.3
Nonstandard Schemes
A major advantage of having exact finite difference schemes for differen tial equations is that various questions related to the issues of consistency, stability, and convergence [3; 4; 5; 6] do not arise. However, the a priori
20
Nonstandard Finite Difference
Schemes
construction of an exact scheme for an arbitrary differential equation is es sentially not possible. To be able to do so is tantamount to knowing the general solution of the equation of interest. Also, the structural complexity of many differential equations is such that even if exact schemes were found it would be very difficult, if at all, to verify the result. Thus, these are clear advantages to determining general principles for the construction of finite difference schemes for which it can be known in advance that the impor tant properties of the solutions to the differential equations are also shared by the corresponding solutions of the difference equations. Particular in stances of such properties include positivity conditions, special solutions with predetermined stability behavior, and boundedness of solutions [6; 7; 10; 16]. Based on both detailed analytical and numerical studies of exact finite difference schemes for a variety of differential equations, the following rules provide a concise summary of the major features "discovered" during these investigations. These rules can provide guidance for the construction of nonstandard finite difference models of differential equations. There are two important points related to the application of these rules. First, for a given differential equation, the rules generally permit a number of nonstan dard schemes. In other words, at the present time, uniqueness does not exist in the determination of nonstandard schemes. Second, while these schemes are not exact, they will in many situations give difference equa tions that are superior to conventional ones for the purpose of providing numerical solutions. It should be strongly emphasized again that the rules to follow are based on the observation of the structural forms that arise in exact schemes for differential equations whose solutions are known. The as sumption is made that such features are generic to all exact finite difference schemes. Rule 1. The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives of the differential equations. Comment 1. If the order of the discrete derivatives are larger than those occurring in the differential equations then "ghost" or "spurious" [3] solu tions can appear. The following example illustrates what takes place when this rule is violated. Using a central difference approximation for the decay equation
S—•
have the properties given in Eqs. (2.102) and (2.103). These functions may also depend on the various parameters that appear in the differential equations. Rule 3. Nonlinear terms should, in general, be replaced by nonlocal dis crete representations. Comment 3. For the Logistic ODE, the nonlinear term u 2 is replaced by Ujfe+iUfc. However, other more general forms may be used, an example being u2 -¥ 2{ukf
- uk+1uk.
(3.7)
Rule 4. Special conditions that hold for the solutions of the differential equations should also hold for the solutions of the finite difference scheme. Comment 4. Numerical instabilities can arise because the discrete equa tions do not satisfy a principle or condition that is of critical importance for the corresponding solutions of the differential equations. An important example is the condition of positivity that must be satisfied by many sys tems in the natural and engineering sciences. If the discrete equations allow their solutions to become negative, then numerical instabilities will occur. The existence of a first-integral is another important condition for many systems. A system of linearly coupled conservative oscillators is another important example. Here the first-integral is the total energy. Rule 5. The scheme should not introduce extraneous or spurious (special) solutions. Comment 5. Many finite difference schemes will generate special solutions that do not correspond to any solution to the original differential equations. See the discussion under Comment 1 for an illustration of this. For ODE's, many of the standard numerical integration methods will have spurious fixed-points whose location and stability properties depend on the value of the step-size [2]. Rule 6. For differential equations having N(> 3) terms, it is generally use ful to construct finite difference schemes for various sub-equations composed of M terms, where M < JV, and then combine all the schemes together in an overall consistent finite difference model.
Applications
23
Comment 6. Differential equations containing many terms may be diffi cult to analyze from the standpoint of constructing finite difference schemes using the five previously stated rules. Rule 6 helps in this task by allowing information obtained in the construction of schemes for simpler differential equations to be used to build up a fully discrete model for the equations of interest. A nonstandard finite difference scheme is a discrete model of a differ ential equation that is constructed according to the above six rules. In general, nonstandard schemes are not exact schemes; however, they do of fer the prospect of obtaining finite difference schemes that do not possess the usual numerical instabilities. While the current rules do not provide a unique discrete representation to a particular differential equation, their use along with a knowledge of important features of the solutions of the differential equation greatly restricts in practice the possible discrete mod els. The next section is devoted to the construction of nonstandard schemes for a number of ordinary and partial differential equations that have been used to model important phenomena in the natural and engineering sci ences. 1.4
Applications
The eleven applications discussed in this section illustrate both the power and weaknesses of the current formulation and implementation of nonstan dard finite difference methods. 1.4.1
First-Order
Scalar
ODE's
Consider a scalar first-order equation
The fixed-points are the constant solutions and are denoted by u; they are solutions to the equation /(«) = 0.
(4-2)
In the construction of a nonstandard scheme for Eq. (4.1), a critical aspect is the selection of a denominator function, (h), which appears in the discrete
24
Nonstandard Finite Difference Schemes
derivative, i.e., du
uk+i
dt
- ujb
(4.3)
(«)
and corresponds to the smallest time scale. This result allows Eq. (4.6) to be rewritten as 4>(h,T') = r * ( l - e-hlT').
