Applications of nonstandard finite difference schemes

4> =

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<*+a)

e-i*hk

iw k+1

e~

^

> = 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)«2

= 0,

(2.43)

and Uk+i

- 1pUk

wk+i - ipwk

(2.44a)

Wk,

4>

-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-^)<-i,n-i

(2.87)

Exact

Schemes

17

A final example is a pair of equations modeling wave motion with spher­ ical symmetry [19]. The time-dependent and steady-state wave equations that possess spherical symmetry take the forms [20] d2(f>

/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

VW

is the exact scheme for the second equation in Eq. (2.91). The exact scheme for Eq. (2.89) is obtained by replacing wm in Eq. (2.94) by rmipm. After some algebraic manipulation the following result is obtained

V>m+i -2V> m +y> m _i / 2\ / V> m+ i-V> m -i \ 2 + Btaa 4 E (^Sin (^) UA(*)(*) ( f );

2

_ (2.95)

Nonstandard Finite Difference Schemes

18

Now the first of Eq. (2.91) is the 1-dim wave equation. Its exact finite difference scheme is [2] u" + 1 - 2u" + u " _ 1 (At) 2

*m+l

*um ^ " m - 1

(2.96)

(Ar) 2

and the step-sizes satisfy the relation At = Ar.

(2.97)

Substituting uJJ, = r m ^ into Eq. (2.96) and rearranging the various terms gives the exact scheme for Eq. (2.88) n-1 / 2 \ {4&+i - K-i \ _ 0m+1 " 2 C + 4>\ \rmJ \ 2Ar ) (At)2 (2.98) A direct calculation shows the unidirectional wave equation having spherical symmetry du u du

d

-

^
^ + C-i 2 (Ar)

x

m+r-+8r-=°>

( 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


where %l> and <j> depend on the step-size, h = At, and satisfy the conditions V>(/i) = l + 0(/i 2 ),

(2.102)


(2.103)

Nonstandard Schemes

19

These functions may also depend explicitly on various parameters which appear in the differential equations. Note that these considerations lead to the following generalization of the usual definition of the derivative [23] dn=_L^u[t dt h-*>

+

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—•

<">

Nonstandard

Schemes

21

gives "A+l ~ W*-l

2/i

-u*.

(3.2)

All solutions of Eq. (3.1) decrease in magnitude smoothly to zero. The general solution of Eq. (3.2) is [14] uk=A(r1)k

+ B(r2)k,

(3.3)

where A and B are arbitrary constants, and ft2-

ri, a = -ft ± \ / l +

(3.4)

The ri,2 are functions of ft and have the following properties: r_(ft)<-l,

forO
r_(ft) = -2ft + O Q J ,

0
r+(/l) =

for ft large,

forO
2^+° ih)'

for

* large-

(3.5a)

(3.5b)

(3.5c)

(3 5d)

-

As a consequence of these properties, it follows that the second term of the right-side of Eq. (3.3) oscillates with an amplitude that exponentially increases with k, while the first term decreases monotonically to zero. Since this behavior holds for any step-size, ft > 0, the central difference scheme for the decay equation has numerical instabilities regardless of the value of ft. The first term models the actual solution behavior of the decay equation, while the second term is an "extra solution" which is created by the use of a second-order discrete approximation to the first-order derivative appearing in the differential equation. Using a central difference scheme for a nonlinear first-order ODE may not lead always to unbounded solutions, however, numerical instabilities will exist [2]. Rule 2. Denominator functions for the discrete derivatives must, in gen­ eral, be expressed in terms of more complicated functions of the step-sizes than those conventionally used.

22

Nonstandard Finite Difference Schemes

Comment 2. The discrete first derivative takes the form du uk+i - il>uk * -* * '

(36)

where rp and 4> 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)


such that the (linear) stability properties of the fixed-points in the difference equation are exactly the same as those of the differential equation. It has been shown [24] that this can be done. Let Eq. (4.2) have /-real solutions, where / may be unbounded. Define Ri to be du

(4.4)

u=fi(')

where (u^'; i = 1,2,...,/} is the set of fixed-points, and R* as R* =max{\Ri\;i

= l,2,...,I}.

(4.5)

A particular <> / that can be used is [22; 24] 1_e-R-h

4,(h,R') = - ^

,

(4.6)

which has the properties (h, Rm) = h + 0(R*h2), O<0(/i,iT)<-^.

(4.7a) (4.7b)

The result of Eq. (4.6) can be given a physical interpretation which leads to a deep fundamental understanding of what it represents. For a dynamical system where t is the time, the Ri have units of inverse time. Therefore, a set of time scales can be defined by means of the relations

Ti

= i~'

(4 8)

-

where T* is given by

T' = ±>

(«)

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<(h,T')
(4.11)

In this formulation, the denominator function can be interpreted as a "renormalized" or "rescaled" step-size such that its value is never larger than the smallest time scale of the system. Since many of the mechanisms that give rise to numerical instabilities have their genesis in using a step-size that is larger than some relevant physical time scale, the above method of choosing <j> 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<2> = 0,

u<3> = 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[uk + (uky}-

(4 20)

-

Using the fact that 4>(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<3) = 1 when k -t oo. This behavior is exactly the same as that of the corresponding solutions of Eq. (4.12). Further, the nonstandard scheme of Eqs. (4.19) has no numerical instabilities for any value of the step-size h. However, accurate numerical solutions require that 0 < h <3C 1; but, for any h > 0, the qualitative properties of the solutions to this scheme will be exactly the same as those of the differential equation. Additional applications of this method for constructing nonstandard finite difference schemes appear in Mickens [26] where the Logistic, sine, Monod, and cubic equations are investigated. 1.4.2

A Photoconduction

Model

The charge carriers in intrinsic semiconductors can exhibit complex behav­ ior in their kinetics, particularly when photoconductivity is dominated by the presence of one or more types of traps and/or recombination centers caused by defects or impurities in the material. For the case of a single

Applications

27

type of trap close to the conduction band such that the ejected electrons are thermal, the photoconducting process is modeled by the following three nonlinear, coupled differential equations [27] — = G — ain(Nt — m) + jim — C\n, at

(4.22a)

—— = cnn(Nt — m) — Somp — 71m, at

(4.22b)

~!r = G- 60mp - c2p. (4.22c) at The n, m and p represent, respectively, the electron, trapped electron, and hole population densities. The process is described by the seven parameters (G, a\, Nt, 71, c\, C2, SQ) whose values are known [27]. Initial work by Serfaty de Markus [27] indicated that a fixed step fourth-order Runge-Kutta (RK) scheme produced numerical solutions with chaotic behavior. It was also found that adaptive integration techniques appropriate for a stiff system, such as a Gear or Moulton- Adams scheme removed this complex behavior yielding the expected smooth well-behaved curves for n(t), m{t), and p(t) versus time. Additional investigations by Serfaty de Markus and Mickens [28] showed that the use of a direct nonstandard scheme produced numerical solutions with the proper dynamical behavior expected from physical considerations of the photoconducting system. This discrete model used a forward Euler representation for the discrete derivative du dt

uk+i - uk 4>

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><*i mk)nk l + «aiNt + ei)

(4.25c)

they can be solved for + (£71 mk '

(4 26a)

-

l + <j>(aink +<)oPfc+7i)

1 + <j>(S0mk + c 2 ) A direct calculation shows that the fixed-points of Eqs. (4.26) are the same as those of the ODE's in Eqs. (4.22). Further, they have the same linear stability properties. Both of these results hold for any value of the step-size h. This problem is very stiff. Using the published values [27] of the seven parameters in Eq. (4.22), it turns out that the photoconducting system has a large number of time scales whose values were in the range l s e c < T * < 10 5 sec.

(4.27)

The smallest scale was determined by 71; consequently, T in the denomi­ nator function, Eq. (4.24), was chosen to be [28] T = —.

(4.28)

A large number of numerical integrations of Eqs. (4.22) using Eqs. (4.26) showed that the use of this nonstandard forward Euler scheme produced numerically stable and well behaved solutions, despite the considerable stiff­ ness of the system. As a check on these results, several standard and a

Applications

29

nonstandard central difference representation were used. For these discrete representations, numerical instabilities arose. The major conclusion of this work was that simple nonstandard discretizations can be used to model certain types of coupled, nonlinear, first-order systems of ODE's. 1.4.3

The Duffing

Oscillator

A technique for numerically integrating oscillatory equations can be con­ structed by using exponential methods [29]. Consider the nonlinear, secondorder ODE #u w

= = // (( «« )) ,,

du(0) ™~j= « i , t

u(0)=ti o> «(0) = «o,

(4-29)

where, in general, f(u) is a nonlinear polynomial. The exponential or cosine method uses the following discrete equation [29] uk+i -2uk

+ uk-i

Kty

= f{uk)

>

(4 30)

-

where ,, . cos-i/^z-1 V>* = 77\ « (§)

,,df 17, ■ du u=uk

z = h

,,„..* - )

4 31

Note that since cosv'^-l

14\

. i

,

(y/—z\

—JIJ——u;- (—)

(4.32)

Eq. (4.30) can be written as Ufc+l -2uk

+ U*_i

/ r=^ (^)sin'(^)

= -/(«*)•

( 4 - 33 )

For the Duffing equation [30], Eq. (4.29) is

£? = -«-«'.

(4.34)

where e is a real parameter. A variety of perturbation methods have been applied to Dumng's equation to calculate the small amplitude periodic so­ lutions [31]. This equation can also be solved exactly in terms of Jacobi

30

Nonstandard Finite Difference Schemes

elliptic functions [31]. A "standard" finite difference scheme for Duffing equation, using the result of Eq. (4.33), gives / ( u ) = —u — eu 3 ,

du

= -l-3eu2,

(4.35a)

z = -h2{l + 3eul), ujt+i - 2uk + Ufc_i s 2

T

(4.35b)

(4.35c)

+ ufc + tu\ - 0.

(i+fc?) - [(f) v*+&=z The above scheme can be combined with a nonstandard modification to yield the following discrete form of Duffing's equation

(ITl^) sin2 [(f)^^ T ^]

H

2

;

The major difference between this scheme and that of Eq. (4.35c) is that u 3 was replaced by the nonlocal expression 2 / u * + i + w*-i "*( 2 )

(4.37)

For conservative oscillators, this preserves invariance under time reversal [32; 33; 34; 35]. An interesting special case of the Duffing differential equation is

£u + u 3 = 0.

(4.38)

dt2

Since no small parameter occurs in the equation and also because there is no linear term in u, perturbation methods cannot be used to study the pe­ riodic solutions. However, the method of harmonic balance can be applied and very good analytical approximations to the periodic solutions can be calculated [33]. A nonstandard scheme for Eq. (4.38) is Ufc+i - 2ufe -I- Uk-i sin 2 (/t^/Jufc)

2 /"*+! +«*-!

+ M " " " o " " •)

"V

2

0.

(4.39)

Applications

1.4.4

Mixed Parity

31

Oscillator

A single degree-of-freedom, conservative system can be described by a single coordinate, x(t) and an energy function [8; 9] .o 7YIX

E(x,x) = —z- + V(x) = constant,

(4.40)

where m is the mass, V(x) is the potential energy, and x = dx/dt. The equation of motion can be obtained by taking the total time derivative of E(x, x). Doing this gives dt

.. dV mx + —— x = 0, dx

(4.41)

or dV mx + — = 0.

(4.42)

The last equation is precisely what is obtained using either the Hamiltonian or Lagrangian approaches to classical mechanics [8; 9]. Note that both Eqs. (4.40) and (4.42) are invariant under the transformations t -t —t (time reversal), t —¥ t + to

(time translation),

(4.43) (4.44)

where Eq. (4.44) implies and is implied by the conservation of energy prin­ ciple. In general, the discrete modeling process begins not with an energy func­ tion, but with the differential equations of motion [5; 6]. Proceeding this way generally gives difference equations that do not have a first-integral. Since the original differential equations do satisfy a conservation law, i.e., have a first-integral, it is desirable to have this feature built into the discrete scheme, otherwise numerical instabilities will arise. One way to do this is to apply the nonstandard modeling rules directly to the energy function. For the case of a single degree-of-freedom system, the following steps can be implemented: (i) Replace (x, x) by (x,x) -> [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 <-► k — 1, i.e., E{xk,Xk-\)

=

E(xk-i,xk).

(4.47)

This constraint requires the nonlinear terms in the differential equa­ tion to be replaced by nonlocal representations in the finite differ­ ence model. It should be indicated that Eq. (4.47) is the discrete analogue of the invariance of the energy function under the opera­ tion of time reversal. (iii) The equation of motion is obtained by applying the A operator to E(xk,xk-i) and equating this to zero, i.e., AE(xk,xk-i)

(4.48)

= 0.

To illustrate these procedures and show what occurs when they are not followed, consider the linear harmonic oscillator equation x + w x = 0.

(4.49)

The energy function is

«**>-(£)<*>" + (£)

(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 [
, +

fXk+l + Xk 2 U Xk +a I 3

+

P*l(Xk+l+2Xk~1)=0.

+Xk-l\

) Xk (4.58)

Note that this is linear in Xk+i, consequently, the finite-difference scheme is really explicit. This means that Xk+\ can be directly expressed as some function of Xk and Xk-i, i.e., xk+i =F(xk,xk-i),

(4.59)

where F(xk,Xk-i) can be found by solving Eq. (4.58) for Xk+i. Also, ob­ serve that all the nonlinear terms in the differential equation are represented nonlocally in this difference equation.

Nonstandard Finite Difference Schemes

34

The procedure discussed above can be generalized to the case of Ncoupled, nonlinear oscillators. This work is reported in Mickens [35]. 1.4.5

A Cubic Reaction

Problem

in

Neurophysiology

Price, et al. [36] used an explicit second-order accurate finite difference scheme for the initial value problem [37] du — =au(l-u)(u-a), at

u(0) = uo,

(4.60)

where a and a are parameters, to construct an implicit scheme for the PDE ^- = H^+au{\-u){u-a),

(4.61)

where 0 > 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
(4.87)

Now the nonlinear, first-order difference equation ykVk-i + AkVk + BkVk-i = Ck,

(4.88)

Applications

39

where (AktBk,Ck) are known functions of A;, can be solved exactly [14]. Equation (4.85) has this form with the following identifications Ak = -(j^j,

Bk = ^ - ^ ,

Ck = -2A.

(4.89)

The values for the wave speed c and the integration constant A may be determined from Eq. (4.85) by requiring its solution yk to satisfy the conditions Lim yk = ui, k—*oo

Lim yk = u 2 .

(4.90)

k—» — oo

See Eq. (4.80b). Since ui and u 2 must be fixed-points of Eq. (4.85), it follows that f - 2cy + 2A = 0,

(4.91)

y2 - ( u i +u2)y + uxu2 = 0.

(4.92)

where Eq. (4.90) implies

Comparison of the last two equations gives H i ± ^

c =

A =

^l_

(4.93,

In summary, the asymptotic values of yk determine both the traveling wave velocity and the integration constant A. Note that c is exactly that given by the ODE. Following the procedure of Mickens [14, Section 6.3], the following ex­ pression is obtained for the solution to Eq. (4.85)

where 1 —Ui0

A=

r+(

1 — U2

B =

r_

lz**f

(4.95)


and £> is an arbitrary constant. The requirements r+ > 0,

r_ > 0,

u2>«i,

(4.97)

40

Nonstandard Finite Difference Schemes

force (/i) to satisfy the constraint 4>(h) <—, 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

(4.102)

and this implies 0
(4.103)

Putting all this together gives for yk 111 =

u2Xk + ui : A* + l '

(4.104)

This is the traveling wave solution to the discrete model given by Eq. (4.85). An easy calculation shows that taking the limits h->0,

k -> oo,

hk = | =

fixed,

(4.105)

gives Vk -» /(«)■

(4.106)

The nonstandard scheme for the second-order equation determining the traveling wave solutions, i.e., Eq. (4.80), can be gotten by applying the

Applications

41

difference operator, A, to Eq. (4.85). T h e resulting equation is [47] Vk+i ~ 2yk + Vk-i

_

Vk+l - Uk-l 2ip{h)

Vk

24>(h)1>(h)

~

— c

Vk ~ Vk-i

(4.107)

where ip(h) has the property 0{h2).

t/>(/i) = 2h +

1.4.8

Linear

Advection-Diffusion

(4.108)

Equation

A reasonable linear approximation to many phenomena in acoustics and fluid dynamics is the mathematical model given by the P D E , in normalized form [5; 7], ut + ux =

buzx,

(4.109)

where the constant b is positive. This equation has both linear advection and diffusion. In general, the independent variable is such t h a t it must be non-negative. This positivity property will be used later t o obtain a functional relationship between the step-sizes At and Ax, where xm = (Ax)m,

tk = (At)k,

t 4 - «(xm,*i).

(4.110)

To begin, note t h a t Eq. (4.109) has the following three sub-equations [48] ut+ux

= 0,

ux = buxx,

ut = buxx.

(4-111)

T h e first two equations have known exact finite difference models; they are [2] uk+\

_

k

At

+

Mm ~ " m - l

Aar

tc

= b

- u :m - l = 0, Ax

Ax = At,

"m+l - 2«m + " m - l 6( e A*/6 _ l ) A i

(4.112)

(4.113)

A finite difference scheme t h a t incorporates the features of both of the sub-equations for Eq. (4.109) is M-i _ "m

At

uk u

m

uk ,, - 2uk + uk t C - u :m - l = b + Ai ft(eAx/6 _ 1 ) A a .

(4.114)

42

Nonstandard Finite Difference Schemes

with Arc = At. Solving for u j ^ 1 gives <4 +1 = # 4 + 1 + (1 - a - 2 / J ) < + (a + / J ) < _ i ,

(4.115)

where At a = Ax '

„

a

0 = e^/6 - 1' ^

(4.116)

The positivity condition, ujj, > 0 => u j ^ 1 > 0, is enforced if 1 - a - 2/3 > 0,

(4.117)

or e Ax/6

At <

_ j

e Ai/6 +

j

Ax.

(4.118)

Note that when 6 = 0, this relation reduces to At < Ax. However, for this case, Eq. (4.114) reduces to Eq. (4.112) and At = Ax. Thus, there is obtained a functional relation between the step-sizes given by At =

e Ai/6

_

1

eAx/b

+

i

Ax.

(4.119)

For small Ax, Eq. (4.118) becomes At <

(Ax) 2 26

(4.120)

This is the relation that generally applies to the diffusion equation [3; 5; 6]. Also, the condition given in Eq. (4.117) forces the solutions u£, to satisfy a max.-min. condition [6]. 1.4.9

A Combustion

Model

The following diffusion PDE with a nonlinear reaction term ut = uxx + u 2 ( l -u),

(4.121)

provides a model for combustion in 1-dim [25]. Physical meaningful solu­ tions have the property that 0 < u(x,0) < 1 =^ 0 < u(x,t) < 1.

(4.122)

Applications

43

A possible nonstandard scheme for Eq. (4.121) is =

At

"m+l + «m-l

( « ^ + i ) 2 + («S,_i) a ' «

+

i )

2

+

« - i )

2

,A+1

„*+l

" m - f l - 2M m + Mro-1

(4.123)

(Az) 2

Below it will be shown that this scheme preserves the positivity condition given by Eq. (4.122). The scheme of Eq. (4.123) was constructed by making the following discrete replacements in Eq. (4.121) U2 =

2u

2 _

u

2 ^

{ u l + i f

+

{ u l i ?

«+l)2

+

_ ^km+l+<-^

«-l?

uL+1>

,fc+l

(4 J 2 4 )

(4.125)

One way to reason why these structural forms are used is to realize that Eq. (4.121) is invariant under x -> —x. The discrete analogue is to have the difference equation invariant under (fc 4- 1) ** (fc — 1); see references [34; 35]. Equation (4.123) is linear in ujj+x and by solving for it, the following expression is obtained R(ukm+1 + «*,_!) + At[(nkm+1)2 + « _ x ) 2 ] + (1 - 2R)ukm m

1 + ( f ) K + i + 04 + i) 2 + < - i + « - i ) 2 ]

(4.126)

where R

At (Ax) 2 "

(4.127)

Note that if u^ is non-negative, then u^1"1 will also be non-negative if 1 - 2R > 0.

(4.128)

Nonstandard Finite Difference Schemes

44

The selection of the equality gives the following functional relation between the step-sizes (Ax)2 At = ¥=2- .

(4.129)

Thus, a nonstandard finite difference scheme for Eq. (4.121) is k+l

=

(5)

fel

+ U ^ - l ) + ^ [(»m + l) 2 + « - l ) 2 ]

(4.130)

provided the condition of Eq. (4.129) holds. Let 0 < um < 1, for all m and fixed Jfc. Then it follows from Eq. (4.130) that 0 < u^1"1 < 1, for all m and fixed k. The proof of this result makes use of the following fact: for w real, 0<wtu2< —-—.

(4.131)

It should be clear that the particular discrete forms for the nonlinear terms, given in Eqs. (4.124) and (4.125), were selected to insure that the positivity condition was satisfied by the solutions of the difference equation. Similar types of analyses have been applied to other nonlinear, reaction-diffusion, PDE's [49; 50]. These equations model 1-dini phenomena and can be rep­ resented as follows ut = Duxx + / ( u ) ,

(4.132)

where u(x, t) = [ui(x, t),u 2 (x, <),... ,UAT(X, t)]1, D is an N x N diagonal matrix of constant diffusion coefficients, and /(u) is a column matrix of nonlinear reaction terms. 1.4.10

Influence

of Spatial Discretizations

for

PDE's

Nonstandard finite difference schemes provide the possibility of constructing discrete models of PDE's for which the elementary numerical instabilities do not occur. This section summarizes the results of studies that have been done on how particular space discretizations can influence the occurrence or not of numerical instabilities [51; 52; 53]. The equation investigated is a modified Korteweg de Vries PDE ut + uux + euuxxx = 0,

(4.133)

Applications

45

where e is a positive parameter. Equation (4.133) has a special solution that can be calculated using the method of separation of variable, i.e., let u(x,t) = f(x)g(t),

(4.134)

then f(x) and g(t) satisfy the ordinary differential equations g'+cg2=0,

£/'" + / = c,

(4.135)

where c is the separation constant and the primes denote differentiation with respect to x for the f(x) equation, and with respect to t for the g(x) equation. The solutions to these equations are f(x) = ei cos (^)

+ C 2 S i n

9(t) =

(^)

+ cx + e3

1 ct + e4 '

(4.136)

(4.137)

where (ei,e2,e3,e,j) are arbitrary integration constants. Thus, the separa­ tion of variables (SOV) solution is d cos ( ^ . j + e2 sin (-^ j +cx + e3 u{x,t) —

(4.138)

ct + e4

The central idea, in the calculations to follow, is to study the SOV solutions to nonstandard discrete models of Eq. (4.133) and compare them to the result of Eq. (4.138). If they do not agree, in at least their qualitative properties, then strong a priori evidence is provided for the existence of numerical instabilities in the finite difference schemes [2]. Two nonstandard models are considered: Model A uk+i

_

At +eu

k

+u t+i ' "

U

fc+i

u~

- i tm - l di

m+2-3Mm+l+3<-U^_1

d3

Model B k

At

m

k m-l

di

= 0;

(4.139)

46

Nonstandard Finite Difference Schemes

+eu* + 1

"m+l ~ 3»m + 3 < _ ! - < _ 2 d3

= 0.