(4.10)
Applications
25
Note that for any h > 0, (f> is bounded, i.e., 0 eliminates these instabilities [24]. Stated another way, the use of such a denominator function allows the step-size h to be much larger than one normally used because as far as the discrete equations are concerned it is the effective step-size <j> that determines the stability and not the ac tual step-size h. The following example illustrates this feature. Additional equations are treated in detail in the references [2; 22; 24]. An ODE that provides an elementary model for combustion [25] is ^ = u 2 (l - u), u(0) = u0 > 0. at Here /(u) = u 2 (l — u) and three fixed-points exist fid) = u = 0,
u = 1,
(4.12)
(4.13)
and R1^R2
= 0,
R3 = 1,
R* = 1.
(4.14)
The denominator function is 0(/i) = 1 - e~\
(4.15)
and the discrete derivative is dt
1 - e~h
The physical requirement is that u(t) > 0. The corresponding condition for uk is uk > 0 => u fc+1 > 0.
(4.17)
Note that since «o > 0, all solutions to Eq. (4.12) monotonically go to the value u^ = 1. These two conditions can be satisfied, as it will be seen, if
26
Nonatandard Finite Difference Schemes
the following representations are used for the nonlinear terms u2 -)• 2(u*) 2 " uk+1uk u3->u*+i(ufc)2.
(4.18a) (4.18b)
Making these substitutions into Eq. (4.12) gives Uk+ Uk ^)
= 2(«*) 2 " u k + i u k - uk+1(uk)2.
(4.19)
Since this expression is linear in uk+i, it can be solved to obtain the result _ Uk+l
-l
(1 + 2(h) has the property 0 < (h) < 1,
h > 0,
(4.21)
it can be easily proved that the 1-dim map of Eq. (4.20) has the following features: (i) It has three fixed-points located at u^ = vS2^ = 0, and u^ = 1. (ii) For uo > 0 and any h > 0, all solutions uk monotonically approach u
u={n,m,p),
(4.23)
with
(-!)]•
0 = T 1 — exp
(4.24)
The linear and nonlinear terms, on the right-side of Eqs. (4.22), were di rectly modeled such that positive values for (nk,mk,Pk) implies positive values for (njt+i,mfc + i,pjb + i). This requirement follows from the physical property that (n,m,p) are particle densities and by definition are nonnegative. Any finite-difference scheme that allows negative solutions will have
Nonstandard Finite Difference Schemes
28
numerical instabilities. The particular scheme selected was - * + 1 , — - = G- {aiNt)nk+i
= (aiNt)nk
- ainkmk+i
Pk+i ~Pk
4>
+ ainkmk
+ ■yimk - citik+i,
(4.25a)
- 8oTnk+ipk - limk+u
(4.25b)
= G - 60mkpk+i
- c2pk+i.
Since Eqs. (4.25) are linear in (nk+i,mk+i,pk+i), and the following expressions obtained n
*+1 =
G + ( 1 + 4> [xk,(xk -xk-i)/(h)),
(4.45)
Nonstandard Finite Difference Schemes
32
where h = At, tk — hk, and {h) has the property 0(h2).
(h) = h +
(4.46)
(ii) Replace E(x, x) by E(xk, z * - i ) , where the discrete energy function is to be symmetric under the interchange k -(£)" + (£)
(4.50)
A discrete energy function satisfying the above requirements is given by the expression E{xk,xk
->-G)
Xk -
Xk-l
(T) XkXk-l-
4>(h)
(4.51)
Applying the A operator to E gives the equation of motion Zjfc+l
-
2zjfc + Z j f c - 1 2 — -I- U) Xk
(4.52)
0.
im?
A discrete energy function that does not satisfy all of the above require ments is the standard scheme Ei(xk,xk-i)
■G)[
Xk
-Xk-l
4>{x)
2
_2
(4.53)
Applications
33
Calculating AEi = 0, we obtain the following equation of motion Xk+l
~,MMXk~l + 2 ( XM " '" ) (X~h±1^IlL) = °(4-54) h m )l \*k+i - *k-\) \ 2 ) Observe that the equation is nonlinear. This undesired feature is a conse quence of modeling the nonlinear term x2 by the local representation x\. Thus, the energy function E\ (xk , z/t-1) does not have the property of being invariant under k o k — 1. In contrast, Eq. (4.51) does have this property, i.e., x2 is modeled by XkXk-iA much more interesting case is the unforced, undamped Duffing's equa tion with a quadratic term x + u2x + ax2 + fix3 = 0.
(4.55)
The energy function is E(x,±) = Q ) (x)2 + ( £ ) x2 + ( | ) z 3 + f^j
x\
(4.56)
A discrete energy function for Eq. (4.56) is
£(**,**-i) = ( O f ^ T " ] + ( y ) »*«*-! From AE = 0, the following equation of motion is obtained Xk+i - 2a;* + Xk-i [ 0. The latter equation models certain phenomena in neuro physiology [38]. The particular numerical scheme they used for Eq. (4.60) was „ k+1
{2 - ah[a - (uk)*}}uk ~ 2 - ah[2uk - 3(ufc)2 + 2auk - a) '
K
'
where h = At. To obtain this equation, they employ a symmetric nonlinear representation of the nonlinear terms in Eq. (4.60). (See Eq. (2.20) of reference [36].) They selected this form because it is chaos free, of higher order than a simple forward Euler scheme, and gives numerical solutions that converge to the proper fixed-points at u = 0 and u = 1, for positive values of a and 0 < a < 1. However, the actual convergence of their scheme can be either monotonic or oscillatory depending on the values of uo, a, a and h. Using a nontrivial denominator function, Mickens [39] showed that the above scheme could be easily transformed into one that is chaos free, has a local truncation error of second-order, and converges monotonically to the correct fixed-point for all values of the step-size. First, Eq. (4.60) was changed into a more symmetrical form by the following change of variables and parameter selections t -* i = at,
a = -1,
uk ->• Ufc.
(4.63)
Substitution of these into Eq. (4.60) and dropping the bars gives ^=u(l-u2).
(4.64)
Applications
35
Note that nothing essential has changed. Both Eqs. (4.60) and (4.64) have three fixed-points, the "outer" two are stable, while the one in the "middle" is unstable. Thus, topologically the solutions of these two equations are equivalent. The denominator function for the scheme of Price et al. [36] is just the standard one, i.e., 4>P{h) = h,
(4.65)
however, Mickens used the nonstandard form [39] 1 - e~2h {h) = <j>M(h) =
.
(4.66)
Thus, the following scheme was obtained, for Eq. (4.64), u*+i =
(5-e-2/,) + (l-e-2',)(ufc)2 i «* • (3 + e - 2 h ) + 3 ( l - e - 2 h ) ( u * ) 2 _
(4-67)
Applying the analysis of reference [24] to Eq. (4.67) shows that the solu tions of this 1-dim map have exactly the same qualitative properties as the corresponding solutions to Eq. (4.64). A consequence is that used as a numerical integration method, Eq. (4.67) is a discrete model of Eq. (4.64) that has no numerical instabilities. 1.4.6
Time-Independent
Schrodinger
Equations
A variety of systems in the natural and engineering sciences can be mod eled by linear, second-order ODE's. The general equation can always be transformed to a new equation having the form [40] ^
+ f(x)u = 0.
(4.68)
A particular example of such an equation is the time-independent Schrodinger equation [41]. A nonstandard scheme was constructed for Eq. (4.68) by Mickens and Ramadhani [42]. Their starting point used the fact that the ODE cPu -T-J + Au = 0,
A = constant,
(4.69)
36
Nonstandard Finite Difference Schemes
has the following exact finite difference scheme [2] Mra+l - 2 u m + U m _ i
(*)*»'(¥)
(4.70)
+ Aum = 0,
where h = Ax and um = u(xm). This result is correct whether A is positive or negative if use is made of the relation sin(i0) = isinh(0),
i = yf^.
(4.71)
The Mickens-Ramadhani scheme for Eq. (4.68) is gotten by replacing A in Eq. (4.70) by fm - f(xm). Doing this gives «m+i - 2u m + u m _i
G t ) - " *2 ^J
(4.72)
+ fmUm = 0.
Using the trigonometric identity 2sin 2 0 = 1 - c o s 20,
(4.73)
allows Eq. (4.72) to be rewritten as U m + i + U m _i = 2 COS (/l\/7m)J
u
(4.74)
Chen et al. [43] then generalized this scheme to one that they called the combined Numerov-Mickens finite difference scheme (CNMFDS): 1 +
h2f,m + l 12
"m+l +
1+
= 2 [cos (hy/JZ)] 1 +
h2f,m - l 12
h2f„ 12
"m-l
(4.75)
They carried out numerical studies and compared the Numerov [44], MickensRamadhani [42], and CNMFDS representations for Eq. (4.68). The follow ing conclusions were reached: (i) The Mickens-Ramadhani scheme is (formally) of 0(h2), while the Numerov scheme is 0(h*). However, the Mickens-Ramadhani scheme performs much better than the Numerov scheme for large values of h. (ii) The Mickens-Ramadhani scheme is an exact finite difference model for f{x) = constant. This is not the situation for the Numerov method.