(4.140)

In these equations d\ and d2 are, for the moment, unspecified denominator functions with the following properties di = h+ 0(h2),

d3 = h? + 0(/i 4 ),

h = Ax.

(4.141)

However, previous work on the Burgers' equation [2] shows that dx = h = Ax.

(4.142)

Note that Models A and B differ only in how they represent the third-order space derivative: forward 3rd-order scheme backward 3rd-order scheme

uxxx -»

, u* m+a

u IXX -►

u * ,. m+1

+ .

«* , — ,

+ . . . - u* , ,, =* .

(4.143a)

(4.143b)

The SOV method can also be applied to partial difference equations [14]. For this case, um is written as ukm = fmgk.

(4.144)

Substitution of this into Model A gives the following equations satisfied by fm and gk 9k+i ~9k = (aAt)gk+lgk,

/m+2 - 3/m+1 + f 3 + ^

fm - (l + ^j

(4.145)

fm-i = a,

(4.146)

where a denotes the separation constant. Equation (4.145) has the solu­ tion [2; 14] 9k = -r±—, ** = (At)*, (4-147) ctk + e4 where the integration constant is taken to be e\ and a is set equal to c. With this notation, it follows that gk=g(tk),

(4.148)

Applications

47

where g(t) is the expression in Eq. (4.137). The solution to Eq. (4.146) is determined by the characteristic equa­ tion [14]

A'-3A' + ( 3+ ^)A-( 1 + A) = 0 ,

(4 . 149)

where fm is a linear combination of terms having the form [14] fm ~ Am.

(4.150)

Examination of Eq. (4.149) shows that Ai = 1. Thus, A2 and A3 are solu­ tions to the equation 2A

(>+£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 -

<4-154)

The corresponding characteristic equation can be written in the form (A - 1) A2 - (2 - ^

A+1 =0.

(4.155)

Nonstandard Finite Difference Schemes

48

To fix dz write the second term in the brackets as 1 - § = cos ( - ^ eh

.

(4.156)

Solving for d3 gives d3 = (4e/i) sin2 ( ^ ) .

(4.157)

Consequently, the three roots of the characteristic equation are Ai = 1,

. {%)■

A2 = A; = exp [-=

(4.158)

Choosing the coefficients properly to help compare the results with fm has the form fm = ei cos I -j= J + e 2 sin I - ^ J + cxm + e 3 ,

f(x),

(4.159)

and it follows that fm = /(arm).

(4-160)

Based on these calculations we find that the Model B nonstandard finite difference scheme has a SOV special solution which is exactly equal to the corresponding SOV solution of the modified Korteweg de Vries PDE. Since the only difference between the Models A and B was how the third spatial derivatives were represented, the conclusion is that in the construc­ tion of nonstandard schemes attention must be paid as to how the spatial discretizations are made. Other work on nonstandard schemes are in agree­ ment with this result [2]. While this type of procedure, comparison of special solutions, does not insure the absence of numerical instabilities, any scheme that gives special solutions having properties not shared by the original differential equa­ tions, will certainly have instabilities. Finally, it should be indicated that the above analysis does not provide for restrictions on the time step-size relative to the space step-size. However, based on general considerations, it is expected that At < C ( A i ) 3 , where C is a constant.

(4.161)

Future Directions

1.5

49

Future Directions

This chapter has provided a broad introduction to the general philosophy behind nonstandard finite difference schemes and illustrated, by way of various examples, how the at present rather primitive modeling rules can be applied to nontrivial differential equations. It is clear from the presentation that the particular nonstandard scheme to be used for a differential equation will depend on what properties of its solutions are desired to be reproduced in the resulting discrete model. It therefore follows that before any attempt be made to construct a nonstandard scheme a full as possible knowledge of the properties of the solutions to the differential equation be obtained. There are a number of fundamental issues that need to be resolved. The first is to place the currently known nonstandard modeling rules on a firm mathematical basis, i.e., derive them from "first principles." Second, it is important to see if any additional rules exist. Third, the role of special conditions such as positivity needs to be closely examined; in particular, its role in eliminating certain types of numerical instabilities. The same considerations apply to systems for which first integrals exist. Two research areas in which nonstandard methods have not made sig­ nificant inroads are its application to PDE's that have either nonlinear advection or nonlinear diffusion terms. Many systems in the biological and chemical sciences are described by equations having such features [38; 54]. Another research topic is related to the numerical integration of or­ dinary differential equations where the periodic solutions have neutral sta­ bility. A prime example of such a set of equations are the Lotka-Volterra differential equations [38; 55]

— =x-xy,

-r = -y + xy.

(5.1)

Even with nonstandard techniques, it is very difficult to construct finite difference schemes that give the proper periodic solutions behavior about the fixed-point at (x, y) = (0,0); see, for example the paper of Sanz-Serna [56]. Based on past experiences, the future appears to hold great promise for both understanding nonstandard methods and in their application to significant problems.

50

Nonstandard Finite Difference Schemes

Acknowledgments The research reported here was supported in part by grants from DOE and the MBRS-SCORE Program at Clark Atlanta University.

Bibliography

51

Bibliography

[I] R. E. Mickens, "Exact solutions to a finite difference model of a nonlinear reaction-advection equation: Implications for numerical analysis," Numer­ ical Methods for Partial Differential Equations 5 (1989), 313-325. [2] R. E. Mickens, Nonstandard Finite Difference Models of Differentials (World Scientific, Singapore, 1994). [3] F. B. Hildebrand, Finite-Difference Equations and Simulations (Prentice-Hall, Englewood Cliffs, NJ, 1968). [4] J. M. Ortega and W. G. Poole, Jr., An Introduction to Numerical Methods for Differential Equations (Pitman, Marshfield, MA, 1981). [5] M. B. Allen III, I. Herrera, and G. F. Pinder, Numerical Modeling in Science and Engineering (Wiley-Interscience, New York, 1988). [6] D. Greenspan and V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering (Addison-Wesley, Redwood City, CA, 1988). [7] D. Potter, Computational Physics (Wiley-Interscience, New York, 1977). [8] L. A. Pars, A Treatise on Analytical Dynamics (Ox Bow Press, Woodbridge, CT, 1979). [9] S. B. Palmer and M. S. Rogalski, Advanced University Physics (Gordon and Breach, New York, 1996). [10] E. S. Oran and J. P. Boris, Numerical Simulation of Reactive Flow (Elsevier, New York, 1987). [II] R. E. Mickens, "Pitfalls in the numerical integration of differential equa­ tions," in W. E. Collins et al. (editors), Analytical Techniques for Material Characterization (World Scientific, Singapore, 1987), pp. 123-143. [12] V. V. Nemytski and V. V. Stepanov, Qualitative, Theory of Differential Equa­ tions (Princeton University Press, Princeton, NJ, 1969). [13] J. A. Murdock, Perturbations: Theory and Methods (Wiley-Interscience, New York, 1991), Appendix D. [14] R. E. Mickens, Difference Equations: Theory and Applications (Chapman and Hall, New York, 1990).

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[15] S. L. Ross, Differential Equations (Xerox, Lexington, MA, 1974, 2nd edition). [16] J. David Logan, Nonlinear Differential Equations (Wiley-Interscience, New York, 1994). [17] E. Zauderer, Partial Differential Equations of Applied Mathematics (WileyInterscience, New York, 1983). [18] R. E. Mickens, "Exact finite-difference schemes for two-dimensional advection equations," Journal of Sound and Vibration 207 (1997), 426-428. [19] R. E. Mickens, "Exact finite difference schemes for the wave equation with spherical symmetry," Journal of Difference Equations and Applications 2 (1996), 263-269. [20] J. Irving and N. Mullineux, Mathematics in Physics and Engineering (Aca­ demic Press, New York, 1959). [21] R. E. Mickens, "Exact finite difference scheme for a spherical wave equation," Journal of Sound and Vibration 182 (1995), 342-344. [22] R. E. Mickens and A. Smith, "Finite difference models of ordinary differential equations: Influence of denominator functions," Journal of the Franklin Institute 327 (1990), 143-145. [23] J. Marsden and A. Weinstein, Calculus I (Springer-Verlag, New York, 2nd edition, 1985), Section 1.3. [24] R. E. Mickens, "Finite-difference schemes having the correct linear stability properties for all finite step-sizes II," Dynamic Systems and Applications 1 (1992), 329-340. [25] R. E. O'Malley, Jr., Thinking About Ordinary Differential Equations (Cam­ bridge University Press, New York, 1997), Section 6.5. [26] R. E. Mickens, "Discretization of nonlinear differential equations using ex­ plicit nonstandard methods," Journal of Computational and Applied Math­ ematics 110 (1999), 181-185. [27] A. Serfaty de Markus, "Numerical crisis found in the fixed step integration of a photoconductor model," Physical Review E 56 (1997), 88-93. [28] A. Serfaty de Markus and R. E. Mickens, "Suppression of numerically in­ duced chaos with nonstandard finite difference schemes," Journal of Com­ putational and Applied Mathematics 106 (1999), 317-324. [29] M. Hochbruck, C. Lubich, and H. Selhofer, "Exponential integrators for large systems of differential equations," SIAM Journal of Scientific Computing 19 (1998), 1552-1574. [30] G. Duffing, Erzwungene Schwingungen bei veraderlicher Eigenfrequenz (Vieweg, Braunschweig, 1918). [31] R. E. Mickens, Nonlinear Oscillations (Cambridge University Press, 1981). [32] R. E. Mickens, "Construction of a finite difference scheme that exactly con­ serves energy for a mixed parity oscillator," Journal of Sound and Vibration 172 (1994), 142-145. [33] R. E. Mickens, Oscillations in Planar Dynamic Systems (World Scientific, Singapore, 1996a), see Chapter 4. [34] R. E. Mickens, "Properties of finite difference models of nonlinear conservative

Bibliography

53

oscillators," Journal of Sound and Vibration 124 (1988), 194-198. [35] R. E. Mickens, "Construction of finite difference schemes for coupled nonlin­ ear oscillators derived from a discrete energy function," Journal of Differ­ ence Equations and Applications 2 (1996b), 185-193. [36] W. G. Price, Y. Wang, and E. H. Twizell, "A second-order, chaos-free, ex­ plicit method for the numerical solution of a cubic reaction problem in neurophysiology," Numerical Methods for Partial Differential Equations 9 (1993), 213-225. [37] E. H. Twizell, Y. Wang, and W. G. Price, "Chaos-free, numerical solutions of reaction-diffusion equations," Proceedings of the Royal Society of London, Series A 430 (1990), 541-564. [38] J. D. Murray, Mathematical Biology (Springer-Verlag, Berlin, 1989). [39] R. E. Mickens, "Comments on 'A second-order, chaos-free, explicit method for the numerical solution of a cubic reaction problem in neurophysiology'," Numerical Methods for Partial Differential Equations 10 (1994), 587-590. [40] H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics (Cambridge University Press, London, 3rd edition, 1956). [41] E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961). [42] R. E. Mickens and I. Ramadhani, "Finite-difference scheme for the numerical solution of the Schrodinger equation," Physical Review A 45 (1992), 20742075. [43] R. Chen, Z. Xu, and L. Sun, "Finite-difference scheme to solve Schrodinger equations," Physical Review E 47 (1993), 3799-3802. [44] S. E. Koonin, Computational Physics (Addison-Wesley, Redwood City, CA, 1986), see Section 3.1. [45] R. E. Mickens, "Asymptotic properties of solutions to discrete Coulomb equa­ tions," Computers in Mathematics and Applications 36 (1998a), 285-289. [46] R. E. Mickens, "Asymptotic properties of solutions to discrete Airy equa­ tions," Journal of Difference Equations and Applications 3 (1998b), 231239. [47] R. E. Mickens, "Exact solutions to difference equation models of Burgers equation," Numerical Methods for Partial Differential Equations 2 (1986), 213-218. [48] R. E. Mickens, "Analysis of a new finite-difference scheme for the linear advection-diffusion equation," Journal of Sound and Vibration 146 (1991), 342-344. [49] R. E. Mickens, "Relation between the time and space step-sizes in nonstandard finite-difference schemes for the Fisher equation," Numerical Methods for Partial Differential Equations 13 (1997), 51-55. [50] R. E. Mickens, "Nonstandard finite difference schemes for reaction-diffusion equations," Numerical Methods for Partial Differential Equations 15 (1999), 201-214. [51] S. Roy Choudhury, "Some results on the stability and dynamics of finite difference approximations to nonlinear partial differential equations," Nu-

54

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merical Methods for Partial Differential Equations 9 (1993), 113-117. [52] S. Roy Choudhury and J. Neal, "Characterization of true versus numeri­ cally generated chaos via Lyapunov exponent calculations," International Journal of Applied Science and Computation 5 (1998), 156-178. [53] R. E. Mickens, "Influence of spatial discretizations on nonstandard finite difference schemes for nonlinear PDE's," International Journal of Applied Science and Computation (accepted for publication). [54] J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, New York, 1983). [55] L. Edelstein-Keshet, Mathematical Models in Biology (McGraw-Hill, New York, 1987). [56] J. M. Sanz-Serna, "An unconventional symplectic integrator of W. Kahan," Applied Numerical Mathematics 16 (1994), 245-250.

Chapter 2

Nonstandard Methods for Advection-Diffusion- Reaction Equations

Hristo V. Kojouharov* and Benito M. Chen*

2.1

Introduction

Advection-difFusion-reaction equations are among the most widespread in many fields of engineering and science [32; 33]. Mathematical models that involve a combination of advective and reactive processes appear in mod­ els of atmospheric pollution control, age-structured population dynamics, problems involving enhanced oil recovery and electric current through a semi-conductor. In hydrology, equations of this type govern the fate and transport of reactive pollutants and bionlm-forming microbes in porous media. They are commonly used to predict and control the extend of con­ tamination in vadose zones and groundwater systems [11; 12; 13; 22]. Unfortunately, there are only few cases for which analytical solution to the advection-diffusion-reaction equations exists. This fact has motivated an extensive research into numerical techniques for accurate and efficient solution of the aforesaid equations. While classical techniques, such as stan­ dard finite-differences and Galerkin finite-elements, work well for transport problems that are dominated by diffusive movement, they suffer from severe •Department of Mathematics, Arizona State University, P.O. Box 871804, Tempe, AZ 85287-1804 (hristoCmath.la.asu.edu). 'Department of Mathematics, University of Wyoming, P.O. Box 3036, Laramie, WY 82071-3036 (chenCcuervo.uwyo.adu). 55

56

Nonstandard Methods for Advection-Diffusion-Reaction Equations

nonphysical oscillations or excessive numerical diffusion when advection, associated with the velocity field, dominates the diffusive effects. Gener­ ally, one can select a numerical scheme that eliminates one of the problems at the expense of increasing the effect of the other. Spurious oscillations near sharp concentration fronts are common in approximations that have second-order spatial accuracy [19]. First order schemes, such as standard upstream-weighting methods, can stabilize advective flows by suppressing the numerical oscillations but they suffer from grid-orientation problems, and may introduce excessive amount of numerical diffusion that can smear sharp fronts of solutions [2]. Due to the hyperbolic nature of the advection-diffusion-reaction equa­ tions, Eulerian-Lagrangian methods have been successfully applied to solve advection-dominated transport problems. Many such schemes have been developed, including the modified method of characteristics [17], the mod­ ified method of characteristics incorporating streamline diffusion [4], the Eulerian-Lagrangian localized adjoint method [9; 36; 37], and the finitevolume Eulerian-Lagrangian localized adjoint method [20], to name a few. Each of these methods produces non-oscillatory numerical solutions, re­ duces the numerical diffusion, and has greatly improved time truncation errors compared to the classical Eulerian methods. However, EulerianLagrangian schemes formally derived to date are applicable mainly to trans­ port equations with linear or linearized reaction terms [29; 36]. Nonlinear reaction terms play a significant role in many applications involving bacte­ rial growth and contaminant biodegradation in porous media. In contrast with linear reactions, nonlinear reactions may contain thresholds, and er­ rors near these thresholds can affect significantly the qualitative structure of solutions [29]. The objective of our research work is to develop reliable, accurate and efficient numerical methods for nonlinear advection-diffusionreaction equations. We consider the model advection-diffusion-reaction equation dc ^-+v-Vc-V-(D-Vc) at

=

r(c),

(2.1)

for the unknown function c(x,t) in a spatial domain fi C R d , d = 1,2, or 3, over a time interval J = (0,T]. The initial condition for the problem is c(x, 0) = / ( x ) , where / is a given function. Of particular interest to us are applications of the above equation to the modeling of groundwater solute transport and biofilm growth in porous media. In the above context, c

Introduction

57

denotes the concentration of a dissolved substance or the density of biofilmforming microbes within the groundwater, v is the Darcy velocity of the fluid mixture, D is the hydrodynamic dispersion tensor, and r(c) represents the total rate at which the solute is produced via reactions and sources [l]. We propose new Eulerian-Lagrangian numerical methods that efficiently handle the numerically difficult advection-dominated transport problems with nonlinear reactions. The non-standard techniques are based on a nonlocal numerical treatment of nonlinear reaction terms and more so­ phisticated discretizations of the time derivative. We define the numerical solution of the advection-reaction part of the transport equation using an "exact" time-stepping scheme. This enables us to follow the transport and track sharp fronts much more accurately than with the standard numerical schemes. The relative importance of advection, and biological and chem­ ical reactions are directly incorporated into the corresponding numerical schemes. Having dealt with the most difficult part of the transport prob­ lem, only the smoothing property of the diffusion term remains. Then, standard finite differences or Galerkin finite elements are well suited for solving the diffusion part. The non-standard methods are relatively easy to implement and have much greater computational efficiency as compared to standard numerical schemes. The appropriate time step size for a particular model problem can be determined by physical considerations, rather than stability, convergence or consistency reasons. We were able to confirm, theoretically and through a set of numerical experiments, that the non-standard schemes are reliable and propagate sharp fronts accurately, even when the advection and re­ action processes are highly dominant and the initial data are not smooth. Error analysis shows that the proposed new methods produce oscillationfree numerical solutions with zero local truncation error with respect to time. An outline of the chapter follows. In the next section, the non-standard numerical method for solving one-dimensional advection-diffusion-reaction equations is developed. First, "exact" time-stepping schemes are derived for a variety of diffusion-free partial differential equations with nonlinear reaction terms. The semi-discrete procedure is then combined with finite difference spatial-approximations of the diffusion term to obtain the new non-standard method. In Section 2.3, one-dimensional convergence results are presented. For the "exact" time-stepping schemes a zero local trunca­ tion error with respect to time is obtained. However, an interpolation of

58

Nonstandard Methods for Advection-Diffusion-Reaction Equations

the approximate solution values, such as the piecewise-linear polynomial and piecewise-cubic spline interpolations, is necessary to evaluate the con­ centration at the backtrack characteristic points. As a consequence, the non-standard numerical solution error is proved to be a combination of the errors due to the chosen interpolation technique and the finite difference scheme used to discretize the spatial derivatives in the diffusion term. To demonstrate the performance of the proposed new method, numerical re­ sults for one-dimensional advection-dominated reaction-diffusion problems as well as the diffusion-free, logistic growth equation are presented in Sec­ tion 2.4. Based on the ideas from the one-dimensional case, in Section 2.5, the non-standard method for solving multi-dimensional advection-diffusionreaction problems is developed. Error analysis of the proposed new method in multiple dimensions is also studied. Section 2.6 contains a few concluding remarks summarizing our findings.

2.2

Non-Standard Methods in One Dimension

We propose a new Eulerian-Lagrangian numerical method for solving the one-dimensional advection-diffusion-reaction equation

dc

dc

d

/„dc\

d-t+Vdi-di{Ddi)=ric)-

Eq. (2.2) is a parabolic partial differential equation which changes to hyperbolic if the hydrodynamic diffusion/dispersion coefficient D is set to zero. Motivated by the hyperbolic case, we define the numerical solution of the advection-reaction part of the transport Eq. (2.2) using an "exact" timestepping scheme. This enables us to follow the transport and track sharp fronts much more accurately than with the standard numerical schemes. Having dealt with the most difficult part of the transport problem, only the smoothing property of the diffusion term remains. Then, standard finite differences or finite elements are well suited for solving the diffusion part.

Non-Standard Methods in One Dimension

2.2.1

Advection-Reaction

59

Equations

Our goal is to construct an "exact" time-stepping scheme for the advectionreaction part of Eq. (2.2): dc dc « + « t o =»■(«).

(2-3)

where x e il — [0,L] and t 6 J = (0,T]. The initial condition for the problem is c(x, 0) = f(x), where / is a given function. To introduce the concept of "exact" time-stepping schemes let us con­ sider the following numerical scheme Cm(x) =

F(Cm-l(x),At,m),

where At is the time step size and Cm(x) is the numerical solution at time mAt. Assume that it has a solution Cm(x) = G(C°(x), At,m) = G(f(x),

At,m).

The numerical scheme is said to be an "exact" time-stepping scheme if the relationship Cm(x) — c{x,mAt) holds for arbitrary time step size At and at every spatial location x [31]. 2.2.1.1

Logistic Growth Reaction Terms

Consider the advection-reaction Eq. (2.3) with the logistic growth reaction term r(c) = Ac(l - c/K). The two parameters, A and K, are positive constants, where K is called the carrying capacity of the environment [33]. Before analyzing the self-limiting logistic growth equation !

+

, g

= Ac(l-c/10,

(2.4)

subject to the initial condition c(x, 0) = / ( z ) , we express it in simpler terms by introducing the new quantities c = c/K,

i = x/L,

v = v/L,

f =

f/K.

Dropping all tildes to simplify the notation, we obtain the following sim­ plified advection-reaction equation

<£+v(x,t)^- = \c(l-c),

(2.5)

60

Nonstandard Methods for Advection-Diffusion-Reaction Equations

subject to the initial condition c(x, 0) = f(x). It can be easily observed that the new concentration function c(x, t) has equilibria at 0 and 1, and the new spatial domain is fi = [0,1]. The nonlinear partial differential Eq. (2.5) can be easily solved using the method of characteristics [24]. The general solution assumes the form <2"6>

+ M ( l _- e1- -A*t )W / ( /, )^» *'■*>=.-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.

<£+v(x,t)^

= \c + ncN.