37
Applications
(iii) The CNMFDS is of 0(/i 4 ), just like the Numerov method, and is an exact finite difference method for f(x) = constant. It also out-performs the Numerov scheme for large step-sizes. Additional mathematical results on these nonstandard schemes was ob tained by investigating the asymptotic (m large) properties of their solu tions and comparing them to the corresponding asymptotic solutions of Eq. (4.68). The test problem was a transformed Bessel's equation. It was shown [42] that both the Mickens-Ramadhani and CNMFDS expressions agreed with the results from the ODE up to terms 0 ( z ~ 2 ) . See also refer ences [45; 46]. 1.4.7
Traveling
Wave
Solutions
The Burgers PDE provides a useful elementary model for the study of nonlinear fluid behavior [16] and as a test equation for evaluating numerical integration schemes [5; 6]. This equation is ut + uux = uxx,
u = u(x,t),
(4-76)
and has a special type of solution called traveling waves [16]. For these solutions, u(x, t) takes the form u(x,t) = f{x-ct),
(4.77)
where f(z) has a second derivative, the wave speed c is a priori unknown, the boundary conditions are u(-oo,£)=u2,
u(+oo,t)=Ui,
(4-78)
with ui and u 2 positive constants, and u 2 > ui. Let z = x-ct,
(4.79)
then f(z) satisfies the following nonlinear ODE
-c/'+ //' = /", / ( - c o ) = u2,
/'=£,
(4.80a)
/(+00) = «i.
(4.80b)
Nonstandard Finite Difference Schemes
38
This equation can be solved to yield the solution
M
=
J
r(:
u u1
•
(4.81a)
l + exp[fc^ c=HL±^.
(4.81b)
The task to be considered is the construction of a nonstandard finite difference scheme for Eqs. (4.80) such that it can also be solved exactly. To proceed, first observe that Eq. (4.80a) can be integrated once to give f
f' =
-l-cf
+ A,
(4.82)
where A is an arbitrary integration constant. With the change of variable z= - ,
(4.83)
Eq. (4.82) becomes ^
= f2-2cf
+ 2A.
(4.84)
Applying the nonstandard finite difference rules [2], with an emphasis on the nonlocal representation of nonlinear terms, the following discrete model is obtained y
* " ^ ~ ' = VkVk-i ~ 2cyk-i + 2A,
(4.85)
where y* is an approximation to f(zic) and zk = hk,
h = Az,
(4.86)
and the, at present unknown, denominator function 0 satisfies 0,
r_ > 0,
u2>«i,
(4.97)
40
Nonstandard Finite Difference Schemes
force (/i) to satisfy the constraint 4>(h) 0. (4.98) u2 A particular explicit functional form for cj>(h) that satisfies Eqs. (4.87) and (4.98) is 1_
p-h"2
<j>(h) =
.
(4.99)
U2
The constant D can be calculated by selecting an arbitrary value for y0. A possible choice is [16] Ul + U 2 Vo=
2
(4.100)
'
which leads to D_
1 - "2'
(4.101)
Since u2 > «i, it follows from the condition of Eq. (4.98) that 0 +£H
(4.151)
y^j
(4-152)
+ \ l + 7±
=0;
therefore A 2 =A5 = l + i \ - t and it follows that |A2|2 = |A3|2 > 1.
(4.153)
The latter condition implies that fm increases exponentially with m for large values of m. This clearly is not consistent with the result of Eq. (4.138). The conclusion is that the finite difference scheme given by Model A has a numerical instability and consequently should not be used to determine numerical solutions for Eq. (4.133). Carrying out the same calculations for Model B gives the result of Eq. (4.145) for gk- The solution is Eq. (4.147) and, as for Model A, 9k = g{tk)- The equation determining fm is
/m
Y"" 1 + (£) u™+* -3^+3/»-i
- /— 2 )= c -
=.-A. X
where s is the initial condition for the following ordinary differential equa tion
i.e. s = x(0). After solving the above initial value problem, the parameter s can be represented as a function of the variables a; and t. Substitution of s into t h e expression (2.6) gives the exact solution of the problem (2.5) for a given velocity field v(x,t). As a first case, we consider velocity fields independent of the space variable x, t h a t is v = v(t) = P n _ i ( t ) . Here, Pn_! (t) = an-it"-1
+ an-2tn~2
+ ■ ■ ■ + ait + aQ
is a polynomial of (n — 1) order, where {ai}"^ 1 a r e g i y e n real constants. The solution of the ordinary differential Eq. (2.7) is given by x(t) = Pn(t) + s, where
Pn(t) = f v(T)dr = ^ V + *L± f »-i + • • • + 2Lt2 + aQt. Jo
n
n -1
2
Substitution of s = x — Pn(t) into the expression (2.6) yields AT t\ c ^> l> ~ e-xt +
f{X (1
~ Pn(t)) _ e - « ) / ( x - Pn(t))'
O R\ ( ^'
Here, c(x, t) is the analytical solution of Eq. (2.5) corresponding to the space-independent velocity field v(t) — P„_i(t). Comparison of the analyt ical solution at time t (2.8) with the analytical solution at time t + At gives
Non-Standard
Methods in One
Dimension
61
the following relationship c(x, t + At) =
^-
e-xAt
+ (1
c(x-[Pn(t + At)-Pn(t)),t) _ e - A A t ) ^ _ [p n ( f + At) - Pn(t)}, t)"
Based on it, we construct the "exact" time-stepping scheme
C n+1( )
"
J (Atr ( ^ )
= AC
""(*m)(1 " Cm+1W>
(2-9)
where the denominator function is given by S(At) = (eXAt - 1)/A and the backtrack point xm has the following expression xm = x - [Pn((m + l)At) -
Pn(mAt)].