(2.17)

The above equation can be viewed as a Bernoulli type ordinary differ­ ential equation in the characteristic direction. The exact solution have the

Non-Standard Methods in One Dimension

65

following expression

fN-l(e-aP"W(x e-HN-i)t

+ H(e-MN-i)i

+ b/a) - b/a)

_ l ) / ^ - i ( e - a P „ U ) ( x + b/a) - b/a)

(2.18) where f(x) = c(x,0) is the initial condition to the problem (2.17). Similarly to the case of logistic growth reaction terms, we can construct the new "exact" time-stepping scheme C"+'(*)-Cm(*m) HAT)

_ -

xrm(fm)

AC (x ) \N-l

+fl

m+1 (c {x)) X m+1 N

{C

(x))

-

,„m,=m^N-l

(cm(xXAtm)ym

{e C (xm))r/:^^,

N-2

Y^ eXkAt (Cm+1(x)

-

eXAtCm(xm))

k=0

(2.19) where 6(At) — (eXAt - 1)/A and the backtrack point xm has the expression (2.12). Second, consider the transport Eq. (2.3) with the following reaction term r(c) = A c + ^ r , where N > 0 is a positive integer number, and A and y. are real constants, i.e. dc . , dc fi v(x,t)— = = \c+^. -++ Xc+v { x , t ) -

(2.20)

The general solution to the above advection-reaction equation satisfies the relation cN+1(x1t)

= fN+1(e-aP"W(x

+ b/a) - b/a)ex(N+1»

+ ^ c ^ + O ' - 1).

(2.21) Then, the "exact" time-stepping scheme for solving the transport problem

66

Nonstandard Methods for Advection-Diffusion-Reaction Equations

(2.20) is given by the representation Cm+l(x)-Cm{xm) S(At)

\Cm(xm)

=

(Cm+1(x))"+1

(eXAtCm(xm))

-

N

lb=0 XAt

Here, 5(At) = (e pression (2.12)

m

— 1)/A and the backtrack point x

(2.22) has the same ex­

xm = e -°t' p "« m+1 ) At »- p "< mAt »l(a; + b/a) - b/a. N A For the case of a reaction term r(c) W = „ . . , where PN(C) W = > ajC*. P*(c)' " ^ the exact solution to the corresponding transport Eq. (2.3) has the expres­ sion

E77Tn

c < + 1

M = ^ + E 7 7 ^ T : r + 1 ( e - ^ W ( x + 6 / a ) - 6 / a ) . (2.23)

The "exact" time-stepping scheme for solving the advection-reaction equation dc

,

.dc

A

,

,

can be similarly derived to yield Cm+l(x)-Cm{xm) At

" t=0

f (C m + 1 (x)) < + 1 -(C m (x m )) i + 1

-. (2.25)

(i + l ) ( C m + 1 ( x ) - C m ( x , n ) )

Remark 2.4 When first order reaction terms Ac are used as a part of the nonlinear reactions r(c), there are two distinctive features in the imple­ mentation of the "exact" time-stepping scheme — the denominator func­ tion S(At) has the more complicated expression 6(At) = (eXAt - 1)/A, and high-power reaction terms are modeled non-locally involving the weighted

Non-Standard Methods in One Dimension

67

advective concentration values

{(e^Cm(xm)Y}lo, where M is a positive integer number. Remark 2.5 In the absence of first order reaction terms, the denominator function 6(At) is equal to the time step size At and modeling of nonlinear reactions r(c) using the "exact" time-stepping method can be viewed as either an arithmetic mean or as a harmonic mean of terms M

[(Cm+1(x))M

'(C m (x m )) i }_

For example, let us consider the fourth order reaction term r(c) — c 4 . Substituting A = 0, /x = 1, and N = 4in the transport Eq. (2.17) yields dc

dc

4

-+v(x,t)-=c. Then, the timestepping scheme (2.19) can be written in the following two ways Cm+1(x)-Cm{xm) At

2.2.2

(Cm+1(z))3(Cm(xm))3 (Cm+1{x))

+Cm+1(x)Cm(xm) 3

1 (C^+^x))2

3Cm+1(x)Cm(xm) 1 Cm+l{x)Cm{xm)

Advection-Diffusion-Reaction

Equations

+

+

{Cm{xm)f

1 (Cm(xm)f

Without loss of generality, we confine our discussion to the development of the non-standard method for solving the diffusive transport Eq. (2.2) with logistic growth reaction terms. Applying the "exact" time-stepping scheme to the logistic growth advection-diffusion equation yields the following im­ plicit in nature, semi-discrete procedure Cm+1(x)-Cm{xm) S(At)

-|(c^

M

)=

A c m

(i"'»<1-c"

, + ,


68

Nonstandard Methods for Advection-Diffusion-Reaction

Equations

To complete the construction of the new non-standard method, we need to introduce an approximation technique for discretizing the spatial deriva­ tives involved in (2.26). Let us consider a finite difference approximation of the diffusion term. The finite element method can be used as successfully as the finite difference method. However, we shall not present it here since the general idea of constructing the non-standard method is the same. In the general case, time-dependent, variable-spacing grids can be writ­ ten in the following way 7™ = { x £ \ < , . . . , x £ J ,

(2.27)

x? - x?_x = Ax? > 0,

where X™ = 0 , x^m = 1 and the superscript m denotes that the grid is employed at time mAt. Let the approximation to the diffusion term be given by

+

£(- '^) /im+1 £>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)A<)C>0(e-aPn((m+l)A<)(a.

+

tyfl) _ tyffl) '

Imposing the initial condition for the model problem as an initial condition for the "exact" time-stepping scheme, i.e. C°(x) = f(x), gives Cm+1{x) _

/(e-aPn((m+l)At)(a. +

b

/Q) _

b / q )

e-A(m+l)At + (! _ c-A(m+l)A*)^(c-afl.((m+l)A*)(a: + 6/0) - 6/a) '

(2.30) In Section 2.2.1.1, the general solution to the partial differential Eq. (2.5), in the case of space-dependent velocity field (2.10), has been derived as c(x 1

t)

' '

=

/(«-■*■<'>(*+ 6 / « ) - 6 / « ) e-* + (l-e->*)f(e-*''*W(x + b/a)-b/ay

3 [

'

Error Analysis of the Non-Standard Method

71

Comparison of solutions (2.30) and (2.31) gives the following equality Cm+1(x)

= c(x, (m + I) At).

(2.32)

The above relationship is satisfied for every positive integer m and for arbitrary time step size At, which allows us to conclude that the "exact" time-stepping scheme (2.9) obeys a zero local truncation error with respect to time. 2.3.1.2

Zero Local Space-Truncation Error

Let us assume that each backtrack point {x™} lies at a grid point at every time level m, i.e. x?eym,

te{0,...,Arm+1},

m>0,

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)\<—.

\Pn((m +

Without loss of generality, we assume that xj" 6 (ij(j),x J ( i ) + 1 ). Estimating the factor u\ (x) we have: | utix) \= | (x? - xj(i))(x™ - xj{i)+1)

| < aAxi'Ax,

(2.37)

where 0 < a < 1 and aAx = Zj(i)+i — x™. Hence, Eq. (2.33) yields the following estimate at the backtrack point xj"

\f{x?)-f{i?)\

< f |/"(C) I AxAx/ (2.38) <

2

AT|/"(C)|min(Ax ,AxAt),

where AT is a positive constant. The above estimate (2.38) holds at each time level of the discretization scheme (2.34). Since the global error in the numerical solution of the nonstandard method is only due to the accumulation of interpolation errors at each time-iteration level (recall the results in Section 2.3.1.1), bounds of the error can be easily obtained using the triangle inequality. This line of reasoning proves the following theorem: T h e o r e m 2.1 Let the solution of the model problem (2.5) belong to the space C 2 ([0,1]) x C([0,T]), and let the approximate solution dm+1 be de­ fined by the difference method (2.34), where Cm(x^)

= =

C m (e- 0 l p "(( m + 1 ) A t )- p -( m A t )](xi + 6/a)-fe/a) (h{Cr})(i?),

the piecewise-linear interpolant of {d™}.

Then there exists a constant

Error Analysis of the Non-Standard Method

75

C > 0, such that the global error of the "exact" scheme satisfies the bound ||c(-,fm+1)-Cm+1(-)||00

<

(m + l)LCmin(Ax 2 ,AxA0

<

/Ax2 rLCmin'

V At

(2.39)

,AxY

where T is the final moment in time and L = max 0
d2ck dx2

Here, || • H^ denotes the norm

|| /IU = sup | f(x) I, *€[0,1]

for the vector space C 2 ([0,1]) of all real valued functions / , denned on the interval [0,1], whose derivatives up to the second order are continuous. In order to increase the accuracy of the approximation, let us consider a cubic spline interpolation of the advective concentration C m (xJ"). The following error estimate [3] holds for the moments m* of a complete cubic spline s: Theorem 2.2 Suppose that f € C 4 ([0,1]), and let 7 = { x 0 , x i , . . . ,x/v} be a grid on [0,1] having mesh size Ax. Denote bym€ TlN+1 the vector of moments rrn of the complete spline s interpolating f on 7, and let f € 1ZN+1 be the vector of true second-derivative values /"(x<) at the nodes ofj. Then || m - f \\x = max | m< - /"(*,) | < | | | /<4> I ^ A x 2 . 0<:
£

To obtain a bound of the error | /(x[") -s(xj") |, we follow the "bootstrap" strategy proving an estimate for

i/( 3 )(xD- s ( 3 ) (xr)i, then using the result to argue for | /"(xj") - s"(x™) |, and so forth.

76

Nonstandard Methods for Advection-Diffusion-Reaction Equations

Using (2.35) and the definition of a cubic spline, we get f{3)(*?)

~ s(3)(z?)

<

\

Ax

mj(<)+1-/"(xj(i)+1) Ax

* m j(<) -

f"(xj{i)) Ax

(i)

f(x jW+1 )-/"(xD _ /"(^t))-/"(xr) _ (3) Ax

Ax

J

l {

'

We estimate the terms labeled (/) and (77) separately. Theorem 2.2 implies that

(/)< 3 | l / y%3|l/«'ILA* To estimate (II), we use (2.36) and Taylor expansions about the point x™, finding points 771,7/2 € (£;(«)'^'(•'z+i) for which (77) =

J-(, - *n/(3,(*n + J ^ 4Ax^ / ( 4 ) ( m ) A i JW+I - ±( AxXm

) gr)a w -*D/ (8) (*D - ^ ( ' i ( i Ax / fa) - /(3)(xD A.

2 2 < i l ^ i ^ ( ( | A x ) + (Ax < *) ).

Combining results gives I / ( 3 ) (x,^) - sW(x?)

I < K^f*

^ ( ( A x ^ ) 2 + (Ax) 2 ), Ax

where K is a positive constant and AXJ* has the expression (2.36). Having already analyzed f^ — s^, we get to / " — s" and so forth by integrating. Applying the fundamental theorem of calculus and the triangle inequality one can easily obtain the following estimate: I / ( x - ) - s(x?) I < K\\ /«> 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
c

*

m+l) '

(2.49) cm+l

=

m+1 _

Cm+1

=

m a x

^m+l

_

m+1)

By the choices (2.49) of the indices i\ and 12, the quantity in the second brackets of Eq. (2.48) is nonpositive for i = i\ and nonnegative for i — %2This implies that C + 1 > C™ + K*t)e?i+1 C + 1 < C + s(AtK+1

+6(At) [Ric?*1,^) + H*t) [R(c™+1,c?2) -

-

R(C^+1,Cm(x^))} R(CZ+1,Cm(x%))]. (2.50)

80

Nonstandard

Methods for Advection-Diffusion-Reaction

Equations

Then, for any j € { 1 , . . . , N - 1} we have |Cf +1 |

<

^ m a x j c r + ^ e r +6(At) [R(cT+1,cT)

<

max

~

- R(C?+1,Cm(x?))]

I C I +H&)

Ki
'

+6(At)

i<

1

I e" , + 1 I

max

'

|

Ki
m a ? ] | R(
- R(C?+\Cm(x?))

\. (2.51)

Eq. (2.51) implies that max

,-m+l IC I <


*

'

max

"~

I C I +S(At)

I em+1

max

Ki<7V-l

'KKW-l'

+<5(Ai) X

| i?(c t m + 1 ,cf)

max Ki
V

'

'

*

-^(cr+Sc"1^))!. (2.52) Applying the Mean Value Theorem for the function R £ (^([O,1] x [0,1]), we can further obtain max

IC

I <

l
' '

'

-

max

I £ " I +S(At)

1<»<JV —1 ' *

+S(At)L (

m+l I e?

max

l
max

| C,m+1 I +

\l
max l
\C\), '

'

7

(2.53) where L = max I

sup

9iJ dz\

sup [0,l]x[0,l]

dR dzo

(2.54)

\JO,l]x[0,l]

Rearranging terms in (2.53) yields [1 - L6(At)}

max

I Cm+1 I <

[1 + LS(At)) -rS(At)

max max

| C | e"1"1"1 |.

Error Analysis of the Non-Standard

Method

81

Provided that L6(At) < 1, and dividing both sides of the above inequality by 1 — L6(At) we have ,
whereM=

i-^L)-

To evaluate the advective concentration C7m(x™), some type of interpo­ lation of the approximate solution values {CJ1} should be used. In order to relate max I CJ" I to max I G™ L it is necessary to select the interpolation rule. In the analysis that follows K will denote a generic positive constant. The first choice to be made in this section is piecewise-linear interpola­ tion, the operator for which we shall denote by I\. Then max

| C l< max | £» I +

\
l
max | c(xittm)

- J 1 c(-,r*)(x i ) | .

\
(2.56) Thus, it follows from (2.56) and the estimate (2.38): d2J dx2

\c{xi,tm)-ric(;tm){xi)\
min(Ai , AxAt),

previously obtain in Section 2.3.1.3 (Theorem 2.1), that max

I C,m+1 I <

[1 + 2ML] (

+K +M

d2cm dx2

max | C min(Ax 2 ,Aa;At)j

max I e" ,+ 1 I . Ki
By Taylor series expansions, it is easy to show that M =

6(At) = 1 - LS(At)

O(At),

and [1 -I- 2ML] < (1 + EAt), where E is a positive constant. Then, it

82

Nonstandard Methods for Advection-Diffusion-Reaction

Equations

follows that max

I C+1 I <

(1 + EM)

max

„ / d2cm

| C . (Ax2

max

+

a22„0 c«

\

I eT+l \) At. (2.57) = I I { C ( X J , 0 ) } , we

If the approximate solution C°(x) is given by h{C°} have \C?\
.

min(Ax 2 , Ax At).

2

ax

Thus, iterating the inequality (2.57) we arrive at the estimate max

| Cr+1 I

KKJV-l ' '

< V (1 + EAt)kKAt

max

0
*=o

+ V S(1 + EAtfKAt *—< fc=0

. f Ax

(£•-)

min '

dx2

max

'

< £ e ™ ' f f At (Dl v *=o

d2Ck

Ki
I ef+l I *

min ( * £ , A x ) + x

7

max —

| e ^ 1 |) '

< (m + l ) e C T * A t ^ m i n ( ^ , A x ) + ^ m a x ^ | e™+1 | )

< T e ^ t f (Dl

min ( ^ , A x ) + Z& (Ax 2 + A t ) ) ,

where T is the final moment in time, DL. = max 0
5x 2

Error Analysis of the Non-Standard

Method

83

and Qicm+\

D2 = max

dx*

d2C

, sup oo [O.lJxJ

dc

sup

8T2

[0,l]xJ

dr

sup [0,1]x[0,1]

dR dzi

This line of reasoning proves the following theorem: Theorem 2.3 Let the solution of the model problem (2.41) belong to the space C 4 ([0,1]) x C([0,T]), and let the approximate solution dm+1 be defined by the non-standard numerical method (2.43), where Cm{x?)

=

Cm

(e-olP.Um+l)*t)-P.(mAt)](x.

the piecewise-linear interpolant of {Cj m }. Q > 0, such that II c(-,tm+1)

tyQ) _

+

6/fl)

Then there exists a constant

- C m+1 (-)lloo < QS (min ^ - , Ax) + At),

(2.58)

where S = max I max I0
d2ck dx2

a4cm+1 oo

dx*

, sup

d2c »

2 oo [0,l]xJ 8T

sup [0,1] xJ

dc dr

sup

dR

[0,l]x[0,l] dZ2

To increase the accuracy of the numerical approximation, one can use a high-degree polynomial interpolations. Our second choice in this Section is the cubic spline interpolation of the advective concentration C m (xJ"). The following error estimate (2.40) holds for a complete cubic spline 5 interpolating a function / € C 4 ([0,1]): | f{x?)

- s(x?) I < K\\ /«> ILmintAz 4 ,

Ax3At),

where e-alP„({m+l)At)-PnlmAt)](x.

+

fe/fl)

_

6/fl

84

Nonstandard Methods for Advection-Diffusion-Reaction

Equations

is the backtrack point associated with the grid point X{ 6 7. Then, substi­ tution in inequality (2.55) of the estimate max

I C l<

Ki<JV-l

d\ dx*

| C1 I +K

max KKW-1

* '

min(Ax 4 ,Aa: 3 At)

yields

I Cm+l

max

<

(1 + EAt) max I CT I

~

Ki<JV-l 4 m

/ a c

. /Ax4

A

,\

max I e"' + 1<»
+1

\) At. 7

where £ is a positive constant. Similarly to the case of piecewise-linear interpolations, we can obtain the following global error estimate for the new, non-standard method (2.43) with cubic spline interpolations

iic(-,<m+i)-cm+i(-:

< QS (Ax 2 + A t ) ,

where Q is a positive constant, and 5 = max I

max

\ 0<*<m+l

8x*

, sup oo [0,l]xJ

d2c dr2

sup [0,l]xJ

9c dr

dR sup [0,1] X [0,1]

dzi

Remark 2.7 The initial-value problem and the advective-diffusive problem with Neumann boundary conditions can be treated by simple variants of the proposed method, as in [17]. Error estimates can be easily carried through in an analogous fashion.

2.4

One-Dimensional Numerical R e s u l t s

We now present a set of numerical experiments to demonstrate the per­ formance of the proposed new method. Our main focus here is to explore the ability of the non-standard method to solve the numerically challenging advection-dominated transport Eq. (2.2) with nonlinear reaction terms, and to compare the results with an analytical solution and with results obtained from other numerical methods developed for partial differential equations.

One-Dimensional

2.4.1

Advection-Reaction

Numerical Results

85

Equations

The governing equation examined here is the simplified logistic growth advection-reaction Eq. (2.5) dc

^t+v(X,t)-

. dc

, .„

=

Xc(l-c).

The Lagrangian equation Dc/Dt = Ac(l — c) has two steady equilibrium states, namely c = 0 and c = 1, due to the logistic growth reaction term r(c) = Ac(l — c). The carrying capacity K = 1 determines the size of the stable state population. All initial concentration values c(x, 0) in the inter­ val (0,1) grow monotonically to the stable equilibrium c = 1. However, any small negative excursions in the concentration are forced to grow without bound in magnitude, due to the unstable equilibrium c = 0. Non-physical oscillations, caused by the implementation of numerical methods that do not recognize the hyperbolic nature of the problem (2.5), can be fatal to the numerical solution, leading to eventual machine blowup. For comparison purposes, all figures display the corresponding solutions at the same final time T = 0.35. In each plot, the initial condition and analytical solution are given with solid lines, while the numerical solutions with dashed lines or symbols. A uniform spatial grid 7 = {xo, xi1 • • •»£100} with 101 grid points and grid size Ax = 0.01 is employed. The initial growth rate is chosen as A = 10. The first numerical experiment involves an advective transport of a nar­ row initial step function under the space-independent velocity field v = v(t) = t3 + 0.8t2 + 0.5t + 1. We impose the following initial condition

The purpose of this experiment is to compare the non-standard method results with the analytical solution of the problem, and with the results ob­ tained by using the implicit modified method of characteristics (MMOC) and the implicit finite difference method. To calculate the advective concen­ tration C m (x r a ), in the nonstandard numerical procedure (2.9), piecewiselinear (LNM) and cubic spline (CSNM) interpolations are implemented.

86

Nonstandard Methods for Advection-Diffusion-Reaction Equations

Figure 2.1 shows the non-standard numerical solution (o) with adjusted time step sizes Ar m , that guarantee a perfect match between the backtrack points {x["} and the grid points {z™ = tAx}j°° (see Section 2.3.1.2). We denote the non-standard method with adjusted time step sizes by ATSNM. Numerical results obtained with the LNM (—.) and the CSNM ( — ) , using a time step size At = Ax/2 = 0.005 are also presented in Figure 2.1. Figure 2.2 shows the poor performance of the traditional implicit finite difference method (—.) and the implicit MMOC ( — ) , when applied to models of a pure advective transport of discontinuous step functions. The second and third experiments involve an advective transport of a highly localized, discontinuous plume. The initial condition corresponds to the "cutoff" Gaussian distribution centered at xo with standard deviation 1/y/a, i.e.,

c(x,0) ■{ = < ^ x p { - a ( l " a:o)2} ' I * " X0 I * 1 / 1 6 1

(2.60)

0, otherwise where xo = 0.2 and a = 800. In the second experiment we use a time step size Ati = Ax/2 = 0.005. Representative velocity field of the correct magnitude is chosen as v(x,t) = (0.5x 4- l)(0.8t 2 -I- 0.5t + 1). Figure 2.3 compares the numerical results obtained with the LNM (—.) and the CSNM ( — ) procedures to the analytical solution. Also displayed in Figure 2.3 is the non-standard numerical solution (o) obtained on a partially irregular spatial grids, that guarantee exact modeling of the back and the front of the discontinuous plume (2.60) (see Section 2.3.1.3). We denote the non-standard method on such irregular grids by IRGNM. Figure 2.4 demonstrates the diffusive-like effects that appear in the im­ plicit finite difference (-.) and the implicit MMOC ( — ) approximations, enhanced by the stable equilibrium c = 1. The purpose of the next numerical experiment is to verify that large time steps can be taken without sacrificing the solution's accuracy of the nonstandard method. Additional runs are made with a time step size twice as large as the time step size in the second experiment, i.e. A<2 = Ax = 0.01. Again, the IRGNM solution (o) matches almost exactly with the analytical solution of the problem (Figure 2.5), while the implicit finite difference procedure (—.) leads to a significantly distorted numerical solution (Figure 2.6). We now turn to a set of numerical experiments to demonstrate the performance of the proposed new method for problems with smooth initial

One-Dimensional Numerical Results

87

Fig. 2.1 Numerical solutions of the logistic growth CR equation generated by the nonstandard method with adjusted time step size At m (o), and with cubic spline (—) and piecewise-linear (—.) interpolations of the advective concentration after 70 time steps. The solid lines are the initial contaminant plume (left) and the analytic solution (right) at T = 0.35 with v(t) = t 3 + 0.8«2 + 0.5* + 1.

0.4 0.5 DISTANCE

0.6

Fig. 2.2 Numerical solutions of the logistic growth CR equation after 70 time steps generated by the implicit modified method of characteristics (—) and the implicit finite difference (—.) method.

88

Nonstandard Methods for Advection-Diffusion-Reaction Equations

Fig. 2.3 Numerical solutions of the logistic growth CR equation generated by the nonstandard method on partially irregular spatial grids (o), and with cubic spline ( — ) and piecewise-linear (—.) interpolations of the advection concentration after 70 time steps with Ati — Ax/2. The solid lines are the initial contaminant plume (left) and the analytic solution (right) at T = 0.35 with v(x,t) = (0.5i + l)(0.8t2 + 0.5t + 1).

Fig. 2.4 Numerical solutions of the logistic growth CR equation after 70 time steps generated by the implicit modified method of characteristics (—) and the implicit finite difference (—.) method.

One-Dimensional Numerical Results

89

■

^ IB^da

inmiHiifli

0

0.1

02

03

0.4 0.5 DISTANCE

t

0.6

0.7

0.8

0.9

1

Fig. 2.5 Numerical solutions of the logistic growth CR equation generated by the nonstandard method on partially irregular spatial grids (o), and with cubic spline ( — ) and piecewise-linear (—.) interpolations of the advective concentration after 35 time steps with At2 = A i . The solid lines are the initial contaminant plume (left) and the analytic solution (right) at T = 0.35 with v[x, t) = (0.5x + l)(0.8t2 + 0.5t + 1).

Fig. 2.6 Effects of increasing the time step size on the implicit modified method of characteristics (—) and the implicit finite difference (—.) numerical solutions of the logistic growth CR equation. Concentration profiles are computed after 35 time steps.

90

Nonstandard Methods for Advection-Diffusion-Reaction Equations

data. The experiments involve advective transport of a highly localized, continuous plume. We impose an initial condition that corresponds to the Gaussian distribution: c(x,0) = exp{-<7(x-x 0 ) 2 }, where a = 800, and xo = 0.2. In the following two experiments, the velocity field is chosen as v(x, t) = (x + 0.8)(0.1r + 1). In the first experiment, a time step At = Ax/2 — 0.005 is used, leading to a Courant number Cu « 0.9 (Figures 2.7 and 2.8). Figure 2.7 compares the analytical solution with the results obtained using the LNM ( — ) and CSNM (o) procedures after 60 time steps. Figure 2.8 demonstrates the poor performance of the traditional linear finite element ( — ) and the implicit finite difference (—.) methods over the narrow initial distribution. Additional runs are made with time step At twice as large as the original one (Figures 2.9 and 2.10). The corresponding Courant number in this case is Cu as 1.9. Again, the CSNM procedure (o) matches almost exactly the analytical solution (Figure 2.9), whereas the implicit finite differences (—.) and finite elements ( ) lead to serious numerical dispersion, after 30 time steps (Figure 2.10). 2.4.2

Advection-Diffusion-Reaction

Equations

We now turn to a set of numerical experiments to demonstrate the per­ formance of the proposed new method applied to the diffusive advectionreaction transport Eq. (2.2): dc

di +

dc

d / _ dc \

v

D

, .

r{c)

Tx-d-x{ Tx)= -

The two example problems presented here involve propagation of highly localized, discontinuous plumes. The velocity field is v(x, t) = (0.5x + l)(0.8i 2 + 0.5r + 1), and hydrodynamic dispersion coefficient is chosen as D(x, t) = 0.002(x2 + 2x + 50The time step size is set to be equal to the uniform mesh size, i.e At = Ax = 0.01. All figures display the corresponding solutions at the same final moment in time T = 0.35.

One-Dimensional Numerical Results

0.1

0.2

0.3

0.4 0.5 0.6 DISTANCE (X/L)

0.7

91

0.8

0.9

1

Fig. 2.7 Numerical solutions generated by the non-standard method for the cases of cubic spline (o) and linear (—) interpolations of the advective concentration, after 60 time steps. The solid lines are the initial contaminant plume (left) and the analytic solution (right) at t = 0.3.

:I m : ■'if 1...1..1..1

g0.6

\ t\ ......u

!

7

-'

' i f : L./../...J;

1 V V l..:....>...\j \ :

I I I : 1 ' / :

;

/:.;..:

0.2 i s

i

■

0.1

0.2

0.3

!

j

j

/:

;

V A v : v

;

-

\...\.J

V J

.

:V

i

0.4 0.5 0.6 DISTANCE (X/L)

i

i

0.7

1

0.8

0.9

Fig. 2.8 Numerical solutions generated by the linear finite element ( — ) and implicit finite difference (—.) methods after 60 time steps. The solid lines are the initial contam­ inant plume (left) and the analytic solution (right) at t = 0.3.

92

Nonstandard Methods for Advection-Diffusion-Reaction Equations

0.1

02

0.3

0.4 0.5 0.6 DISTANCE (X/L)

0.7

0.8

OS

Fig. 2.9 Numerical solutions generated by the non-standard method for the cases of cubic spline (o) and linear (—) interpolations of the advective concentration, after 30 time steps with At = Ax. The solid lines are the initial contaminant plume (left) and the analytic solution (right) at t = 0.3.

I I

'

/

1A

/

1

\

Is II 2

\

''I 1 1

0.4-

5

1 1

-0.2

0.1

1

0.2

'

0.3

'

1

0.4 0.5 0.6 DISTANCE (X/L)

1

-

*■ v\xx

*xy

1

\

\

v

s

^

1 1

\... \ . , . \ N

1

\

0.7

0.8

0.9

.

X

1

Fig. 2.10 Effects of increasing the time step on the linear finite element ( — ) and im­ plicit finite difference (—.) numerical solutions. Concentration profiles are computed after 30 time steps.The solid lines are the initial contaminant plume (left) and the analytic solution (right) at t = 0.3.

One-Dimensional Numerical Results

93

The purpose of the experiments is to compare the new, non-standard method solutions with the "almost exact" solution of the problem, and with the numerical results generated by the implicit modified method of charac­ teristics (MMOC) and the implicit-central finite difference (FD) method. The "almost exact" solution is obtained by numerically solving the trans­ port problem (2.2) on a very fine mesh, with spatial grid size Ax = 0.001 and time step size At = Az/10. To calculate the advective concentration Cm(xm) in the non-standard numerical procedure (2.28), piecewise-linear (NSM.PL) and cubic spline (NSM.CS) interpolations are implemented. In each plot, the initial condition and the "almost exact" solution are given with solid lines, while the numerical solutions with dashed lines or symbols. The initial condition in the first numerical experiment corresponds to the Gaussian distribution (2.60). Homogeneous Dirichlet boundary con­ ditions are used. We examine the advection-diffusion-reaction Eq. (2.41) with a logistic growth reaction term. The nonlinear reaction term r(c) has the form r(c) = 10c(l - c), where the intrinsic growth rate A is chosen as A = 10, and the environmental carrying capacity is K — 1. Figure 2.11 displays the non-standard numerical solution obtained with the NSM.CS (o) and with the NSM.PL ( — ) . The piecewise-linear interpolations of the advective concentration introduce small diffusion-like effects in the nu­ merical solution, while the cubic spline interpolations lead to a near perfect match between the non-standard numerical solution and the "almost exact" solution of the transport problem. Figure 2.12 demonstrates the diffusive behavior of the implicit-central FD (—.) and the implicit MMOC ( — ) approximations, which is additionally enhanced by the stable equilibrium c = 1 of the diffusion-free transport equation. The second experiment involves the advective-diffusive transport Eq. (2.2) with nonlinear reactions: dc dc _ . ___ . +V D d-t dx--dx-( dx\ d-X dxj =Wc 20<

+

1 5C

- '

where r(c) — —— + 1.5c. We impose the following initial condition

c ( x

' 0 ) - \ 0.02,

otherwise

(261)

centered at xo = 0.2 with a = 800. Neumann no-flow boundary conditions are used.

94

Nonstandard Methods for Advection-Diffusion-Reaction

Equations

Figure 2.13 compares the numerical results obtained with the NSM.CS (o) and with the NSMJ'L ( — ) procedures to the "almost exact" solu­ tion of the differential equation. The poor performance of the traditional implicit-central FD method (—.) and the implicit MMOC ( — ) can be seen in Figure 2.14.

2.5

Non-Standard Methods in Multiple Dimensions

In this section, we develop effective non-standard numerical methods for solving the multi-dimensional advection-diffusion-reaction Eq. (2.1). The new Eulerian-Lagrangian methods are based on observations for the onedimensional case. 2.5.1

Development

of the Non-Standard

Method

We confine our discussion of the derivation of the new non-standard meth­ ods to two-dimensional advection-diffusion-reaction equations of the form (2.1). Non-standard numerical methods for solving transport problems in three dimensions can be similarly develop without any principal difficulties, and we shall not discuss them here. Throughout the main part of this section, the subject of our examination will be the normalized, self-limiting logistic growth equation: dc — + v-Vc = Ac(l-c),

(2.62)

subject to the initial condition c(x, 0) = / ( x ) . The non-linear partial dif­ ferential Eq. (2.62) has a solution of the form [24]: f(s) c x

( .*) = ^ 7 v

//

xt

w w / x< Xi

(2-63) v

' e~ + (1 -e- )f(s) ' where the two components of s = (sx,sy), sx and sy, are the initial condi­ tions for the following ordinary differential equations *£±=vx(x,t)

and

^=Vy(y,t),

(2.64)

respectively. Consider the linear-in-space, polynomial-in-time velocity field v(x, t) = (vx,vy), where each of the velocity components has an expression of the

Non-Standard Methods in Multiple Dimensions

0.4 0.5 DISTANCE

95

0.6

Fig. 2.11 Numerical solutions of the CDR equation with the logistic growth reac­ tion term r(c) = 10c(l - c), generated by the non-standard methods NSM.CS (o) and NSM.PL ( — ) , with Ax = At = 0.01. The solid lines are the initial con­ taminant plume (left) and the "almost exact" solution (right) at T = 0.35 with v(x,t) = (0.5i+ l)(0.8t2 +0.5t + l) and D(x,t) = 0.002(x2 - r 2 i + 5t).

0.4 0.5 DISTANCE

0.6

Fig. 2.12 Numerical solutions of the CDR equation with the logistic growth reaction term r(c) = 10c(l — c), generated by the implicit MMOC (—) and the implicit-central FD method (-.).

96

Nonstandard Methods for Advection-Diffusion-Reaction

Equations

Fig. 2.13 Numerical solutions of the CDR equation with nonlinear reactions r(c) — 0.05c + 1.5c, generated by the proposed new method NSM.CS (o) and NSM-PL ( — ) , with Ax = At = 0.01. The solid lines are the initial contaminant plume (left) and the "almost exact" solution (right) at T = 0.35 with v(x, t) = (0.5i + l)(0.8t 2 + 0.54 + 1) and D(x,t) = 0.002(x* + 2x + 5t).

0.4 0.5 DISTANCE

0.6

t'ig. 2.14 Numerical solutions of the CDR equation with nonlinear reactions r(c) = 0.05c+ 1.5c, generated by the implicit MMOC ( — ) and the implicit-central FD method

Non-Standard Methods in Multiple Dimensions

97

form (2.10), i.e., vx(x,t)

= (aix + bi)Pni-i(t)

and

vy(y,t) = (a2y + b2)Pna-i(t).

(2.65)

Then, Solution (2.63) has the expression C(X t}

'

/ ( e - ^ W • (x + b / a ) - b / a ) ~ e-H + (1 - e - " ) / (e-P»W • (x + b / a ) - b / a ) '

(2-66)

where b / a = (61/01,62/02)) e —Pn(t) =

Je-t/'.1(0)e-«i'.JW))

and

P „ , ( 0 = /" P n , - i ( r ) d r ,

i = 1,2. Based on it, we construct the two-dimensional "exact" timestepping scheme for solving Eq. (2.62) [28]: C m + 1 ( X )

^ " ( * m ( x ) ) = AC""(x-(x))(l - C - + 1 ) .

(2.67)

Here, C m (x) denotes the numerical solution at location x = (x, y) and time mAt, the denominator function S(At) = (eXAt — 1)/A is the same as in the one-dimensional case, and the backtrack Cartesian point xm(x) =

(x">(x),ym(y))

has the following coordinates: xm = xm(x) = e -»i['*-i((">+i)A0-i , -,(mA0I( z

+

^/aj) _ ^ / ^ (2.68)

ytn _ ym(y)

=

g ~ Oj [P„ „ ( ( «,+ 1 ) A t) ~ P„ „ ( 77, A t)] ,"y

+

^/j^) _

fc^.

Remark 2.8 There are no principal differences between the implementa­ tion of the "exact" time-stepping scheme to the one- and two-dimensional logistic growth transport problems. The logistic growth reaction term is modeled in the same nonlocal fashion as in the one-dimensional scheme (2.11). Similar results can be shown for the two-dimensional "exact" timestepping scheme applied to transport problems with nonlinear reactions that fall into the three groups of reaction terms discussed in Section 2.2.1.3 [28]. Similarly to the one-dimensional case [27], applying the "exact" timestepping scheme (2.67) to the logistic growth advection-diffusion-reaction

98

Nonstandard Methods for Advection-Diffusion-Reaction

Equations

equation: dc — + v • Vc - V • (D • Vc) = Ac(l - c)

(2.69)

yields the following implicit in nature, semi-discrete procedure: Cm+1(x)-Cm(xm(x)) _ _ T

V-(D-+1-VC'"+1(x)) (2.70) =

ACm(xm(x))(l-Cm+1(x)).

To complete the construction of the two-dimensional non-standard method, we need to introduce an approximation technique for discretizing the spa­ tial derivatives involved in (2.70). Without loss of generality, let us consider a finite difference approximation of the diffusion term. Combining the semidiscrete procedure (2.70) with a difference approximation S^(Dm+16xCm+l)ij of the diffusion term, yields the new non-standard method for solving Eq. (2.69): /jm + l _ (7"»fv rn \

'J'

XAI

, ^

- * x ( D m + 1 * x C m + 1 ) i , j = A C m ( x - ) ( l - C £ + 1 ) , (2.71)

where x £ = x m (xi,j/j) and C?tj = /(x?,y°). Here, C™+1 denotes the grid value of the approximate solution at the point x™+1 = (x™+1, y^+i) at the advanced time level (m +1), and the backtrack Cartesian point x™7- has the the following coordinates:

2.5.2

x -.m = c - . l [ P . I ( ( m + l ) A O - P » I ( m A I ) ] ( x m + l

+ 6 j

y.m

+

_

Error

e-o2[P„2((m+l)A0-P„2(mA0](ym+l

/

a i

^j^)

) _ _

&J/a

,_

^ / ^

Estimates

In this section, we sketch the error analysis for the non-standard numerical method applied to the logistic growth advection-diffusion-reaction Eq. (2.69). First, consider the "exact" time-stepping scheme (2.67) applied to the diffusion-free logistic growth Eq. (2.62). Rearranging terms in Scheme (2.67)

Non-Standard Methods in Multiple

Dimensions

99

yields: rm+l

( \_ W

e

G""(x m (x)) - ^ + (l-e-*A')Cm(x"»(x))'

l

;

where x m (x) is the backtrack Cartesian point with coordinates (2.68). Re­ cursively applying Relation (2.72), as in Section 2.3.1.1, one can easily obtain the following relation [28]: Cm+1(x) C° ((x™ o x™- 1 o . . . o x 1 o x°)(x)) _

e-A(m+l)At

+ (1 _ e-A(m+l)At^O (( x m

0 xm-l

o . . . o x 1 O X°)(x)) '

Here, (x m o x"*- 1 o . . . o x 1 o x°)(x) = e - a P »« m + 1 ) A «) • (x + b / a ) - b / a , where (fog) denotes the composite function defined by (f°g)(z) = f(g(z)). Imposing the initial condition for the model problem as an initial condition for the "exact" time-stepping scheme, i.e. C°(x) = / ( x ) , gives Cm+1(x) / (e- aP -(('"+ 1 ) A *) • ( x + b / a ) - b / a ) -

e-X(m+l)At +

(! _ e-A(m+l)At)y (e-aP„((m+l)At) . ( x

+ b/a)

_

b/a)

"

(2.73) Comparison of Solution (2.73) and the exact solution (2.66) yields: Cm+1(x,y)=c(x,y,(m

+ l)At).

(2.74)

Now, consider the new non-standard method (2.71) for solving the dif­ fusive, logistic growth, advection-reaction Eq. (2.69) on a fixed uniform spatial grid lx = {xo,xi,...,xN}, ly = {yo, yi, • • ■, VM},

Xj - Xj_i = Ax > 0 Vj~ Vj-i = Ay > 0.

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. Without loss of generality, we present the error analysis of the new numerical method (2.71) applied to transport Eqs. (2.69) with diagonal

100

Nonstandard Methods for Advection-Diffusion-Reaction

Equations

hydrodynamic dispersion tensors: D =

Dx 0

0 A,

Consider the two-dimensional analog of the centered, second spatial differ­ ence approximation of the diffusion term V • (D • V C m + 1 ( x ) ) [21], i.e., <J*(D"* +1 * x C m+1 ) ili = 6,(D?+16sC?+1)i

+

S9(D^l6vCr+1)j,

where each of the spatial coordinate approximations has the expression (2.42). Note that for the exact solution c(x,y,t), with cj^" 1 = c(xi,yj,tm+1) m and c^j ~ c(x™,y™,t ), the following relationship holds: Cm"M _ £tn

_ \ j ~ *«(D m + 1 *,c m + 1 )i,,- = Ac&(l - c™/ 1 ) + e™/ 1 ,

^XAt

(2.75)

where e™*1 is the truncation error of the non-standard method at time-level (m + 1). By definition m+l

„m+l

xm

) + v ( x i J > r + 1 ) - V c ( :( x W , * m + i ) ) - [<5*(Dm+1<*xCm+1k; - V • (D(x i j -,t m + 1 ) •

c(x,tj,tm+1))]

-[R(cZ+\cZ)-R(cZ+\cZ+% (2.76) where i?(zi,2 2 ) = Az2(l - *i), with R e C1^,!} x [0,1]). Using Taylor series expansions and the Mean Value Theorem, calculation shows that |e£+1|

Qicm+l

=

Q*cm+l

HLM + +o <£H (&->l><)

o(max{

(2.77)

Non-Standard Methods in Multiple

Dimensions

101

where dc'/dr and d2c*/dr2 are some evaluations of the first and the second tangential derivatives along the characteristic segment between (x^j, < m+1 ) and ( x j j , t m ) , respectively. For the difference £ = c — C between the exact and the approximate solution, we have

+ 5(At) [Ric™;1,^)

o(At)

fnm+1

fl(C£+1,C»(x£.))]

-

f/-m+l

,-m+l \ ,

nm+l

/ym+1

,-m+l \1

(2.78) where 6(At) = (eXAt - 1)/A, and ^ = 0. Similarly to the one-dimensional case, assuming all functions in (2.41) are spatially H-periodic [15; 17], using that ReC1 ([0,1]x[0,1]), and provided L6(At) < 1, one can further obtain: max

| C £ + 1 | < [ 1 + 2ML]

max

|^

|+Af

max

| e £ + 1 |, (2.79)

where M =

, . . . , and the constant L has the expression (2.54). 1 - LS(At) In order to relate max I C?1; I to max I C7\ I, it is necessary to l
*'3

l
'•'

select an evaluation rule for the advective concentration Cm(x™,yjm). The first choice to be made in this Section is piecewise-linear interpola­ tion of the approximate solution values {C,™}, the operator for which we shall denote by Ifl\. Then max |<£.| < max |C£ | +

max

| c(XiiJ,r) - /r^(-,*m)(*ij) I

1<J,J
(2.80) To estimate the piecewise-linear interpolation error introduced in the twodimensional non-standard method (2.71), we make use of the fact that tensor-product interpolation schemes inherit the error estimates from the interpolation schemes used in the individual coordinate directions. We

102

Nonstandard Methods for Advection-Diffusion-Reaction Equations

begin the analysis with a lemma [28]. L e m m a 2.2 Let ai, 02, 61, and 62 be positive real constants. Then the following inequality is satisfied: maxjmin (al,aibi)

,min (02,0262)} < min {max (a 2 , a?,),max(ai,a 2 )max(61,62)}

Using the triangle inequality one can prove the following theorem [3]: T h e o r e m 2.4 Let ^x and ~fy be grids on [a, b] and [c,d], respectively, and suppose that~yx has mesh size Ax andjy has mesh size Ay. LetM(-fx) and A r (7 v ) be piecewise polynomial spaces with associated interpolary projections If and I\, respectively. Suppose that S is a vector space of functions f : [a,b] x [c,d] —> H such that 7f ,I\ : S -> S and that we have interpolation error estimates of the form II /(•,») - /f/(",y)lloo < CA:r p ,

|| f(x, •) - / ? / ( * , OIL < CAy".

Then the tensor-product space M{jx) Iflff obeying the estimate

® -W(Ty) contains an interpolant

Wf-IflUWoo^CmaxiAx^Ay"). Thus, it follows from Theorem 2.4, Lemma 2.2, and the estimates: c(-,j,)r"'+1)-/fc(-)y,r+1)||00
c(x,.><"

+1

y

)-/1 c(*,-,*

m+1

)lloo<^

d\ Bx2

min(Ax 2 ,AxAf)

d2cm dy2

min(Aj/ 2 , Aj/At)

that c ( v > t r a + 1 ) -IflM;;tm+l)\\oo

< KGmm(Ax2,AxAt).

(2.81)

Here, G = max [ max \0
d2Ck

dx

2

, max 0<*<m

d2ck dy2

and

K —

max(Kx,Ky).

Combining Inequalities (2.79) and (2.80), and Estimate (2.81) we arrive at the following estimate for the difference £ between the exact and the

Non-Standard Methods in Multiple

Dimensions

103

approximate solution [28]: max

,-m+l | C,T

< ( l + £At)

max

\Q\

+ K(Gmin(^-,Ax) \ \ At < r e £ T / C (Gmin (^-,

+ )

max | e™+1 \) At, l
Ax) +

max

| e™,+1 A , (2.82)

where £ is a positive constant, and T is the final moment in time. This line of reasoning proves the following theorem: T h e o r e m 2.5 Let the solution of the model problem (2.69) belong to the space C 4 ([0,1] x [0,1]) xC([0,T]), and let the approximate solution C™+1 be defined by the non-standard numerical method (2.71), where

the piecewise-linear interpolant of {C™,}. Then there exists a constant Q > 0, such that \\C(;;tm+1)-Cm+l(;

< QS f min (^-,AxJ

+ At\,

(2.83)

where S — max [ max ( \0<*<m \

d2c" dx2

d2ck dy2

) •

sup

d2C Qr2 , sup

10,l]xJ

d*cm+l dy*

Q*cm+1 4

dx

[0,l]xJ

dc 8T

sup [0,l]x[0,l]

dR dzo

To increase the accuracy of the numerical approximation, one can use a high-degree polynomial interpolations, such as cubic splines. Using Estimate (2.40) for the cubic spline interpolation error, we can

104

Nonstandard Methods for Advection-Diffusion-Reaction

Equations

obtain the following inequality: max

left I <

max +

| Cft I | c(xitj,tm)

max

- SZSv3c(;tm)(xij)

\

1
<

max

|C

I + KGmin (Ax 4 , Ax 3 At) ,

(2.84) where S3S3 denotes the piecewise-cubic spline interpolation operator [3] and G = max

max

a4c*

d4Ck

, max

\0
dy*

0
——

)

Similarly to the case of piecewise-linear interpolations, we can obtain the following global error estimate for the new non-standard method (2.71) with cubic spline interpolations: || c(-, •, tm+l) - Cm+1(-, OIL < QS (Ax 2 + At) , where Q is a positive constant, and S = max

max

I

0
dx

4

1 00

dy*

) , sup 00/

[0,l)xJ

sup [0,l]xj

2.6

dc dr

d2c dr2 ) sup)

dR

[0.1|x[ 0,1] 8Z2

Summary

Non-standard numerical methods have been developed for solving tran­ sient advective-diffusive transport equations with nonlinear reaction terms. Some of the important features of the proposed new methods are the nonlo­ cal modeling of nonlinear reactions and the use of more complicated denom­ inator functions. We were able to confirm, theoretically and through a set of numerical experiments, that the non-standard schemes are reliable and propagate sharp fronts accurately, even when the advection and reaction processes are highly dominant and the initial data are not smooth. They

Summary

105

have much greater computational efficiency as compared to standard nu­ merical schemes. Error analysis shows that the proposed new methods pro­ duce oscillation-free numerical approximations with zero local truncation error with respect to time. Large time steps can be taken without affecting the accuracy of the numerical solution. The "exact" time-stepping schemes can be easily combined with the operator-splitting ideas to produce a new class of non-standard numerical methods that accurately approximate the numerically difficult advection-reaction part of arbitrary model problems.