Here, Cm(x) denotes the numerical solution at location x and at time mAt. The left-hand side of the numerical scheme (2.9) can be viewed as a nonstandard backward difference approximation of the characteristic derivative Dc
dc
,
. dc
di+v{x>t)dx->
Di =
while the right-hand side represents a nonlocal modeling of the reaction term r(c) = Ac(l - c). As a second case, we consider velocity fields linear in the space variable x and polynomial in time, that is v = v(x,t)
= (ax + 6)P„_i(t),
(2-10)
where a and b are given constants, and Pn-i(t) is the same polynomial as in the first case. For a velocity field v(x, t) polynomial in the space variable, when a solution of Eq. (2.7) exists, implementation of the "exact" timestepping scheme is similar to the linear case. However, it is more difficult and we shall not present it here. The solution of the ordinary differential Eq. (2.7), in the case of velocity field (2.10), is given by the expression x(t)=eaP"w(s where Pn(t) = / 0
V(T)CIT.
+
b/a)-b/a,
Substitution of
s = e-aP"(t)(x
+
b/a)-b/a
62
Nonslandard Methods for Advection-Diffusion-Reaction Equations
into the expression (2.6) yields c(x, t) =
e -At +
f(e-aP"W(x + b/a) - b/a) (! _ e-* t )/(e— p -(«>( 1 + b/a) - b/a)'
Similarly to the case of space-independent velocity field, we can con struct the new non-standard method Cm+1(x)-C™(x™) 6(At)
_xr,m,^M = \Cm{xm)(l
M l . - Cm+1).
(2.11)
Here, the denominator function 6(At) = (e AAf — 1)/A is the same as in the scheme (2.9), but now the backtrack point x m has the more complicated expression i™ = c-a[P«((m+l)A*)-P.(mAt)](a;m+l
+
&/a)
_ bJa
(2.12)
Remark 2.1 It is easy to generalize the "exact" time-stepping scheme to variable time step sizes Atm. The only difference in the implementation is that the denominator function S(At), from the above scheme (2.9), should have the form 6(At) — (e A A ' m - 1)/A. Here, At™ is the time step size between the old time level m and the new time level (m + 1). Remark 2.2 In the case of a constant velocity field v, the backtrack point x m coincides with the corresponding point x from the discretization of Eq. (2.5) when the modified method of characteristics [17] is applied, i.e. xm = x ~ vAt — x. With the additional requirement of time step size At equal to the uniform spatial grid size Ax, an "exact" time-stepping scheme for Eq. (2.5) has been first developed by Mickens [30]. 2.2.1.2
Linear Reaction Terms
Here, we briefly examine the implementation of the "exact" time-stepping scheme for solving the advective transport equations with linear reaction terms. First, consider the transport Eq. (2.3) with a constant reaction term r(c) - n, i.e,
Non-Standard Methods in One Dimension
63
Using the method of characteristics, we can find the general solution of the above Eq. (2.13) in the form c(x,t) = f(s)+nt,
(2.14)
where s is the initial condition to the ordinary differential Eq. (2.7). Solving the initial value problem (2.7) for the case of space-dependent velocity fields (2.10) and substituting the parameter s = e-aP^l\x
+
b/a)-b/a
into expression (2.14) yields c(x, t) = f(e-aP"W(x
+ b/a) - b/a) + fit.
Similarly to the case of logistic growth reaction terms, we can construct the "exact" time-stepping scheme Cm+l{x)-Cm{xm)
_
At
~^
where 2m
=
e - 0 [P„((m+l)A()-P„(mA()] (l m+l
+
ft/fl)
_6
/ f l
In this case, the denominator function is equal to the time step size At. Second, consider the first order reaction term r(c) = Ac. Then, the governing Eq. (2.3) becomes £ + » ( « , * ) ! = Ac
(2.15)
The general solution to the above Eq. (2.15) assumes the form c(x,t) =
f(s)ext,
where s is again the initial condition to the ordinary differential Eq. (2.7). The "exact" time-stepping scheme for solving the transport problem (2.15), in the case of space-independent velocity field, is as follows C™+\x) - C " ( s " ) _ 6(A7) ~XC
xrm,.m) {X h
Here, the denominator function 6(At) has the more complicated expression 6(At) = (eXAt - 1)/A.
(2.16)
64
Nonstandard Methods for Advection-Diffusion-Reaction
Equations
For the case of general linear reaction terms r(c) = fi + Ac, the "exact" time-stepping scheme for solving the advective transport equation 9c , ,, dc _ + l,(M)_ =
/i + Ac
can be similarly derived to yield Cm+1{x)-Cm(xm) 6(At)
= (i +
\Cm(xm),
where the denominator function S(At) has the expression (2.16). Remark 2.3 When first order reactions are used in advective transport problems, there are two major differences between the implementation of standard finite difference schemes and the "exact" time-stepping scheme for solving the corresponding models — the first order reaction terms are mod eled explicitly at the backtrack points {xm} and the denominator function S(At) has the more complicated expression (2.16). 2.2.1.3
Nonlinear Reaction Terms
"Exact" time-stepping schemes exist for a variety of advective-reactive transport problems [27]. Here, we present the new discretization ideas applied to three groups of nonlinear reaction terms for which analytical solutions of Eq. (2.3) exist. First, consider the transport Eq. (2.3) with the following reaction term r(c) = \c + ncN, where N > 2 is a positive integer number, and A and \i are real constants, i.e.
m+l ° i + l
Ax'm + l
i+
fim+l °t
m + l _ £>m+l ^ t
m + 11
Ar AxT!"!"
i
*-*
Llz
Ax
s~tm+l °»-l m+1
i+l
where .m+l
D™Y = D
m+l +. X'i+1
,(m + l)A* ,
«+5
and Ax, m+1 + Ax™*1
are the hydrodynamic dispersion coefficient located at the center of a space increment and the arithmetic mean grid size Ax"*"*!1. Combining the semi-discrete procedure (2.26) with the above spatial approximation of the diffusion term yields the new non-standard method
Error Analysis of the Non-Standard
Method
69
for solving the logistic growth advection-diffusion-reaction Eq. (2.2) Cm+l
_ pm^m)
S(At) f~im+\ £)m+l ° i + l
Ax™+1 * '
*+i
_
sim+l ^t
ATm+1
s~*m+\
D\
s~tm+l
Ax, m + l
1
=
\Cm(x?)(l-C™+1),
(2.28) where C° = f(x°). Here, C t m + 1 denotes the grid values of the approximate solution at the point x™+l G -ym+1 at the advanced time level (m + l). In the case of space-dependent velocity field (2.10), the backtrack point x™ associated with the grid point x™+l G 7 m + 1 has the expression xtm = e- 0 [ p "« m+1 > At >- p »( mA ](x, m+1 + b/a) - b/a, and C m (iJ") i s the numerical solution at the point x™; for the moment we leave the definition of the advective concentration C m (x[") unspecified. 2.3
Error Analysis of the Non-Standard Method
We now analyze the new non-standard method. Since the error estimates in the analysis of the proposed new method are independent of the non linear reaction term present in the transport equation, we will discuss the convergence of the numerical solution only for the case of logistic growth reactions r(c) = Ac(l - c). It will be shown that large time steps can be taken without affecting the accuracy of the numerical solution. 2.3.1
Advection-Reaction
Equations
Convergence of the numerical solution to that of the advective transport Eq. (2.5) is demonstrated in two parts — with respect to the time variable t and with respect to the spatial variable x. It will be shown that the error in the numerical solution is equal only to the error introduced by the interpolation technique used to evaluate the advective concentration Cm(xm).