106

Bibliography

Bibliography

Allen, M.B. (1988) "Basic mechanics of oil reservoir flows", Multiphase Flow in Porous Media, M.B. Allen III, G.A. Behie, and J.A. Trangenstein, Lecture Notes in Engineering 34, C.A. Brebia and S.A. Orszag, Eds., SpringerVerlag, New York, 1-81. Allen, M.B. and F. Furtado (1997) "Computational Methods for Porous-Media Flows", Advances in Fluid Mechanics: Fluid Transport in Porous Media, J.P. du Plessis, Ed., Computational Mechanics Publications, Southampton, UK. Allen, M.B. and E.L. Isaacson (1997) Numerical Analysis for Applied Science, John Wiley & Sons, New York. Allen, M.B. and B. Liu (1995) "A modified method of characteristics incorporat­ ing streamline diffusion", Numer. Methods Partial Differential Equations 11, 155-174. Arbogast, T. and M.F. Wheeler (1995) "A characteristics-mixed finite element method for advection-dominated transport problems", SIAM J. Numer. Anal. 32, 404-424. Bailey, J.E. and D.F. Ollis (1986) Biochemical Engineering Fundamentals, McGraw-Hill, New York, NY. Bear, J. (1972) Dynamics of Fluids in Porous Media, American Elsevier, New York. Celia, M.A., I. Herrera, E.T. Bouloutas, and J.S. Kinder (1989) "A new numerical approach for the advective-diffusive transport equation", Numer. Methods Partial Differential Equations 5, 203-226. Celia, M.A., T.F. Russell, I. Herrera, and R.E. Ewing (1990) "An EulerianLagrangian localized adjoint method for the advection-diffusion equation", Adv. Water Resour. 13:4, 187-206. Celia, M.A. and S. Zisman (1990) "An Eulerian-Lagrangian localized adjoint method for reactive transport in groundwater", Computational Methods in Water Resources VIII, Vol. 1: Computational Methods in Subsurface Hy-

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drology, G. Gambolati et al., Eds., Computational Mechanics Publications, Southampton Boston, 383-392. Characklis, W.G and K.C. Marshall (1990) Biofilms, John Wiley and Sons, Inc., New York. Chen, B.M. and H.V. Kojouharov (1999) "Non-Standard Numerical Methods Ap­ plied to Subsurface Biobarrier Formation Models in Porous Media", Bulletin of Mathematical Biology 61:4, 779-798. Chiang, C.Y., C.N. Dawson, and M.F. Wheeler (1991) "Modeling of in-situ biorestoration of organic compounds in groundwater", TYansport in Porous Media 6, 667-702. Currie, I.G. (1993) Fundamental Mechanics of Fluids, 2nd edition McGraw-Hill, Inc. Dawson, C.N., T.F. Russell, and M.F. Wheeler (1989) "Some Improved Error Estimates for the modified method of characteristics", SI AM J. Numer. Anal. 26, 1487-1512. Douglas, J.Jr., F. Furtado and F. Pereira (1997) "On the numerical simulation of waterfiooding of heterogeneous petroleum reservoirs", Computational Geosciences 1:2, 155-190. Douglas, J.Jr. and T.F. Russell (1982) "Numerical methods for convectiondominated diffusion problems based on combining the method of character­ istics with finite element or finite difference procedures", SIAM J. Numer. Anal. 19, 871-885. Ewing, R.E. and T.F. Russell (1982) "Efficient time-stepping methods for miscible displacement problems in porous media", SIAM J. Numer. Anal. 19, 1-66. Gray, W.G. and G.F. Pinder (1976) "An analysis of the numerical solution of the transport equation", Water Resour. Res. 12:3, 547-555. Healy, R.W. and T.F. Russell (1993) "A finite-volume Eulerian-Lagrangian lo­ calized adjoint method for solution of the advection-dispersion equation", Water Resour. Res. 29:7, 2399-2413. Huyakorn, P.S. and G.F. Pinder (1983) Computational Methods in Subsurface Flow, Academic Press, New York. James, G.A., B.K. Warwood, A.B. Cunningham, P.J. Sturman, R. Hiebert, and J.W. Costerton (1995) "Evaluation of subsurface biobarrier formation and persistence", Proceedings of the 10th Annual Conference on Hazardous Waste Research, Great Plains/Rocky Mountain Hazardous Substance Research Center, 82-91. John, F. (1991) Partial Differential Equations, Springer-Verlag, New York. Johnson, C. and J. Saranen (1986) "Streamline diffusion methods for the incom­ pressible Euler and Navier-Stokes equations", Math. Comp. 47:175, 1-18. Kojouharov, H.V. and B.M. Chen (1998) "Non-Standard Methods for the Convective Transport Equation with Nonlinear Reactions", Numer. Methods Partial Differential Equations 14:4, 467-485. Kojouharov, H.V. and B.M. Chen (1999) "Non-Standard Methods for the Convective-Dispersive Transport Equation with Nonlinear Reactions", Nu-

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mer. Methods Partial Differential Equations, accepted. Kojouharov, H.V. and B.M. Chen (1999) "Non-standard Eulerian-Lagrangian Methods for Multi-Dimensional Reactive Transport Problems", SIAM J. Numer. Anal., submitted. Liu, B., M.B. Allen, H. Kojouharov and B. Chen (1996) "Finite-element solution of reaction-diffusion equations with advection", Computational Methods in Water Resources XI, Vol. 1: Computational Methods in Subsurface Flow and Transport Problems, A.A. Aldamaet al., Eds., Computational Mechan­ ics Publications, Southampton Boston, 3-12. Mickens, R.E. (1989) "Exact solutions to a finite-difference model of nonlinear reaction-advection equation: Implications for numerical analysis", Numer. Methods Partial Differential Equations 5, 313-325. Mitchell, A.R. and D.F. Griffiths (1980) The Finite Difference Method in Partial Differential Equations, John Wiley & Sons, New York. Morton, K.W. (1996) Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, London. Murray, J.D. (1993) Mathematical Biology, Springer, Berlin. Peaceman, D.W. (1977) Fundamentals of the Numerical Reservoir Simulation, Elsevier Scientific Publishing Company, Amsterdam. Russell, T.F. and M.F. Wheeler (1983) "Finite element and finite difference meth­ ods for continuous flows in porous media", Frontiers in Applied Mathemat­ ics, Vol. 1: The Mathematics of Reservoir Simulation, R.E. Ewing, Ed., SIAM, Philadelphia, 35-106. Vag, J.E., H. Wang, and H.K. Dahle (1996) "Eulerian-Lagrangian localized ad­ joint methods for systems of nonlinear advective—diffusive-reactive trans­ port equations", Adv. Water Resour. 19:5, 297-315. Wang, H., R.E. Ewing, and M.A. Celia (1995) "Eulerian-Lagrangian localized ad­ joint methods for reactive transport with biodegradation", Numer. Methods Partial Differential Equations 11, 229-254.

Chapter 3

Application of Nonstandard Finite Differences to Solve the Wave Equation and Maxwell's Equations

James B. Cole*

Abstract We develop practical nonstandard finite difference (NSFD) algorithms to solve the wave equation and Maxwell's equations, and use them to simu­ late wave propagation and scattering in nonuniform media with irregular boundaries. To test the accuracy on actual problems we consider the case of Mie scattering and compare with known analytic solutions.

3.1. Introduction In this chapter we introduce nonstandard finite differences (NSFD) in the context of practical algorithms to simulate wave propagation and scattering phenomena. We need to solve either the wave equation or Maxwell's equa­ tions. In inhomogeneous media with irregularly shaped boundaries there are no general analytical solutions, and numerical calculations are the only alternative. A straightforward and natural approach is to approximate the differen* Institute of Information Sciences and Electronics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan (coleOis.tsukuba.ac.jp). 109

Application of Nonstandard Finite Differences

110

tial equation to be solved with finite differences (FD), and then try to solve the resulting difference equation. The first derivative can be approximated by

f{x)*idmf{x) = ±M,

(3.1)

where dx is a difference operator, and h = Ax is the discretization. There are three basic difference operators defined by, dxf{x)

= f(x + h)-f(x),

(3.2a)

dxf(x)

= f(x)-f(x-h),

(3.2b)

dxf(x)

= f(x + h/2) - f(x - h/2).

(3.2c)

Operators (3.2a-c) are called forward, backward, and central difference operators, respectively. Although lim.h->o(dx/h)f(x) — f'(x) for all choices of dx, for non-infinitesimal h these approximations give different results in numerical algorithms. Expanding (3.2a) and (3.2c) in Taylor series about x, we find /(x + < 0 - / ( « ) f(x

+

h/2)-f(x-k/2)

= //(i) +

=

^

+

1 hfl{x) !_ h2fH{x)

The approximation error, e = \(dx -dx)f(x)\, FDs is thus

+

1_ h2f)ll{x) +

+

^ (3

3a)

(3 3b)

for the forward and backward

£fb = lhf"(x) + ±h2f"(x) + ..., while for the central FD it is

ec = ±h>f»(x) + .... The forward and backward FDs are said to be first-order accurate, because e/b ~ h, while the central FD is second-order accurate because ec ~ ft2. For |/i| < 1, one expects that ec < £/;,, but exceptions can arise due to the behavior of f"(x) and higher derivatives. The choice of which FD approximation to use is governed not only by considerations of accuracy,

The One-Dimensional

Wave Equation

111

but by other factors, such as algorithm stability, that we will discuss later. Sometimes it is actually better to use a lower accuracy FD. Replacing the derivatives of the differential equation by FD approxi­ mations yields a difference equation, the solution of which, we hope, is a reasonable approximation to that of the differential equation.

3.2. The One-Dimensional Wave Equation 3.2.1. Finite-Difference Time-Domain Algorithm There is an old proverb, "One good example is worth a thousand theorems." In this spirit let's derive an algorithm to solve the one-dimensional wave equation, (dtt-v(x)2dxx)4>(x,t)=0.

(3.4)

The wave function, tj), is the displacement of some physical quantity from its equilibrium position — of a point on a stretched string, or of the air pressure from its ambient value —, and v is the propagation velocity. Rewriting the wave equation in the form dtlip(x, t) - v(x)2dxxxj>(x, t), duip(x,t) can be interpreted as an acceleration, and v(x)2dxxtp(x, t) as a force term. Furthermore, dxxtp{x,t) is proportional to curvature. Think­ ing of tp(x, t) as the displacement of a point on a string, if dxxip(x, t) were zero, the string would be locally straight and there would be zero accelera­ tion because the equal and opposite forces due to the string tension would cancel. The wave equation thus states that acceleration is proportional to curvature. Propagation velocity is a function of the physical properties of the medium, such as its density and elastic modulus. Light, for example, travels with velocity VQ in free space, but in a medium of local refractive index n(x), its velocity is v(x) = VQ/TI{X). Equation (3.4) is thus a model of the propagation of light in a medium in which the index of refraction varies with position. Before proceeding to construct an algorithm to solve (3.4), we need an FD approximation for f"(x). Expanding f(x + h) and f(x — h) in Taylor

Application of Nonstandard Finite Differences

112 series, we find

f'M

~ fi* + h) +

f{x-h)-2f(x)

Defining dxdx = cfc, where dx is the central FD operator (3.2c), we find dlf(x)

= f(x + h)+ f(x -h)~

2f(x),

(3.6)

so we can write

f(^a*) = ^ .

(3-7)

From the Taylor series expansion of (3.5) we find that the approximation error is \{d„-dXI)f(x)\~h2.

e=

Using (3.7) in (3.4), we obtain an FD approximation to the wave equation, (±;%

-v(xf±d^

tf(s,i)

=0.

(3.8)

For now, we write \j) in place of ip to emphasize that the solutions of (3.4) and (3.8) are not necessarily the same. While (3.4) is a differential equation, (3.8) is only one of several possible difference equations that could model it. Later another difference equation model of the wave equation will be introduced. Now let us solve (3.8). Many techniques are available, but for our purposes the most useful one is the finite difference time-domain (FDTD) method [1]. Rewriting (3.8) in the form, %ii>{x,t)=v{x)2-±-dlt and expanding dtV>(x, t), we obtain rp(x, t + At) + i>(x, t-At)-

At 2 2^(x, t) = - ^ - u(x) 2 <£^(x, t).

Solving for i/>(x, t + At), we find j>(x, t + At) = 2xj)(x, t) - ip(x, t-At)

At2 + -r-r v{x)2<%.4>(x, t).

(3.9)

The One-Dimensional Wave Equation

113

Given the two initial values, ^(x,0), and tj)(x, At), ij> can be found at all future times simply by iterating (3.9). It is easy to implement algorithm (3.9) in a computer program. In the range 0 < x < a, let x take the discrete values x = 0, h, 2h,..., Nh, where h = a/N, so that V and v are (N + l)-arrays. As an example, let v(x) = 1 for x < a/2, and v(x) — 1/2 for x > a/2. This models a space in which the index of refraction equals 1 for x < a/2 , and 2 for x > a/2. We can input an initial incident signal in the x < a/2 region and compute tp at all later times without having to explicitly insert the boundary conditions at the interface between the different indices of refraction, provided that V> = 0 initially in the x > a/2 region. This is an important advantage of the FDTD method, especially in two and three dimensions, where the interfaces between different indices of refraction could be a complicated surfaces. Many physical problems involve sources (of light and sound, for example) and absorbing media. The wave equation with sources and ab­ sorption is (dtt - v(x)2dxx)i>(x, t) = v(xfs(x,

t) - 2a(x)dtrp(x, t),

(3.10)

where s(x, t) is the source term and Q(X) is the absorption. Rearranging, we can write dtf4>{x, t) = v{x)2dxxtp{x, t) + v{x)2s(x, t) - 2a{x)df4>{x, t). Here v(x)2s(x, t) can be interpreted as an external driving force, while a(x)dti>(x, t) is a frictional retarding force proportional to velocity. The factor of 2 in front of a is merely conventional. The solution of (3.10) cannot be expressed as a simple function. For l kx ut at the case s = 0, however, an approximate solution is ip(x, t) = e ( - )e- ( we define v — w/k. Substituting into (3.10), it can be verified that ip is a solution if 0 < Q 2 < 1. The lack of an exact analytic solution, even for this slightly more complicated wave equation, shows the importance numerical calculations. An FD model for this equation is ( J L % - v(x)2 1 a^j rj,(x, t) = v(x)2s(x, t) - a(x) ± d'^x,

t), (3.11)

Application of Nonstandard Finite

114

Differences

where d't is a difference operator defined by d'tf(t) — f(t + At) — f(t — At). We must use the approximation ait

^ ~ <*W>(M)

instead of dtijj(x,t) = (1/At)dtip(x,t) because d|i/>(x, £) is evaluated using t and £ ± At, whereas dtip{x, t) is computed using t± At/2. Expanding the FD approximations and collecting 4>(x, t + At) terms we obtain, (1 + a(x)At)^(x, t + At) = 2^(x, t) - (1 - a{x)At)i>(x, t - At) At2 + — v(x)2cPxij;(x, t) + v(x)2At2s(x,

t),

which can be solved to give 4>(x, t + At) = fax, t) + ^

d\i>(x, t - At/2) + j - ^ 3 - v{x)2J*i>(x, t)

+ -^-v(x)2At2s{x,t),

(3.12)

where A±=l±a(x)At,

(3.13)

and we have used the identity 2 — A- — A+, and the fact that dtip(x, t - At/2) = i>(x, t) - rf>(x, t - At). In a computer program to iterate (3.12) rp, A+, A-/A+, v, and s are arrays. Assuming that for 0 < t, ip(x, t) = 0 and s(x, t) = 0, then starting the iteration at t = —At, we have the very simple initial conditions that ip(x,—At) = 0, and 4>(x,0) = 0. This initial set of inputs obeys all the boundary conditions, however complicated, and as the algorithm is iterated the boundary conditions are maintained. Input signals can be generated using source arrays. In two and three dimensions, the advantages of this formulation will become apparent.

The One-Dimensional

3.2.2. Algorithmic

Wave Equation

115

Error

An important question remains unanswered, "Is ip a good approximation to ipV We now proceed to address this question. lim

4>(x,t) = ip(x,t),

(3.14)

then V» converges to ip, but the important question for practical applications, is "What is the solution error for non-infinitesimal h and Ai?". One solution to the wave equation (3.4) is V'oOM) = e l ( f c a ! - w t ) ) which represents a forward moving (+ x-direction) monochromatic wave of fre­ quency / and wavelength A, where u — 2ir/f and k = 2n/X. Velocity is related to u; and k by v = uj/k.

(3.15)

Equation (3.15) is called a dispersion relation. A dispersion relation gives the relationship between propagation velocity, wavelength and frequency. In most physical problems the frequency of a signal propagating in an inhomogeneous medium is constant while its wavelength varies with propagation velocity, thusfe(x)= u>/v(x). Substituting ^o hito the FD approximation to the wave equation, (3.8), we find (o?t-v{x)2^-dfjM^t)=s.

(3.16)

Evaluating e, we find e = 2rpo(x,t)£, where At2 £ = (cos{wAt) - 1) - v(x)2 -rp (cos(kh) - 1).

(3.17)

We now see immediately that limh_vo,At-+o £ = 0, so tp does indeed converge to ip, provided that At/h is finite. Later, we shall see that in fact the value of At/h is not arbitrary, but must satisfy the constraint v(x) ^ < 1. (3.18) h Such a constraint is called a stability condition. Stability conditions will be discussed later.

116

Application of Nonstandard Finite Differences

For non-infinitesimal h and At solutions of the wave equation and its finite difference approximation are not the same because e ^ 0. If could be made to vanish, then tp and xjj would be identical, even when h and At are not infinitesimal. Setting i = 0 in (3.17), we find , 2 At 2 cos(wAt) - 1 V{X) W = cos(Mx)/0 - 1 • In other words, the replacement of v(x)At/h

(3 19)

-

by

, . sin(u>At/2) u i = . ,, , ',' ,

.„ „„. (3.20)

in (3.16), gives an alternate FD model of the wave equation, (%-u(x)2JPx)$(x,t)=0,

(3.21)

that has the same solution as the wave equation itself. Replacing by u(x) in (3.9), we obtain i>(x, t + At) = 2ip{x, t) - i>{x, t-At)

+ u(x)2d2x4>(x, t),

v(x)At/h (3.22)

which is an exact algorithm to solve the one-dimensional wave equation for signals of constant angular frequency in a medium of spatially vari­ able propagation velocity. Comparing v(x)At/h with u(x), we see that in model (3.21) waves propagate with velocity u(x)h/At instead of v(x) on the numerical grid, and the dispersion relation is (3.20) instead of (3.15). Another way to reduce the solution error (3.17) to zero is to impose the constraint v ^ = 1. (3.23) h In this case solution error is zero for all wave forms, that is for arbitrary wave functions of the form ip(x, t) = f(x - vt) + g{x + vt)

(3.24)

not just harmonic signals like V'o- Unfortunately, however, error vanishes only for the case of constant velocity. If v is a nonconstant function of i , the error does not vanish, even if (3.23) is maintained by varying h as a function of x. Thus propagation and scattering problems in nonuniform media cannot be solved by with zero error by algorithm (3.9) with condition

The One-Dimensional Wave Equation

117

(3.23). Algorithm (3.22) can, however, solve such problems for harmonic signals with 2ero error except at the media boundaries (the sources of these errors are discussed in Sect. 3.2.4). Furthermore algorithm (3.22) reduces to (3.9) under the constraint (3.23). The important lesson of this section is that clever FD modeling of dif­ ferential equations can yield large improvements in accuracy with low com­ putational cost. A "direct translation" of the original differential equation into an FD approximation, such as (3,8), is not necessarily the most fruit­ ful approach. Algorithm (3.22), more accurate though it is, might appear to be somewhat ad hoc. Let's try to find a more general approach by reconsidering the FD approximation to the derivative. 3.2.3. Nonstandard

Finite

Differences

A major source of error in any FD model of a differential equation is the modeling of the derivatives. One way to reduce the error is to use a smaller step size, but this increases both computer memory requirements and the number of computations. Referring to the one-dimensional wave equation, rp is a one-dimensional array of length proportional to l//i, and the number of iterations needed to solve a problem is proportional to 1/At. Because of the stability requirement (3.18), h ~ At, so the number of computations required is proportional to l//i 2 . For a d-dimensional wave equation, the computational load therefore rises as l/hd+1. Thus, halving the value of h in a three-dimensional version of (3.9) requires 8 times as much computer memory and 16 times as many computations. Clearly the price of decreas­ ing h is very high. Is there a way to increase accuracy without decreasing hi One way is to use a higher order FD approximation for the derivatives. From Taylor series expansions, it can be shown that the approximation error of

^4

I (/(* + h) + f{x - h)) - i ( / ( x + 2/i) + fix - 2/i)) - \

fix) (3.25)

is e = — ^h4 fWix) H . This is called a fourth-order FD approximation 4 because e ~ /i . An algorithm that, uses fourth-order FDs is, in general, more accurate, but unfortunately it is not only much more complicated and difficult to program, but also uses more computing resources.

118

Application of Nonstandard Finite Differences

Is there a better way? Can we achieve higher accuracy with a low order FD approximation, without decreasing step size or resorting to higher order derivatives? Let's consider a single frequency, / . Analytic solutions to the onedimensional wave equation can be expanded in the form, rP(x, t) = oe i ( * x - w t ) + &ei(**+u't>,

(3.26)

where the functions ex(kx^ut) represent forward- and backward- ( i n ­ directions) moving waves, and w and A; are related by the dispersion re­ lation (3.15). If we could define an FD approximation that gives deriva­ tives of e*(fclTwt) with better accuracy than (3.1), we might be able to construct a better algorithm than (3.9) (of course we already know that one exists). Defining dx by (3.2c), let's compute dxexkx. We find, dxexkx = 2ze,fcxsin(fc/i/2). So if we define

* « " > - ^ -

(3 27)

'

where, s(h) = s(k,h), where s(fc)ft)=2Jin^2))

dxf(x) = f'(x) exactly, for / € S = {sin(fcx),cos(fcx),e ±ifcl }. The FD approximation defined by (3.27) is called a nonstandard finite difference (NSFD) [2], where s is a "correction function" that is chosen to minimize the difference \{dx — dx)f(x)\ with respect to some set of basis functions (in this case 5). One might wonder if an exact FD approximation could always be denned by setting « * ) - ^ § > .

(3-29)

Although for set S, it is possible, in general it is not. First of all s must satisfy lim s(h) = 0,

(3.30)

but even when it does, (3.29) may not always be usable in a practical algorithm.

The One-Dimensional

Wave Equation

119

We can now construct a model of the wave equation using NSFDs. All we have to do is to make the substitutions At -> s(tu, At), and h -> s(k, h) in (3.8).

.^(«,t)__ 2

4sin (wAt/2)/to

2

^(M)

v{x? v

'

Q

(331)

Asm2{kh/2)/k2

Using the fact that v = to/k and solving for ip(x, t + At), we again arrive at algorithm (3.22). Although this algorithm is "exact," there are, however, still errors that arise in a practical model. The velocity is a function of x, which may be imperfectly represented on the numerical grid, and since k(x) is not constant over the grid points x — h, x, x + h, there is an error in the NSFD approximation to dxxf(x), even for / € S. In addition the algorithm is exact only with respect to solutions of the particular angular frequency used in (3.31). Since in a finite computational space, we can never have an exactly monochromatic wave, errors will arise from signal components with angular frequencies that deviate from CJ. Despite these limitations, however, the NSFD algo­ rithm (3.22) gives superior results to the standard finite difference (SFD) algorithm (3.9). Let's now construct an NSFD model of the wave equation with sources and absorption (3.10). The procedure is the same as before, except that we now use the approximation

where s(u), 2At) = 2sm(cjAt)/u}. ^

=

Also replacing A± by

l±q(x)

2 t a n (

^

/ 2 )

,

(3.32)

in (3.12) we obtain A' 1 4>(x, t + At) = 4>(x, t) + -r=- dti>{x, t - At/2) + -j- u(x)2(%rp(x, t) A+

+ -±j-v{x)2s{u},At)2s(x,t).

A+

(3.33)

Notice that A'± converges to A± in the limit At -* 0. Although the NSFD correction function, s(u, At), and the source function, s(x, t), share the

120

Application of Nonstandard Finite

Differences

same symbol, they are not related. Which one is meant should be clear from the context and the arguments. 3.2.4. One-Dimensional

Scattering

There is a (probably) modern proverb, "The Devil is in the details." Despite the simplicity of the algorithms, actually applying them to solve interesting problems requires careful thought. We will not enter into all the devilish details that plague practical numerical simulations just now, but in order to compare the performance of the SFD (3.9) and the NSFD (3.22) algorithms on a realistic physical problem with a known solution, let's simulate the physical process of one-dimensional scattering. Let there be an infinite wave scatter off a one-dimensional slab of width w. Inside the slab the index of refraction is n = ni, while outside n = 1. It can be shown that for

" ( » + 5)£

<334

»

(m = integer) that there is no reflection off the slab, where Ao is the wave­ length exterior to the slab. This is a one-dimensional model of an optical anti-reflective coating. A numerical simulation of an antireflective coating constitutes an easy and sensitive test of algorithm accuracy. Setting h = 1, and At = 1, we formulate the problem on the region 0 < x < N, where N is an integer. Taking A0 = 8, n» = 2, and w = 2 let's now apply algorithms (3.9) and (3.12) to simulate a wave impinging on the slab. The first practical question is how to represent the slab on the grid. One possibility is to let the front and back of the slab coincide with the grid points (representation A), but another possibility is to assume that the slab edges lie between the grid points (representation B). In representation A, the slab is represented by three points with n = rij, while in B, only two points have n = rii. A priori it is difficult to know which one is correct. The next problem is how to simulate an infinite wave, and to discriminate the reflected wave from the incident one. Lacking infinite computer memory, we have no choice but to use a finite wavetrain as the input signal. Centering the slab in the computational grid, and inputting a finite wavetrain of length less than (N — w)/2 allows us to clearly see both the reflected and transmitted waves.

The One-Dimensional

Wave Equation

121

Fig. 1. (a) Incident wavetrain impinges on a slab of refractive index 2.0, and width 2. In representation B, two grid points are interior to the slab.

~\

i

i

i

i

i

r

/\i«»vn»i

r? J

L

J 10

12

14

L 16

Fig. 1. (b) Scattered wavetrain using SFD algorithm and slab representation A.

i

1

\ -2

_,

1

A J

1

1

1

1

r

■ A , ■ - ■ « ■.<->.

L

J 10

I

L

12

14

Fig. 1. (c) Scattered wavetrain using NSFD algorithm and slab representation A.

Application of Nonstandard Finite Differences

122

2.0

n

1

i

2

1

1

1

r

Vwwv\/v—*

-1

0 0

2

4

I

I

I

I

L

6

8

10

12

14

16

18

Fig. 1. (d) Scattered wavetrain using SFD algorithm and slab representation B.

2.0

1

2

1

1

V J 0 0

2

1

1

r

^

I

I

I

I

I

I

L

4

6

8

10

12

14

16

18

Fig. 1. (e) Scattered wavetrain using NSFD algorithm and slab representation B.

The One-Dimensional

Wave Equation

123

Exterior to the slab we choose v = 1 so that both algorithms are exact (with h = 1, At = 1), and inside the slab v = 1/rij = 1/2. Iterating both the SFD and algorithms we obtain the results shown in Fig. 1 for repre­ sentations A and B. We see that the NSFD algorithm with representation B gives the best result. The reflections at the leading and trailing edges of the signal are due to the finiteness of the signal. Even though the NSFD algorithm itself is "exact" there are two sources of error due to the imperfect numerical model. Firstly, the NSFD calcu­ lation of dxxip(x, t) is in error in the neighborhood of the slab boundaries (x = a, x = b) because it uses values of ip from two different regions (in­ side and outside the slab) where the wavelength is different. Secondly, we have modeled an infinite monochromatic wave with finite wavetrain. It can be shown from Fourier analysis that the signal generated by a har­ monic source of frequency / turned on for duration T contains a range of frequencies A / ~ 1/T centered on / . Thus error arises in the NSFD calculation of duil>{x, t) because it is exact only at one frequency. Never­ theless, despite these limitations, the NSFD algorithm gives a much bet­ ter result than the SFD one. For increasingly finer discretization (larger value of A//i) we find that the SFD result converges to the NSFD one, and also that calculations using representation A converge to those using repre­ sentation B. This exercise shows some of the limitations of our model. In represen­ tation B, slabs of width 1.5 < w < 2.5 have the same representation on the grid, and are therefore indistinguishable. Also the wavelength that can be represented on the grid is limited to integer multiples of h. There is the additional constraint that the discretization on a one-dimensional grid must satisfy X/h > 3. Thus the maximum index of refraction that can be represented on the grid for the slab problem is the ratio of the exterior wavelength to minimum possible interior wavelength, where the interior wavelength is A* = Ao/rij. From this we find that

n, < I £ •

(3-35)

Despite the limitations of the model, however, we see that it does give the correct answer to a rather difficult problem with a very low discretiza­ tion — especially inside the slab.

124

Application of Nonstandard Finite

Differences

3.3. The Two- and Three-Dimensional Wave Equation 3.3.1. Standard FDTD Algorithm Now let us extend NSFD concepts to the wave equation in two and three dimensions [3-5]. The homogeneous wave equation is (dtt-t,(x)2V2)V(x,t)=0,

(3.36)

where x = (x,y, z) is the position vector, and V 2 = dxx + dyy + dzz. Following Sect. 3.2, it is straightforward to construct the standard finite difference (SFD) algorithm to solve the three-dimensional wave equation. With obvious extensions of the notation, the SFD approximation to (3.36) is (%-v(xf^-D?)i,(x,t) = 0,

(3.37)

where D\ is a difference operator given by

£? = «£ + ($+
(3.38)

The subscript " 1 " of D\ distinguishes it from another difference opera­ tor that will be introduced later. Here we have, for simplicity, assumed a uniform space grid, such that Ax = Ay = Az = h. Everything that follows could be developed on a nonuniform grid, but it would be more complicated. In close analogy with (3.9), the FDTD algorithm to solve (3.37) is Ai 2 V>(x, t + At) = 2V>(x, t) - 4>(x, t-At) + -j^- v(x)2Dfy(x,

t).

(3.39)

Following Sect. 3.2.2 the algorithmic error can be evaluated by substi­ tuting a solution to the wave equation into its FD approximation. A plane wave solution is given by if>o{x.,t) — e « ( k x - u , 0 ) where k = (kx,ky,kz) is called the wave vector. The magnitude of k, A; = |k|, obeys the dispersion relation (3.15), where k = 2n/\, and A is the wavelength. The solution error is \£ - v(x)2 -g- b\\ V-o(x, t)=e,

(3.40)

The Two- and Three-Dimensional Wave Equation

125

where e — 2tpo(x,t)£, and At2 £ = (cos(u/Ar.)-l)-i>(x) 2 —2 h

J^

{cos{kih) - 1).

(3.41)

t=x,y,z

Following Sect. 3.2.2, we might try to construct an exact algorithm by choosing v(x)At/h such that £ vanishes. Unfortunately this strategy fails because £ now depends on the direction of k. In other words the solution error is anisotropic. We could make e vanish for one particular direction of k, but not for all directions. At fixed frequency the general solution of the wave equation can be expressed as a superposition of plane waves propagating in different directions,

V(x,t)= 2> k e i < k o t - u ' *>,

(3.42)

|k|=fc

where the summation is over all directions of k, for a fixed magnitude A;. Since all propagation directions enter into the solution, an exact algorithm for one propagation direction is of little use. The anisotropy of £ is due to the error of the FD approximation to

Wk'x,

£l= V2

(

~§VkX'

(3.43)

which can be expressed as £i = 2e , k x £i, where -£x = k2 - ±2 h

J2

(cos(fcifc) - 1),

(3.44)

i=x, y,z

which is clearly anisotropic. 3.3.2. Generalized Dimensions

Nonstandard

Finite Differences

—

Two

As we saw in the last section, a simple extension of the nonstandard finite difference (NSFD) concept, will not reduce the solution error of the wave equation in more than one dimension. It is necessary to generalize the NSFD concept in some way. For simplicity, let's first develop the basic concepts in two dimensions. In two dimensions, there are two Laplacian

Application of Nonstandard Finite Differences

126

_

_

iB

WT$ l^H^H^H

raWa

.^^

Fig. 2. Gaussian beam generated by a weighted current source array incident on a dielectric ellipse (left) and after scattering (right).

The Two- and Three-Dimensional

Wave Equation

127

difference operators. The first one, D2, which was introduced in the last section, reduces to

£? =
(3.45)

in two dimensions. A second Laplacian difference operator is defined by 2Dlf{x,y)

= f(x + h,y + h) + f(x -h,y + f(x-h,y

+ h)

+ h) + f(x -h,y-h)-

4f(x,y),

(3.46)

and an alternate FD approximation to V 2 is thus D\ = D^/h2. D2 and t)\ are depicted graphically in Fig. 2. Such graphical representations are called "computational molecules." The size and shape of computational molecules affect the speed and memory requirements of practical com­ puter algorithms, as will be discussed later. This second difference op­ erator gives us an additional "degree of freedom" with which to construct a two-dimensional NSFD approximation to V 2 / ( x , y), by using a weighted superposition of D\ and D2. We write V 2 / ( x , y) = D2f(x, y) = ^

/(*•») >

(3'47)

where D 2 = 7 ^ i + (l-7)^22.

(3-48)

and 0 < 7 < 1. The quantities s and 7 are in general functions of position and of the parameters of the differential equation to be solved. D2 is an example of what we shall call a generalized NSFD (GNSFD). Let us now develop a GNSFD approximation of V 2 e* k x . Since in one dimension d%eikx = 2(cosfc - l)exkx, let's try to find a value of 7 such that D g e i k x S* 2(cosk - l ) e i k x

(3.49)

for all directions of k at a fixed magnitude, k = |k|. For notational econ­ omy we have defined k = kh. The significant quantity is not the abso­ lute wavelength but A = X/h in terms of which k — 2TT/\. Notice that 2(cosJfc - 1) = -4sin 2 (fc/2) = -k2s(k,h)2, where s is given by (3.28). In other words if we could find a value of 7 such that (3.50) is true, at least to

Application of Nonstandard Finite Differences

128

"high" accuracy we could construct a high accuracy approximation V 2 e , k x of the form (3.47). Slightly rewriting (3.48), we seek an approximate solution of Yi{D\ - D\) + D 2 ] e i k x = 2(cosk - l ) e i k x .

(3.50)

Defining Z> 2 e ikx = 2e ikx ao 1 ()b,0), and £ » | e i k x = 2 e i k x a o 2 ( M ) , it is easy to show that a6i(k,0) = cos kx + cos ky— 2, a&i{k,6) = coskxcosky

— 1.

(3.51a) (3.51b)

In polar coordinates k = (kx,ky) = fc(cos#,sin#), and a6\ and ao? can be expressed as functions of k and 9, where 9 is the angle k makes with the x-axis. Equation (3.49) then becomes 7(ao! (k, 9) - afaik, 9)) + ao^ik, 9)=cosk-l,

(3.52)

which has the formal solution 7

(fc,g)=(c°^.-1)-^(>g). a6i(k, 9) — ao2(k,9)

(3.53)

At first sight this solution does not appear to be of much use because 7, which we would like to be independent of 9 is, in fact, a function of 9. It turns out, however, that its ^-dependence is "weak." This means that 7(fc, #o) which satisfies (3.52) exactly at a particular value #o is and is still a "reasonable" approximate solution even at 9 ^ 0o- It can be shown that a value of #o which minimizes the deviation (as a function of 9) of ■Doe*K'x/e*k x from 2(cosfc — 1) is given by cos4 0O = \ ■ We thus define tfa

where

=

(k 0 ) =

cosfci0)cosfc£0) -cosA: 1 + cos kx cos kv ' - cos kx ' - cos ky '

{kx°\kv0))=k{cos9Q,sva90).

(3.54)

The Two- and Three-Dimensional Wave Equation

129

Using 'y(k) to construct DQ we can evaluate the anisotropy of ZJge* x / e i k x and compare with that of £>2eik x / e i k x , D\eiV- x / e < k x . Defining £>t2eik:

(3.56)

2(cosA:- 1),

pik x

and expanding in Taylor series, it can be shown that

£l =

h (-(^) 4 +^( fcft ) 6 ) sin22 *+---.

(3.57a)

£2 = ^ ( ( * > 0 4 - j £ ( ^ ) 6 ) sin2 26 + ■ ■ • ,

£o =

i s k ((fc/l)8 ~ k{kh)W) {\sin4

(3.57b)

29

~{V* ~l) sin2 29)+ (3.57c)

Now £>,V k x = [2{cos(kh) - 1) + ffje*-* (i = 0,1,2), and we find that D t 2 e i k x (i = l,2) is D

i

h2

„ikx .

_ik'X

-t,+p"

(3.58a)

ile Z)2e
tkx

fc2

2

- fc +

sinJ(lfc/i/2)

£o

„tk x

(3.58b)

The deviation of D 2 e < k x = (D$/s(k,/i)2)eikx from W k x = - A : V k x is 8 2 thus of order /i , compared with order h for D2 and D2. We have thus constructed an 8-th order accurate FD approximation by superposing two second-order accurate ones. We can now construct a high accuracy algorithm to solve the wave equation in two dimensions from (3.39) with the substitutions h —¥ s(k, h), At -> s(w, At), D\ -> DQ. In analogy with (3.22) we obtain ^(x, t + At) = 2^(x, t) - i>(x, t-At)+

u(x)2D^(x,

t).

(3.59)

The extension to three dimensions is now obvious. We only need a threedimensional expression for DQ.

130

Application of Nonstandard Finite

3.3.3. Generalized Dimensions

Nonstandard

Differences

Finite Differences

in Three

The generalization of the NSFD concept to three dimensions is straightfor­ ward, but is complicated by the fact that there are now three independent Laplacian difference operators. The first one, D\, is defined by (3.38), the second one is 4Dlf{x,y,z)

= f{x + h,y + h,z + + f{x + h,y-h,z

h)+f{x-h,y-h,z-h) + h) + f(x - h,y + h,z-

+ f(x -h,y

+ h,z + h) + f(x + h,y-

+ f(x -h,y

-h,z

h,z-

h) h)

+ h) + f(x + h,y + h,z - h)

-2f{x,y,z).

(3.60)

The third Laplacian difference operator is constructed by combining twodimensional Laplacians of the form (3.46), for each of the three possible coordinate pairs: 2Dl = D*m')+D*"i+D*1"\

(3.61)

or more explicitly ADlf(x, y, z) = f(x + h,y-

h, z) + f(x-h,y

+ f(x + h,y,z-h)+ + /( x > y + h,z-

+ h, z)

f(x -h,y,z

h) + f(x, y-h,z

+ h) + h)- 3f(x, y, z). (3.62)

Letting each difference operator act on e , k x , we define D?e i k x = 2e ikx ao!(jfe,6),

(3.63)

and find adi (k, 8, cj>) = cos kx + cos ky + cos kz — 3,

do2(fe, 9,
(3.64a)

(3.64b)

2do3(fc, 9, ) = cos kx cos ky + cos kx cos kz + cos ky cos kz — 3, (3.64c)

The Two- and Three-Dimensional

Wave Equation

131

where k = (kx,ky,kz) = fc(sin0cos^>, sin0sin(/>, cos#) and 9 and <j> are the usual spherical coordinates. We now seek a combination of the above difference operators,

£o2 = 2 > 4 2 ,

(3-65)

•=i

with the constraint £ ^ rji = 1, such that 3

A 0 = ^i =Jl? i A i

(3.66)

is isotropic and nearly equal to cosfc — 1. First let's minimize the 9-variation at <j> — 0 of the linear combina­ tions A 1 2 = 712A1 + (1 - 7i2)A 2 and A13 - 713A1 + (1 - 7i3)A 3 by solving 712(601 (A:, 9,0) - ao2(k, 9,0)) + ad^tfc, 9,0) = cos k - 1,

(3.67)

7i 3 (a6i(k, 9,0) - ao3(k, 9,0)) + ao3(k, 9,0) = cosk - 1. Following the previous section we find, V

=

'

V

=

'

(co«fc-l)-«fr(M,0) a O l ( M , 0 ) -aoi(k,9,0) (cOsfc-l)-a63(M,0) a6l(k,9,0)~a63(k,9,0)

(3 6ga) K

'

6g K

'

It is easy to show that 712(^,0) = l(k, 9) (the same function as (3.53)) and that 7i3(k, 9) = 27(fc,0) - 1. Again the 9-variation is minimized with the choice 9 = 9Q. Defining 712(A) = j(k) and 713(A) = 27(A) - 1, we can constrict two difference operators that are nearly isotropic with respect to 9 at = 0, D\2= 7 # i + ( 1 - 7 ) ^ 1 ,

(3.69a)

D 2 3 = (2 7 - l)D\ + 2(1 - 7 )D 3 2 .

(3.69b)

Application of Nonstandard Finite Differences

132

A superposition of these operators can now be constructed that has "small" variation with respect to both 4> and 9, D2 = T}D22 + (1-V)D2U.

(3.70)

Redefining Ai 2 = 7A1 + (1 " 7)A 2 ,

(3.71a)

A13 = (27 - l)Ai + 2(1 - 7 ) A 3 ,

(3.71b)

and taking 6 — n/4 we seek a solution of Tj(a6i2(k,7r/4,0) -aoi3(fc,7r/4,<j>)) + a6i3(k,7r/4,4>) = cosfc - 1,

(3.72)

and find -

=

Ai3_(W4,<M-(cosfc-l) A 13 (A;,7r/4,0)-A 12 (fc,7r/4,
(3 73)

The ^dependence of r)(k, ) turns out to be "weak" and we can find a value, o, that minimizes the (/(-variation. In three dimensions the algebraic anal­ ysis is much more difficult, but numerically we find o = 0.118117T. Since Ao(fc, 6,0) = Ao(fc, 6) (the latter being the two-dimensional version), its be­ havior with respect to is the same. Following the procedure of the previous section it can be shown that the deviation of D^e1* x = (Dg/s(fc,/i) 2 )e , k x from V 2 e , k x = _fc 2 e » k x j s of order hs. Finally solving for on in (3.65) we find m = »7(1 - 7) + (27 - 1), m = ij(i - 7 ) ,

(3.74)

m = 1 - (m + m) ■ In three dimensions the final algorithm has exactly the same form as (3.59). The FDTD algorithm to solve wave equation with sources and attenuation (dtt - v(x)2V2)^(x,

t) = v(x) 2 s(x, t) - 2a(x)$V(x, t),

(3.75)

in two and three dimensions is exactly analogous to algorithms (3.12) and (3.33) with the replacement of a*, by either D2 or DQ.

Discretization

and Stability

133

3.4. Discretization and Stability 3.4.1. Discretization The finite differences of finite difference algorithms are not completely free parameters. If the finite differences are too large, accuracy, as we have seen falls, but there are more fundamental limitations. Rather than developing a general theory, we restrict ourselves to the wave equation. For harmonic waves propagating on a grid, one obvious limitation is the wavelength of the signal. Let x = 0,h, 2h,... for a signal ip(x) = asin(fca:)+6cos(fca:), where k = 2-K/X. If X/h = 1, on grid points ip appears to be a steady state signal given by xj> = b. The same holds when X/h = 2. Only when X/h = 3 can we solve for both a and 6. We thus have the general constraint > 3, (3.76a) h where h is the largest spatial step on the numerical grid. The same con­ siderations apply to the ratio of the time step to wave period, and we have T/At > 3, where T is the wave period. In two dimensions the largest spatial step is no longer simply h because signals can propagate diagonally on the grid. The effective space step is thus h\/2. Inserting this value into (3.76a) and rounding up to the nearest integer, we obtain r

-^>5. (3.76b) h Similarly in three dimensions the effective space step is /i\/3, and (3.76) becomes ^ > 6. (3.76c) h These limitations do not mean that shorter wavelength signals cannot propagate at all. Short wavelengths can propagate, but they cannot be accurately computed. 3.4.2.