70
Nonstandard Methods for Advection-Diffusion-Reaction
2.3.1.1
Equations
Zero Local Time-Truncation Error
Rearranging terms in the "exact" time-stepping scheme (2.9) allows us to obtain the following expression Cm+l{x) _
Cm(e-»;Fn((m+l)Al)-P„(mM)](l +
~ e~
AAt
XAt
m
a p
m
l At
p
b/Q)
_
b/Q)
mAt
+ (1 - e- )C (e- l »U + ) )- "(
))(x
+ b/a) - b/a)' (2.29) m+1 The general solution C (x) - G(C°(x), At,m + 1) can be easily found by recursively applying the above relation (2.29), i.e. Cm+1(x) Cm (e-°[Pn((m+l)At)-P„(mAt))(l +
_ XAt
~ e~
+ (1 - e -
_ _
AAt
)C
m
_ tyQ)
(e-a[P„((m+i)At)-p„(mAt)](,I. + &/0) _ 6 / a )
Cl(e-a[P„((m+l)At)-P„(At)](x + e-AmAI +
6/g)
fe/Q)
_
fe/fl)
(J _ e - A m A t ) C - l ( e - a [ P „ ( ( m + l ) A t ) - P „ ( A t ) ] ( I + fc/a) _ cO^-aP^m+DAt)^
= e-A(m+l)A« +
+
6 / a )
_
fc/a)
b/Q)
(! _ e-A(m+l)A0(e-aPn((m+l)A0,
where £.m
=
e -.[P.((m+l)AI)-/'.<mAt)J( a .m + l +
fc/flj
_
6/fl
In this case, the advective concentration C m ( x m ) can be evaluated exactly, since we know the numerical solution at every grid point at the old time level m. So, for any i 6 { 0 , . . . , iV m+ i} there exists some ji G { 0 , . . . , Nm) such that the following equality holds Cm{x?)=Cm{x™)=C™. Similarly to the proof in Section 2.3.1.1, it can be shown that the rela tionship Cm+1
=
C-m+l^m+l)
+1
=
C(x™
, (m +
l)At)
is satisfied for every i € {0,... ,Nm+i}, every positive integer m, and for arbitrary time step size At. The key in proving the above equality is that at every time level, by applying the relation (2.29), we know the exact value of the advective concentration, since it coincides with the numerical solution at a grid point at the old time level. For general space-dependent velocity fields v(x, t), the assumption that each backtrack point at the future time step backtracks to a grid point at the previous time step is too restrictive to be of any practical interest. However, for the case of a space-independent velocity v = v(t), different time step sizes Atm can be taken to preserve the property of zero local truncation error in space even when uniform spatial grids are considered.
72
Nonstandard Methods for Advection-Diffusion-Reaction Equations
The time step size can be selected at each time level to guarantee that every grid point at the new time level is connected to a grid point at the old time level via a fluid particle path, also called a characteristic. For those cases in which such a time step size cannot be selected exactly, numerical methods for locating zeroes of functions, such as the Newton-Raphson iteration [3], can be used to "pull" each backtrack point x™ "sufficiently close" to a grid point x!£ at every time level m. E x a m p l e 2.1
Let us consider the following backtrack point
x? = zr +1 -[^,(< m+i )-p„(m =
iAx-[Pn(tm
+
Atm)-Pn(tm)},
where Ax is the uniform spatial grid size and tm = At° + At1 + --- + Atm~1 is the time after m time steps. To have the equality xj" = x™, where ji = int(t;(£ m+1 )), one needs to solve exactly or using some iterative methods the following nonlinear equation F(Atm)
= (i- ji)Ax - [Pn(tm + Atm) - Pn(tm)} = 0,
for the time step size Atm. 2.3.1.3
Interpolation Errors
In general, the backtrack points x™ do not lie at grid points xj 1 6 7 m , especially when uniform spatial grids are considered in the model. To eval uate the advective concentrations C m (x™), some type of interpolation of the approximate solution values {C™} should be used. Let us consider a piecewise-linear interpolation at each time level of the non-standard method (2.28) applied to the diffusion-free logistic growth Eq. (2.5). For a polynomial interpolation on a uniform spatial grid, the following lemma [3] holds: L e m m a 2.1 Suppose that f £ CN+l([0,1]), and let f be a polynomial of degree N that interpolates f on a grid 7 = {xo,x\,... ,x/v} on [0,1]. For any point x € [0,1], there exists a point C € (0,1) such that
f(x) - f{x) =
UN(*)/("+1>(0
(N + iy.
Error Analysis of the Non-Standard
73
Method
Here, /( N + 1 > is the (N + 1) derivative of / and u)N(x) := (x - x0){x - n ) . . . ( i -
xN).
Error estimates for the piecewise-linear Lagrange interpolation follow from the estimates for the corresponding global linear interpolation scheme. Lemma 2.1 applies, with N = I, for the piecewise-linear interpolant / on each grid element [x;,Xi+i]. In particular, if the interpolated function / € C 2 ([0,1]) and x 6 [XJ,XJ + I], then there is a point £ 6 (XJ,XJ+I) for which
M-M
= ^f^,
(2-33)
where U!i(x) -
(x-Xi)(x-Xi+l).
To estimate the interpolation error after m time-iterations of the nonstandard difference scheme {
{i} j(At)
= AC™(XD(I - cr+l),
(2.34)
let us consider a fixed uniform spatial grid 7 = {X0,XI,...,XN},
Xi - X i _ i
= Ax
>0.
Set j(i) = {j • I *? ~ Xi | = min | if k
Axi* = \z?-
x j ( i ) | < min
- xk |},
/Ax \ l—,KAt).
For the backtrack point -.m
=
e-a[Pn((m+l)AO-Pn(mAt)l(x.