Stability

Even when the discretization satisfies constraint (3.76), there are still fur­ ther conditions that must be met. All iterative numerical algorithms have

134

Application of Nonstandard Finite Differences

the potential to be unstable. If algorithm A acts on the iterate
(3.77)

where, for simplicity, the spatial dependence is suppressed, discretized time is denoted by a subscript, U2 stands for either u2 or v2At2/h2, and D2 denotes a Laplacian difference operator. Postulating a solution of the form ipt = w* and inserting it into (3.77), we obtain w2 - 2bw + 1 = 0,

(3.78)

where 6 = 1 + U2A(k). For (3.78) to have an oscillatory solution we must have b2 < 1, which implies the constraint U2 < —A(k). The upper bound on U2 is determined by the maximum possible value of (—A(fc)). We thus have the condition U2 <

^ = - , max(-A(ib))

(3.79)

with respect to all possible values of k that can occur — not just the wave number of the input signal, and independently of constraint (3.76).

Discretization

and Stability

In one dimension D2 = a2., A = cos k-l 2, which gives

135

= — sin2(fc/2), and max(—A) =

At v— <1 h

(3.80)

for the SFD. Using the fact that v = u/k, we obtain the same result for the NSFD algorithm. Since propagation velocity is a function of position, v is the largest value that occurs in the problem. In two dimensions, for the SFD algorithm, D2 = D2, we have A is given by (3.51a), and we find that max(-A) = 4, which gives v -^ < -= .

(3.81a)

For the SFD algorithm in three dimensions D2 = D2 and m a x ( - A ) = 6, which gives

v 4^ < 4= •

(3.81b)

h ~ y/3 Again v is the maximum velocity in the problem. For the NSFD algorithms we have D2 = D2. In two dimensions for a signal of wave number k A(fc') = 7(*)Ai(*') + (1 - 7(*))A 2 (*') • Although 7 is computed using the local wave number of the signal, k, the difference operators D2 and D\ act on signals with all possible wave numbers k'\ it is for this reason that we write A(fc'). With respect to all possible values of k' we find for the ranges 1 / 2 < 7 < 1 , 0 < 7 < l / 2 that max(-A) = 47,

(3.82a)

max(-A) = 2,

(3.82b)

respectively. For the NSFD Laplacian difference operator, DQ, we always have 7 < 2/3 because Jim 7(*) = \ ,

(3.83a)

136

Application

of Nonstandard

Finite

Differences

and furthermore 7(jfc > 0) < \ .

(3.83b)

In view of (3.76b) for X/h > 5, we find that

7 (jf\

S 0.648 < 7 < | ,

(3.84)

so the stability condition for practical NSFD algorithms now becomes sm^At/^ «n(*h/2) -

1 / ^ '

v

where we have used the fact that u2

= u2 =

sinVAt/2) sin2(ifc/i/2)

Here fc is the wave number of the input signal. Using (3.83b) the stability condition can be simplified to sin(<jAt/2)

y/3

sm(kh/2) ~ T '

(3>85b)

which is a slightly more severe than (3.85a). In three dimensions we have 3 i=l

with the constraint JZ.ffc = 1, and 0 < rji < 1. Again we seek max(—A). Here the situation is much more complicated, because it is difficult to ex­ press max(—A) as a simple function in terms of the 7^. Let us therefore consider the values of 7 and 7/ for X/h > 6 (constraint (3.76c)), that would actually be used to construct D$. Thus for practical algorithms we find

7 (I) = 0.655 < 7 < I •

(3.86)

Furthermore, using the information that lim T)(k) = - , k-Kl

5

(3.87a)

Discretization

137

and Stability

i7(jb > 0) > \ , 5

(3.87b)

we obtain the range: | < n < 0.413 = v(l)

■

(3.88)

In the 7-77 range defined by (3.86) and (3.88), we find that max(-A) ^ 3.102

(3.89)

which implies the stability condition sm(uAt/2) sm(kh/2)

< 0.802.

(3.90)

Summarizing these stability constraints, it is useful to express them in the form

(391)

TJS*'-

where T is the wave period. The larger the value of c, the fewer the num­ ber of time steps needed to propagate one wavelength for a give spatial discretization. In other words c is velocity-like quantity, that measures the maximum speed at which signal can propagate on the grid. Using the fact that v = A/71, we obtain c = d = 1,

(3.90a)

in one dimension for both the SFD and NSFD algorithms. For the SFD algorithms in two and three dimensions, respectively, we have cfFD = ^

^0.707,

(3.90b)

cfFD = — S 0 . 5 7 7 .

(3.90c)

For the NSFD algorithms c is a function of kh. In two dimensions, rewriting (3.85b) we obtain c?SFD(fc) = I arcsin (^-

sin (^

\ ,

(3.90d)

138

Application of Nonstandard Finite Differences

which can be further simplified using constraint (3.76b) and the fact that c(k) increases with falling k (in other words, c increases with X/h); we obtain c f F D = | arcsin (^

sin ( £ ) " ) S 0.849.

(3.90e)

Similarly in three dimensions we have c£SFD(fe) = I arcsin ( 0.802 sin ( | ) ) ,

(3.90f)

which using (3.76c) simplifies to C NSFD =

6 ^^^(040^

s

0 7g9

(3

90g)

Note that the stability conditions derived here for the NSFD algorithms differ somewhat from references [3]-[5]. It is perhaps more of academic than practical interest, but these stability conditions are not the only ones possible. If we regard 7 and 77 as free parameters (rather than as functions of fe) we can create other difference operators, which when used in the FDTD algorithm to solve the wave equation, have different stability constraints from the ones listed above. For example, in two dimensions if we set 7 = 1/2, (3.85a) becomes 8in(ttAt/2)

sin(*ty2) ^1-

V^>

We can manufacture the same stability condition in three dimensions with the choices 7 = 1/2, 77 = 1/2. NSFD-like algorithms that use these choices would not have the same high accuracy as the true NSFD algorithms, but they would still be more accurate than the SFD algorithms. Finally, for practical algorithms, it is of interest to note that ranges of 7 and 7/ are rather small with respect to the discretization constraints (3.76). In three dimensions, ranges (3.86) and (3.88) imply 7

0.451 < 7?! < — ^ 0.467 15 2 0.133 = — < % < 0.143 15

(3.93a) (3.93b)

NSFD Solution of Maxwell's Equations

139

\ < m < 0.406. (3.93c) 5 The above analysis applies for the FDTD algorithm to solve the ho­ mogeneous wave equation (3.36). What about when there are sources and absorption, and at the interfaces between different media? Here a gen­ eral analysis does not exist because there are so many different problemdependent possibilities. Empirically, however, these stability criteria seem to apply. It should be noted that the superior accuracy of the NSFD algorithm comes at the expense of a greater computational load at each grid point, but it is more than offset by the low X/h ratio that can be used, as well as by a decrease in the number of iterations needed. We will not enter into an error analysis for multifrequency signals, we can always be sure that the NSFD algorithm is never worse than the SFD one. We have found reasonable results over with frequency spreads of about 10% from the center frequency at which the NSFD algorithm is defined.

3.5. N S F D Solution of Maxwell's Equations 3.5.1. The Standard Yee Algorithm. Maxwell's equations in non-conducting media can be written in the form [7] fi(x)6tH(x, t) = - V x E(x, t),

(3.94a)

e(x)6tE{x, t ) = V x H{x, t),

(3.94b)

where we assume linear media such that B = fiH and D = eE, where B , and H are the magnetic field and magnetic intensity vectors, respectively, and E, and D are, respectively, the electric field and electric displacement vectors. The units of electric permittivity (e) and magnetic permeability (fi) are chosen such that in vacuum they are both one — or in other words we use relative electric permittivity and magnetic permeability. Since the other two Maxwell's equations V•D = p

(3.95a)

V•B = 0

(3.95b)

140

Application of Nonstandard Finite Differences

can be derived from (3.94) we only need to solve (3.94). Here p is the charge density. Later we will discuss Maxwell's equations in conducting media with current sources. Applying the curl operator to both sides of (3.94) it can be shown that

( d "-7^ V2 ) E -^Rx){ V

Ve(x) • E' ,VMx) e(x)

VXE| (3.96a)

( a "-i(^ v 2 ) H = ;^x){ v

V/i(x) • H

Ve(x)

Mx)

e(x)

_

„ \ (3.96b)

If |Ve(x)/£(x)| and |V/*(x)//i(x)| are both zero or "sufficiently small," then the right sides of (3.96) vanish, and we are left with the wave equation ( d „ - t ; ( x ) 2 V 2 ) V ( * , t ) = 0, where V" is an electromagnetic field component and v(xY =

1 £(X)/|(X)

(3.97)

Thus in media for which e and fx are constant or nearly constant, it is sufficient to solve the wave equation with respect to the electromagnetic field components of interest. When this is not the case, however, we need to directly solve Maxwell's equations. Let's now set up an FD model of Maxwell's equations. Defining a vector difference operator D = ±dx + ydv + idz ,

(3.98)

a central FD approximation to (3.94) is (3.99a) dtE(x,t)

=

e(x)

n

-}-r^i)xH(x,t),

(3.99b)

NSFD Solution of Maxwell's Equations

141

where x, y and z are unit vectors along the x-, y-, and z-axes, respectively. Expanding dtH(x, t) in (3.99a) we obtain H(x, t + At/2) = H(x, t - At/2) - -}— -^ D x E(x, t).

(3.100a)

Now substituting the updated H-field, H(x, t + A t / 2 ) , into (3.99b), we have dtE(x, t + At/2) = -r^- -^ D x H(x, t + e(x) h

At/2).

Finally expanding
(3.100b)

Notice the "staggered" time grid. Whereas E is evaluated at times t, H is evaluated at times t + At/2. This is because we use central finite differences. In similar fashion, because of the central FD approximations to the spatial derivatives, each component of the E- and H-fields is evaluated at a different point in space. Let xj and x> ' denote the positions where component i(i — x, y, z) of E and H, respectively are computed; it can then be shown that the choices ,(H)

x,y-

(3.101a)

z

2'

'

r (H)

2 ,(H)

*

2)

x—, y —,z 2

xi E > = ,(E) _

x!E> =

(3.101c)

2

~hry'z)

(3.101d) (3.101e)

x

>y-\>z)

x,y,z-

(3.101b)

-

(3.101f)

give a consistent set of FD representations for the spatial derivatives. Sub-

142

Application of Nonstandard Finite Differences

stituting (3.101) in (3.100) for each field component, we obtain

*(*!»>,« + « ) - *(«<«>.. - £ ) - -J^ | ( D x E<*<«> ,,),, (3.102a) J*(x< B \t + At) = J * ( x j E \ i ) + - 4 i j - x e(xj ;) n

(f> X

H^'*))* • (3.102b)

On the right of (3.102a) E(xj , t) denotes the positions about which the central FD approximations to the spatial derivatives in (V x E)j are evalu­ ated. The notation in (3.102b) is analogous. Notice also that the values of £ and n at the appropriate positions must also be used, and thus three sets of values must be specified — one for each electromagnetic field component. In practice, however, this is rarely done and one set is used instead. Algorithm (3.102) is called the Yee algorithm [8]. We will call it the SFD, or simply "standard" Yee algorithm, because it uses standard finite differences.

3.5.2. Nonstandard

Yee

Algorithm

In uniform media the Maxwell's equations reduce to the wave equation because V x V x A - V(V • A) - V 2 A ,

(3.103)

where A is a vector function of position. Similarly, the SFD Yee algorithm reduces to the SFD approximation to the wave equation because D x D x A = D(D • A) - D 2 A .

(3.104)

The error and stability analysis of the Yee algorithm is thus exactly the same as that of the SFD algorithm to solve the wave equation. This leads us to try to devise a version of the Yee algorithm using nonstandard finite differences [9,10].

NSFD Solution of Maxwell's

Equations

143

Simply inserting the NSFD approximation (3.27) will not work because the vector difference operator (3.98) gives D2 =

£

<£ = £ ? ,

(3.105)

i=x,y,z

where D\ is the difference operator for the SFD approximation to the Laplacian. We thus need another vector difference operator B' = Std'x+yd'y + zd'z, such that

i=x,y,z

where D$ is the difference operator for the NSFD approximation to the Laplacian. Unfortunately, the df do not exist, but we can decompose DQ in the form

^EW,

(3-106)

where d^ ' now denotes the central difference operator of (3.2c), d\ ' denotes a new variety of difference operator, d is the problem dimensionality (2 or 3), and the (x,y,z) components are denoted by (1,2,3) respectively. Once we find the d\\ an NSFD version of the Yee algorithm can be constructed by simply making the replacement D —► D^°) in one of the equations of (3.102) while retaining D in the other, and inserting the sub­ stitutions h —>• s(k,h), At -> s(w, At). Here D is given by (3.98) and

D<°> = *4°> + y4°> + *4°>

(3.107)

is a vector difference operator using the <^0). Again considering harmonic signals of angular frequency w, we obtain from (3.102) ^ ., H(x, t + At/2) = H(x, t - At/2) - - ^ - u(x)D ( 0 ) x E(x, t). E(x, t + At) = E(x, t) + - ^ u(x)D x H(x, t + At/2),

(3.108a) (3.108b)

144

Application of Nonstandard Finite

Differences

where u = s(u, At)/s(k, h), or k sin(o;Af/2) w sin(*fe/2)

1 sin(a>At/2) v sin(A:/i/2) '

^

;

v is given by (3.97), and fc, the local wave number is a function of x via (3.15). For notational economy we have suppressed the explicit position vectors for each electromagnetic field component, but they are the same as in (3.102). Let us now determine the difference operators, cq '. For simplicity, we find the operators in two dimensions first. For the partial derivatives, in addition to the conventional difference operators dx ' and dj, , we can also define , and dy ' by 2d~Wf(x, y) = f(x+-,y -f(x-^,y

+

hj+f(x+-,y-h\ + hj-f(x-^,y-hj,

(3.110a)

2dWf{x, y) = f (x + h, y + £J + / (x + h, y - | J -f(x-h,y+±\-f(x-h,y-£).

(3.110b)

It can be shown that 2

J(i)J(3) _ o f ) 2 _ n2 £W=2D2-£?,

(3.111)

where D\ and D% are the two-dimensional Laplacian difference operators given in (3.45) and (3.46). We now define the to be weighted superpo­ sitions of the form

^ U a d T ' + a-«)<*?>,

(3.112)

where a is a parameter to be determined such that (3.106) is true. Substi­ tuting (3.113) into (3.106) and using (3.111), we find that a = i±^, where 7 is given by (3.55).

(3.113)

NSFD Solution of Maxwell's Equations

145

In three dimensions there are now three independent difference opera­ tors for each partial derivative. For dx, the first one is the same as dx. The next two are given by 44 2 ) /(*,V,z) = f (x + -,y + h,z + hj + f (x + -,y + h,z - hj + fix

+ -,y-h,z

+

hj+fix+-,y-h,z-hj

- f (x - -,y + h,z + hj - f (x - -,y + h,z - hj - fix-

-,y-h,z

+ hj -fix-

-,y-h,z-hj (3.114)

4dxVf(x,y,z)

= f(x

+ -,y +

+ f(x+-,y,z - fix-

h,zj+f(x+-,y-h,z\ +

-,y + h,zj

-f(x-±,y,z

hj+f(x+-,y,z-hj -fix-

+ h\-f(x-±,y,z-h\

-,y-h,zj .

(3.115)

A similar pattern holds for dy and dz. We now construct a superposition

for each partial derivative. The weights aj obey the constraint J^ • <*_,- = 1, and they are the same for all the partial derivatives. In three dimensions it can be shown that

J24M2)=^l-Dl

(3.117a)

i=l 3

^dl1)ti|3)-2D2-^. t=i

(3.117b)

Application of Nonstandard Finite Differences

146

Putting (3.116) and (3.117) into (3.106) we find a i = m + 3%+27?3>

(3.118a)

02 = ^772,

(3.118b)

«3 = 3772+2*73,

(3.118c)

where the 77* are given by (3.74). A more convenient form is ai = ^ ( l - 7 ) + 7

(3.119a)

"2 = ^ ( 1 - 7 )

(3.119b)

a3 = l - a 1 - a 2 .

(3.119c)

With the d\ ' now determined, we now have a complete realization of an NSFD version of the Yee algorithm. 3.5.3. Maxwell's

Equations

in a Conducting

Medium

In conducting media where electric currents can exist, second Maxwell's equation (3.94b) becomes e{x)6tE(x,t)

= V x H(x,t) - J(x,t),

(3.120)

where J is called the current density. The total current density is J(x, t) = J< c) (x, *) + ^(x,

t),

(3.121)

where J^"'(x, t) is the source current density — imposed from outside the system, and the J^c'(x,t) is the conduction current density, which arises due to the action of the electric fields on the conducting medium. In a linear conducting medium is given by J
(3.122)

where a(x) is the local conductivity. Inserting (3.121) and (3.122) in (3.120) we obtain e{x)6tE(x, t) = V x H(x, t) - ^{x,

t) - a(x)E(x, t).

(3.123)

NSFD Solution of Maxwell's Equations

147

Approximating the derivatives of (3.120) with FDs we obtain dtE(x, t + At/2) = -j-r -£■ D x H(x, t + At/2) - At - L J <•> (x, t + At/2) - At $ r E(x, t + £(x) £(x)

At/2). (3.124)

Since the quantity d<E(x, t + At/2) is computed using E(x, t + At) and E(x, t) we need to express E(x, t + At/2) in the last term on the right of (3.124) also in terms of E(x,t + At) and E(x,t). The simplest approximation is E(x, t + At/2) = ^ (E(x, t + At)+ E(x, t)).

(3.125)

it

We need not estimate J^(x,t + At/2) in terms of J^(x,t + At) and J ^ ( x , t) because it is a known function that can be computed at any time. Expanding the difference operator on the left in (3.124) we obtain E(x,t + At) = ^E(x,t) A+ "

'

+

l

1

A+ e(x)

5 / 2 )-- AtjW(x,t AtJ(-s)(x,t ^ D x H ( x , t + At/2) n

+ At/2 At/2) (3.126)

where A±=l±^-At.

(3.127)

This is the SFD Yee algorithm. If we assume that the E-fields and the source current are harmonic, Jw(x,0 = jW(x)e-w , E(x,t)

= E{x)e~iu't,

(3.128a) (3.128b)

Application of Nonstandard Finite

148

Differences

then we can construct an NSFD algorithm. For functions of form (3.128), we exactly compute f(t + At/2) in terms of f(t) and f(t + At), f{t + At/2) = - ^ — (f(t) + f{t + At)), s(w, At)

(3.129)

5(w, At) = 2 cos(u>Af/2).

(3.130)

where

Using (3.129) to determine E(x,t + Af/2) we can obtain an NSFD Yee algorithm. Equation (3.126) becomes A' 1 1 E(x, t + At) = -rj- E(x, t) + -jj- - r T [u(x)D x H(x, t + At/2) - a ( w , A t ) j W ( x , t + At/2)],

(3.131)

where A'± = \±-

1 a1 — At,

a' = a- tan(wA*/2).

(3.132) (3.133)

In the case of perfect conductors, we can take the limits as a -* oo. We find lira —

= -1,

(3.134a)


lim — = 0 .

(3.134b)


Exactly the same limits hold for the A'±. In summary the SFD Yee algorithm in linear conducting media is given by (3.100a) and (3.126), while the NSFD version is given by (3.108a) and (3.131). For notational simplicity we have suppressed the details of the position dependence of the different electromagnetic field components given in (3.101).

NSFD Solution of Maxwell's Equations

3.5.4. Electromagnetic

149

Simulations

Electromagnetic (EM) fields obey various boundary conditions at the in­ terfaces of different media. When these interfaces have complicated geome­ tries, it is very difficult or even impossible, to analytically find solutions of Maxwell's equations. If, however, an EM field initially obeys the boundary conditions, then it will continue to obey them at all future times as the algorithm is iterated. The reason for this is that the boundary conditions themselves are derived from Maxwell's equations, so iterating the algorithm enforces the boundary conditions. As an example of how this can be applied to a practical problem, con­ sider the case of scattering off a complicated object. Input a signal at t = 0, before it hits the object. In this case the EM field at the object is initially zero and obeys all the boundary conditions. The boundary con­ ditions are maintained as the algorithm is iterated. The simplest initial condition of all is a zero field. Letting t = 0, At, 2At ■■■ for the E-fields and t = l / 2 ( - A t , 0, At ■ ■ •) for the H-fields, we take E(x, t < 0) = 0, H(x,£ < -At/2) = 0, J<5>{x,« < 0) = 0 at the start of the computation. Using an array of source currents, we can then generate any arbitrary input signal. For example we have used a phased array of current sources to create a steerable Gaussian beam for simulation of Mie scattering (where the wave­ length of the incident signal is comparable to scatterer size). The sources should be turned on slowly to minimize the generation of non-harmonic signals. The solution error of (3.108) with respect to the El-fields is of order h8 compared with h2 for the SFD Yee algorithm. Due to the asymmetric decomposition of D2, of (3.106), we do not obtain high accuracy solutions for both E and H in the same iteration, because the H-fields are computed first and contain an anisotropic scale error. To obtain the H-fields with high accuracy, one could either rearrange the algorithm to compute H last, or compute it from the high accuracy E-fields in an extra computational step. Although we have improved the accuracy of the FD approximations with respect to the periodic electromagnetic field components, we do not necessarily approximate such quantities 6xe{x) any more accurately with NSFD operators.

150

Application of Nonttandard Finite

Differences

Since the parameters rj, 7, u are functions of the medium parameters, which vary with position, the difference operators must also be defined at each position in the medium. In addition u depends on the frequency. Solution error is rrunirnized only at the angular frequency, u>, at which u is defined. For non-monochromatic signals, we might therefore expect that accuracy can be high only near this frequency. It turns out, however, that while solution error does increase away from w, it is still smaller than that of the SFD algorithm, and is still highly isotropic. The tradeoffs needed to handle multiple frequencies are discussed in [3]. It should be noted that the solution error of the SFD algorithm is also both frequency-dependent and anisotropic.

3.6. S u m m a r y We have used nonstandard finite differences (NSFD) to construct high ac­ curacy algorithms to solve the wave equation and Maxwell's equations in nonuniform media. By superposing difference operators that are only sec­ ond order accurate (error ~ h2) we could construct ones that are eighth order accurate (error ~ h8) with respect to harmonic solutions. In addition the stability of the NSFD algorithms is generally greater than that of the standard finite difference (SFD) algorithms, which means that fewer time steps are needed to solve a given problem. We have created practical programs in both FORTRAN 90 and Mathcad and have verified the high accuracy of these algorithms by simulating Mie scattering in one, two, and three dimensions and comparing with analytic solutions that exist for some simple objects [11].

3.7. Conclusion Despite the rather challenging mathematical details in some sections, we hope that the reader will be left with one simple message. The most impor­ tant consideration in the numerical solution of any problem is the model. A direct transposition of the parameters or the original differential equation to the numerical model is not always best. Information about the nature of the solutions (we usually know something) can be used to guide the

Conclusion

151

model building. In the case of the wave equation and Maxwell's equations we constructed high accuracy solutions with respect to harmonic basis functions. The nonstandard approach is not, however, limited to harmonic functions but can be applied to other cases as well.