+
h/a)
_
6/a>
the alternative if At-term in (2.36) arises when ( l _ e-a[Pn((m+l)At)-P.(mAt)l'\
A
+
&\
Ax < 2
(2.35)
(2.36)
74
Nonstandard Methods for Advection-Diffusion-Reaction Equations
Remark 2.6 In the case of a space-independent velocity field v = Pn-i (t), i.e., with a the backtrack point x™ = Xi - [Pn((m + l)At) the alternative KAt-teim
Pn(mAt)},
in (2.36) arises when Ax l)At)-Pn(mAt)\ IU min(Ax 4 , Ax 3 A*).
(2.40)
Error Analysis of the Non-Standard Method
77
Similarly to the case of piecewise-linear interpolation of the advective concentration Cm(x™), we can obtain the global error estimate for the non-standard difference scheme (2.34) with cubic spline interpolations || c(-,tm+l)
- C ^ O I L < TLCmin
Ax3) ,
( ^
where C is a positive constant, T is the final moment in time, and
Detailed discussion on common interpolation techniques and proofs of the aforesaid theoretical results can be found in [3]. Although cubic splines exhibit less of the "wiggly" behavior often asso ciated with high-degree polynomial interpolations, oscillation problems can arise when sharp fronts are present in the medium. Negative, non-physical concentrations can easily occur when the cubic spline interpolation under shoots concentration values near zero. Due to the unstable equilibrium of the advection-reaction Eq. (2.5), even small negative excursions in the concentration can grow rapidly to negative infinity. To prevent such an unstable behavior of the diffusion-free numerical solution, we consider partially irregular spatial grids that guarantee exact modeling of the most problematic spots — the front and the back of a sharp concentration distribution. We adjust the position of only those grid points at the new time level that are usually backtracked near the front and the back of a sharp distribution by connecting them to a nodal point at the old time level via a characteristic curve. By doing so, we actually perform a piecewise-cubic spline interpolation that prevents negative concentration values from appearing. 2.3.2
Advection-Diffusion-Reaction
Equations
Now, consider the new non-standard method (2.28) for solving the logistic growth advection-diffusion-reaction equation
dc
dc +V
d f~dc\ D
. =Xc{1 Cl
8-t d-x-Tx{ d-x)
-
on a fixed uniform spatial grid 7 = {x0,xi,...,xN},
Xi - Zi_i = A i > 0.
.
.
^^
78
Nonstandard Methods for Advection-Diffusion-Reaction
Equations
Error analysis of the new method for time-dependent, variable-spacing grids is similar to the uniform spatial grids case, and we shall not present it here. Denote the centered, weighted second difference in x of a grid function m + l by w 6t{Dm+l6xwm+1)i Dm+lw^+1
- (Dm+l + Dm_V) w™+i + X?™+1ii»™+1 Aia
(2.42)
where
Then, the non-standard method (2.28) on a uniform spatial grid 7 is given by the equation Cm+l
_
C™(x?)
6(&t)
- J a (Z? ra+1 «,C TO+1 ) 1 = \Cm(£?)(l
- Cf+1).
Note that for the exact solution c(x, t), with c™+1 = c(xi,tm+1) c™ = c ( x m , r m ) , the following relationship holds m+l _ gn
S(At)
i- -
fc(0m+1$ecm+1)i
= Ac7(l - c m + 1 ) + e m + 1 ,
(2.43) and
(2.44)
where e™+1 is the truncation error of the nonstandard method (2.43) at time-level (m + l). By definition Q m+1
m + l _ gm
5(&t)
Z-^(Xi,t^)+v(xut^)^(xut^
-[R(c?+\c?)-R(c'?+1,c?+1)}, (2.45) where for the function R(z\,Z2) — Az2(l — z\) we have that R € (^([0,1] x [0,1]), true for all nonlinear reaction terms discussed in Section 2.2.1. Using Taylor series expansions and the Mean Value Theorem, calculation shows
Error Analysis of the Non-Standard Method
79
that Q*cm+1
Ax2 (2.46)
+
OR O (c?+\-) dz-i (
At
where dc* /dr and d2c*/dr2 are some evaluations of the first and the second tangential derivatives along the characteristic segment between (xi,tm+1) m and (x™,t ), respectively. For the difference £ = c — C between the exact and the approximate solution, we have
cr+1 - cr S(At)
St(DSxCm+1)i
„m+l - i?(Cr"+ 1 ,C m (x; n )) + «"
= R(c?+1,c?)
(2.47) where C? = 0. Rearranging terms in the above equation, we have Cm+i =
£m
+1
+ S(At) [i?(c7 ,+1 ,cj n ) -
S{At)e™
+
fl(Cr+1,Cm(*JB))]
S(At) Ax'. 2
(2.48) For ease of exposition, we assume all functions in (2.41) are spatially fi-periodic. As stated in [17; 15], periodic boundary conditions are a rea sonable assumption because in general interior flow patterns are much more important than boundary effects. Suppose that .m+l
=
"
cm+l
_ Cm+l
"
=
*»
m m
/m+1 _
l