152

Bibliography

Bibliography

[1] J. B. Cole, R. Krutar, D. B. Creamer, and S. K. Numrich, "Finite-difference time-domain simulations of wave propagation and scattering as a research and educational tool," Comput. Phys. 9, 2 (1995) 235-239. [2] R. E. Mickens, Nonstandard Finite Difference Models of Differential (World Scientific, Singapore, 1984).

Equations

[3] J. B. Cole, "A nearly exact second order finite difference wave propagation algorithm on a coarse grid," Comput. Phys. 8, 6 (1994) 730-734. [4] J. B. Cole, "A high accuracy FDTD algorithm to solve microwave propagation and scattering problems on a coarse grid," IEEE Trans. Microwave Theory and Techniques 4 3 , 9 (1995) 2053-2058. [5] J. B. Cole, "Generalized nonstandard finite differences and physical applica­ tions," Comput. Phys. 12, 1 (1998) 82-87. [6] S. K. Godunov, Difference Schemes (North-Holland, Amsterdam, 1987). [7] R. C. Booton, Computational Methods for Electromagnetics and Microwaves (John Wiley & Sons, New York, 1992). [8] K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antenna Propagation A P - 1 4 (1966) 302-307. [9] J. B. Cole, "High accuracy solution of Maxwell's equations using non-standard finite differences," Comput. Phys. 11, 3 (1997) 287-292. [10] J. B. Cole, "A high accuracy realization of the Yee algorithm using nonstandard finite differences," IEEE Trans. Microwave Theory and Techniques 45, 6 (1997) 991-996.

Bibliography

153

[11] S. A. Palkar, N. P. Ryde, M. R. Schure, N. Gupta and J. B. Cole, "Finite difference time domain computation of light scattering by multiple colloidal particles," Langumir 14, 13 (1998) 3484-3492.

Chapter 4

Non-standard Discretization Methods for Some Biological Models

H. Al-Kahby*, F. Dannan*, S. Elaydi*

4.1

Introduction

It has been observed for some time that the standard (classical) discretiza­ tion methods of differential equations often produce difference equations that do not share their dynamics. An illustrative example is the logistic difference equations dx

-=/Mi-*). Euler's discretization scheme produces the logistic difference equation x(n + 1) = (ix(n)(l - x(n)) which possesses a remarkably different dynamics such as period-doubling bifurcation route to chaos. A more popular discretization method is to modify the given differential equation to another with piecewise-constant arguments and then to integrate the modified equation. In some instances, 'Division of the Natural Sciences, Dillard University, New Orleans, Lousiana 70122 (halkahbyCaol. com). 'Basic Sciences Department, Faculty of Civil Engineering, Damascus University Dam­ ascus - Syria ^Trinity University, Department of Mathematics, 715 Stadium Drive, San Antonio, Texas 78212 (elaydiCtrinity.edu). 155

156

Non-standard Discretization Methods for Some Biological Models

this produces a difference equation whose dynamics is close to its original differential equation. However, oftentimes this is not the case. Neverthe­ less, many authors [l; 7; 23; 10; 9; 8; 3] find it interesting to study the resulting difference equations. This is not a criticism of these authors' re­ search, since the study of nonlinear difference equations is of paramount importance regardless of whether or not they have connections with differ­ ential equations. But what we are actually saying is that from the point of view of numerical analysis such study is of less importance. This chapter concerns itself with those numerical schemes that pro­ duce difference equations whose dynamics resembles that of their continu­ ous counterparts. The most fruitful methods are those of Mickens [29] (for asymptotically stable systems) and of Kahan [16] (for periodic systems). The chapter is organized as follows. Section 2 establishes the basic stability results for Lotka-Volterra differential systems. Section 3 surveys some classical discretization methods that are widely used and show their shortcomings. Section 4 provides the reader with the essential ingredi­ ents of Micken's nonstandard discretization scheme. In Section 5, Micken's method applied to competitive and cooperative methods where it is shown that the resulting difference equations are dynamically consistent with their continuous counterparts. In Section 6, we prove that the obtained discrete competitive and cooperative models are permanent. Section 7 includes three different discretization schemes for the predator-prey Lotka-Volterra system. It is shown that the positive equilibrium point in each difference system is asymptotically stable. We failed; however, to establish global stability. In Section 8, we investigate the global stability of discrete competitive and cooperative systems. Using the theory of monotone dynamical systems [18; 19], we are able to establish global stability of the positive equilibrium point. In Section 9, we discretize a Leslie predator-prey model using Micken's scheme. It is shown that the resulting difference equation is dynamically consistent with the original differential equation. In Section 10, we discretize a periodic Lotka-Volterra differential system using Kahan's scheme [16]. It is shown that the solutions of the resulting difference equation lie on closed curves surrounding the positive equilibrium point. In Section 11, we consider a Kolmogrove continuous model of cooper­ ative system [13]. This model was discretized in [7] using the method of

Stability of Lotka- Volterra Differential

Equations

157

piecewise-constant argument. Surprisingly, the resulting difference equa­ tion is dynamically consistent with its continuous counterpart. Finally, we pose some open problems and directions for further study in Section 12.

4.2

Stability of Lotka-Volterra Differential Equations

Consider two species A and B. Then we have three types of relationship between them. The first type is competition, in which A and B compete for common resources such as food, living space, etc. Here the presence of B adversely affects the growth of A and vice versa. The second type of relationship between A and B is cooperation, where the presence of B produces positive effects on the growth of A and vice versa. The third and last type is predator-prey or host-parasite relationship. If A is the prey (host) and B is the predator (parasite), then the presence of B produces a negative effect on the growth of A, while the presence of A produces positive effects on the growth of B. Now, let x(t) be the density of species A at time t, and y(t) be the density of species B at time t. Then, the growth rate of species A and B can be modeled by the following system of differential equations: f

=

x{t)[n - anx(t)

- a12y(t)]

) (4.2.1)

f

=

y(t)[r2 - a21x(t) - a22y(t)} J

Eq. (2) is often referred to as a Lotka-Volterra System, and was intro­ duced by Volterra in 1931 [21] and by Lotka in 1920 [12]. Here, n , and r2 represent the intrinsic growth (or decay) rate for species A, and B, respec­ tively, while On and 022 represents the negative effects of squabbles among members of the same species A, and B, respectively. Finally, a^ represents the effect on the growth of species A from species B, and 021 represents the effect on the growth of species B from species A. It is now evident that a n > 0 and 022 > 0. However, for the signs of 012 and a 2 i, we have three cases: Case I: Competitive species: 012 > 0, ai2 > 0. Case II: Cooperative species: ai 2 < 0, a 2 i < 0. Case III: Predator-prey species: ai2 > 0, 021 < 0.

158

Non-standard Discretization Methods for Some Biological Models

If we write Eq. (2.1) in the form dx * = ' < » ■ » >

ft=9(*,y) Then we say that (x*,y*) is an equilibrium point if f(x*,y*) = g{x*,y*) = 0. The equilibrium point {x*,y*) is said to be stable if for any open neigh­ borhood U of (x*,j/*) there exists an open neighborhood V of (x*,y*) such that if (x 0 , j/o) € V, then {x{t,x0),y(t,y0)) G U for all t > 0. If in addition, lim (x(t,x0),y(t,yo)) = (x*,y*) for all {x0,yo) in an open neighborhood 4-»oo

W of (x*,y*), then (x*,y*) is said to be asymptotically stable. If W = R 2 , then (x*,y*) is said to be globally asymptotically stable. Observe that if System (2.1) possesses a positive equilibrium point (x*,j/*) then it must satisfy the equations anx'+a12y'=ri

(4.2.2)

a2ix* + a22j/* = r2

(4.2.3)

Hence . ria22-r2ai2 x = ail<*22

—a

12«21

. r2an-ria2i ,y = ailA22

—

IAIA\

(4.2.4)

012021

We are in a position to state the main stability result for System (2.1). Theorem 4.1 [2], [20] Suppose that System (2.1) has an asymptotically stable positive equilibrium point (x*,y*) then (x*,y*) is globally asymptot­ ically stable if an > 0 and a22 > 0.

4.3

Classical Discretization

There are numerous discretization schemes in Numerical Analysis litera­ ture. The simplest numerical scheme is the forward Euler in which ^ is replaced by *(«+*)-*(«> and % is replaced by v(t+V~vW y w h e r e h is the step size of the numerical method. Making this replacement in Eq.(2.1)

Classical Discretization

159

and letting t = nh, x(t) = x(nh) = x(n), and y(t) = y(nh) = y(n) yields the difference system x(n + 1) = i(n)[l + rih - anhx(n) - ai2hy(n)] ~\ \ y{n + 1) = y(n)[l + r2h - a2ihx(n) - a21hy(n)] )

(4-3.1)

Ushiki [24] observed that the dynamics of Eq. (3.1) differs from that of Eq. (2.1) and for some parameter values may exhibit chaotic behavior. Hence, the search for a better numerical scheme continues. Another popular method is to consider Eq. (2.1) with a piecewise constant arguments [22] as follows: §

=

$

=

x(t)[n - allX([t\)

- a13y{[t\)]

) } y(t)[r2 - a21x([t\) ~ a22y([t})} J

(4-3.2)

where 0
- a 12 j/(n)],

1/(0 = y(n) exp[r2 - a 2 ii(n) -

a22y(n)},

te [n,n + l). If we let t -¥ n + 1 in the preceding system, we obtain the following system of difference equations. x(n + 1) - x(n)exp[r! - o a i ( n ) - ai 2 y(n)] "| \ y(n + 1) = y(n) exp[r2 - a2ix(n) - a22y{n)] )

(4.3.3)

System (3.3) has been investigated by Krawcewicz and Rogers [10] for the case of cooperative systems (a\2 < 0, a 2 i < 0) and by Jiang and Rogers [9] for competitive systems {a\2 > 0, a2\ > 0). In both cases, it was shown that System (3.4) may exhibit a dynamical behavior quite different from its continuing counterpart (2.1). In spite of its deficiency, System (3.3) has been given a lot of attention by several authors including Hofbauer et al [8], Dohtani [3], and Yakubu [23].

160

4.4

Non-standard Discretization Methods for Some Biological Models

Nonstandard Discretization Schemes

One of the main aims of numerical analysis is to find a numerical scheme that produces a difference equations that exhibit the same qualitative be­ havior as its continuous counterpart (differential equations). We say that a difference equation is dynamically consistent with its differential equa­ tion if they both possess the same dynamics such as stability, bifurcation, and chaos. In [29], Mickens developed successful nonstandard discretization schemes that produce what every numerical analyst dreams about, namely dynamical consistency. Here we adapt Micken's method to the setting of biological models mainly of Lotka-Volterra type. To illustrate Micken's gen­ eral scheme, we start with a very simple example, the logistic differential equation ^=x(t){a-bx(t))

(4.4.1)

Integrating Eq. (4.1) from t to t + h yields

We now let t = nh, and x(nh) = x(n) in (4.2). Hence, we have x(n + 1) =

l

+

/ " XW ( e ^ ) M

n )

'

, n£Z+

(4.4.3)

Observe that the solutions of Eq. (4.1) and Eq. (4.3) are equal on Z + , regardless of the value of the step size h. Setting eah - 1 -, V(h) = a Eq(4.2) may be written in the form x(t + h) - x(t) = ax(t) - bx(t)x(t + h) f(h)

(4.4.4)

Notice that (4.4) is similar to the difference equation obtained by the for­ ward Euler's method with two major differences: (i) h in the denominator of the left hand side is now replaced by a function of h, tp(h), (ii) the term

Competitive and Cooperative discrete Models

161

x2(t) is now replaced by x(t)x(t + h). The resulting difference equation is given by «(n + l ) -

J + <■**)*(») 1 + hp{h)x{n - 1)

We now formalize the above steps for the general differential equation

£ = f(t,x(t),y(t)) ) } % = g(t,x(t),v{t) J Step 1: x{t

(4-4.5)

Replace the derivative £ by an expression of the form

{t)

l%) , where Vl(h) = h + o(h), and dft by the expression * ^ g ^ where tpz^h) = h + o(h). Step 2: Vary the nonlinear terms by nonlocal expressions. For example, x2(t) y2(t)

—» —> /

*(*)y(0

—► *(* + Mv(0 \

x(t)x(t + h) y(t)y(t + h) a:(t)y(« + /i)

x(t)y(t)

For Step 1, the main question is how to chose the appropriate functions fi(h) and y>2{h)- At this time, we are unable to give a general method for the selection of these "denominator" functions. However, we will demon­ strate to the reader some special techniques that produce appropriate "de­ nominator" functions [ll]. As for Step 2, the selection of appropriate expressions proved to be simple for competitive and cooperative Lotka-Volterra systems and most challenging for predator-prey models [ll]. While performing Step 2, one should make sure that solutions with non-negative initial values must stay non-negative all the time, i.e., the cone R^_ = {(x,y) : x > 0, y > 0} must be invariant. 4.5

Competitive and Cooperative Discrete Models

We now apply the general scheme outlined in Section 4 to both competitive and cooperative system.

162

Non-standard Discretization Methods for Some Biological Models

1. Competitive Species (ai2 > 0, 021 > 0). The following discretization of Eq. (2.1) is proposed for competitive models. x(t + h)- x(t)
=

-

rja;W

_ aiix{t)x{t

r2x(t) - a2ix{t)y(t

+ h)_

ai2xit

+

h ) y

^

+ h) - a22y(t)y(t + h)

which yields the difference system X(n + 1)= x(n) 1+^(h\\Z7(n\'iLy(n)). y(n + 1 ) = y{n)

2

1+V2(h\\Z

x(n\h-lLvM)>

' (4.5.1) .

n € Z+, with Vl(h) = * % i and
x(t + h) x(t) Vi (0 y(t + h)- y(t)

=

=

ria

.w_0na.Wa.(1

+

r2x(t) - a2ix(t)y(t)

/ l

)_ai2a.WyW

- a22y(t)y(t + h),

which yields the difference System x{n + 1) = x(n)(l + ri
+ an
y(n + 1) = y(n)(l + r2y2{h) - a2i
+ a22
with V l (A) = s ^ = l , ifi2(h) = ^ . The stability of System (5.1) has been thoroughly studied in [ll]. We now perform the same procedure on System (5.2). The first step in our program is to investigate the asymptotic stability of the positive equilibrium point. Theorem 4.2 Suppose that the cooperative system (2.1) has an asymp­ totically stable positive equilibrium point (x*,y*). Then, System (5.2) has

Competitive and Cooperative Discrete Models

163

the same positive equilibrium point (x*,y*) which is also asymptotically stable. Proof: It is easy to verify that Systems (2.1) and (5.2) have the same positive equilibrium point (x*,y*). Linearizing System (2.1) around (x*,y*) yields the linear system dx dt dy dt

(:)

where ( -x*an

-x*a12

\

\

-y*fl22

/

-J/*o2i

The characteristic equation of A is given by A2 + ( I ' O U + j/*022)A + x*y*(ana22

- 012^22) = 0.

By The Routh-Herwitz criterion [15], necessary and sufficient conditions for the asymptotic stability of (x*,y*) are given by a:*an + y*a22 > 0

(4.5.3)

Oll022 - Oi202l > 0

(4.5.4)

Since 012021 > 0, condition (5.6) implies that 011022 > 0. But condition (5.3) rules out the possibility of having o n < 0 and a22 < 0. Thus, a n > 0 and 022 > 0. If r2 < 0, then from the expression for x* in (2.4), we are led to the conclusion that r\ > 0. But, then this would violate the positivity of y* in (2.4). Hence, r\ > and r^ > 0. Now Linearizing System (5.2) around (x*,y*) yields the linear system (^n

+

]\)=B(u(;n]),

\v{n + l) J

(4.5.5)

V «(«) )

where j

1 l+anipi(h)x*

1 \

-a2i
-ai2
B = 1 l+a22
By applying the Schur-Cohn criterion ([4] p. 164), we conclude that a necessary and sufficient condition for the zero solution of System (5.7) to

164

Non-standard Discretization

Methods for Some Biological Models

be asymptotically stable is given by (see [4], p.164) \trB\ < l + d e t 5 < 2 .

(4.5.6)

We leave the verification of (5.6) to the interested reader. This completes the proof of the theorem. Theorem 2 can be extended to competitive systems. For details, the reader is referred to [ll].

4.6

Permanence of Discrete Competitive and Cooperative Systems

The notion of permanence is of paramount importance in population dy­ namics. Roughly speaking, a population of species is permanent if they do not go extinct and they do not grow without limit. We now give the precise mathematical definition. Definition 4.1 System (4.5) is said to be permanent if for any positive solution (x(n),y(n)) these are positive real numbers rrii, Mi, i = 1,2, such that m\ < lim inf x(n)

<

lim s u p , , ^ ^ x(n) < Mi

<

limsup,,..^ y(n) < M2

n—foo

(4.6.1)

m2 < lim inf y(n)

Theorem 4.3 Under the hypothesis of Theorem 2.1, Systems (5.1) and (5.2) are permanent. Proof: We consider the competitive system (5.1). The proof for System (5.2) is given in [ll]. From System (5.1), we have . , „ , n . x(n)(l+r1y1(/t)) —;—;—-, ' ~ 1 + alltfl(h)x{n) '

Xln v + 1) < —

V(n yy

+ 1) <

'-

y(n)(l + r2tp2(h)) TTT—;—rl + a22
Hence it follows from ([4]), (Example 2.39) that limsupi(n) < — = M\, limsupj/(n) < — = M2 n-+oo

Qll

n-*oo

&22

(4.6.2)

Predator-prey Discrete Models

165

To find liminf x(n), we let z(n) = ^ y in System (5.1). This yields z(n + 1)

=

zn[ — —

——

=

M{n)z{n) + 0

+ —

—-

(4.6.3) (4.6.4)

Since the expression for x* in (2.4) gives ria 2 2 - r 2 a 1 2 > 0 or ^ 2 r 2 < r i , lim sup M(n) < — n->oo

v ttM

'

= L < 1.

1+ri^^/l)

Using this in Eq. (6.3) yields lim sup z(n) < n-KX)

=-, which implies that 1 — Li

liminfn-vooa;^) > mi, where T\ - 022 -

mi

T2012

ana 2 2

Similarly, one may show that limsupj/(n) > m 2 , where n—*oo

r2an - a 2 i r i m2 = ana22 4.7

Predator-prey Discrete Models

In this section, we consider the discretization of the predator-prey differen­ tial system (2.1), where ai 2 > 0 and 021 < 0. Assuming that the positive equilibrium (x*,y*) is asymptotically stable, then conditions (5.5) and (5.6) hold. This implies that a n > 0 and 022 > 0. Moreover, the existence of a positive equilibrium (x*,y*) implies that either ri > 0, r 2 < 0 and r 2 a n - ria2i > 0 or J*I > 0, r 2 > 0 and ria2 2 - r 2 ai 2 > 0. Corresponding to the predator-prey differential system (2.1), we have come up with three discrete models which we are going to describe. First Model:

166

Non-standard Discretization Methods for Some Biological Models

x{t

X%x{l)

= nx(t) - anx(t)x(t

+ h)- al2x(t + h)y(t)

' (4.7.1)

V{t+

£(7?W

= Wit)

- a21x(t)y(t)

- a22y(t)y(t + h)

which produces the difference system (4.7.2) r /t a; wCn + 11 - v(")(1+H-a ^2(ft)-°-"y"( ) (")) 2A" f A ; 22VJ:!(/.)y(n)

Here in creating (7.1), we followed the same principle employed in competi­ tive and cooperative discrete models (5.1) and (5.2), respectively. If a\2 > 0, we replace x(t)y(t) by x(n + l)y(n) and if ai2 < 0, we replace x(t)y(t) by x(n)y(n). Moreover, if 021 > 0, we replace x(t)y(t) by x(n)y(n + 1) and if 021 < 0, we replace x(t)y(n) by x(n)y(n). The linearized system of Eq. (7.2) around (x*, y*) is given by f u(n + l) \ _ \v(n + l)J

R

f

«(n) \ "\v(n)J

(4.7.3)

where / 1 _ ailifii(h)x' 1 l+rivi(h)

ai2i(/i)

B

V

°21«, I + O22 V2(h)t/*

1 + 022V>2(/>)V*

/

The matrix B may fail the Schur-Cohn criterion and hence (x*,y*) may not be asymptotically stable. For a concrete example, consider the predatorprey differential system dx = x(t)(l-y(t)) ~dl dy = y(t)(-l+ax(t)-y(t)), dt where a > 0 is the predator digestion constant. The positive equilibrium point (^,1) is asymptotically stable. The discrete analogue that results from our procedure is given by x(n +1)

x(n)(l + ipt(h)) l + tfi{h)y(n)

Predator-prey Discrete Models

V{n + 1)

167

y(n)(l - ip(h) + a2(h)x(n) 1+ **(%(»)

=

Taking 1. This implies that the equilibrium point (^,1) in the discrete model is unstable. Second Model: x(tv + h) — x(t) v ' '• = nx{t) - anx{t)x(t tpi(h)

+ h) - ai2x{t + h)y(t)

y(t + h)- y(t) = r2y(t) - a21x(t + h)y(t) - a22y(t)y(t + h)
> y(n ■+ i) -

(4.7.4)

^ V 1 . V) )

i+o 22V > 2 (/i)y(n)

where we choose
Fx

Fy

<»2l¥>2(h)y' p l + 022V'3('')l/* X

(1-Q2iy>2(ft)v'f») H-a22V>2(/l)v*

with F x

~

_ 1 + Qi2li Wy* 1+rmW '

_

F y

anfi (h)x* 1+rmih)

where Fx = | £ and F y = ^ £ . To check the validity of the Schur-Cohn criterion, we need to verify the following three conditions: (i) d e t B < 1, (ii) 1 - trB + det B > 0, (iii) 1 + trB + det B > 0. Condition (i) is easily verified since

168

Non-standard Discretization Methods for Some Biological Models

d e t B

=

l + an
<

1

+ a 2 2 ¥>2(%*)

Next, we consider condition (ii). 1 + det B - trB

Fx

= 1 + —

1

^ - T T ^ - T - Fx

1 + a,22
F

l + a22V2(%* _1

= ( i - f . ) ( i -1,+. -a 2V2(h)y*

1 + a22tf2 (h)y*

|

2

" a2i
\,^j+,?rr,:^F,>o. 1 + a 2'P2(h)y* 2

Finally, we verify condition (iii). i

. 4. D . A~* B

1 + trB + d e t B

i _L T? .

=

1

1 + Fx +

ai2f2{h)y*

... . ... . i s , l + a22V2(/l)j/* l + fl22V2(/l)y*

, . , . =L + M

+-

1 + a22
F*

1 + a22
M = FX-

1+

a.22f2(h)y'

B a i y 2 W y ' F l + a 2 2 ¥ > 2 (/i)y' y

So, we need to show t h a t M > 0. Now, M

_

1 + a\2tp\{h)y* + Q 2 2 y 2 (/t)y* + Qi 2 yi(/t)j/V 2 (1 + riv>i(ft))(l + o 2 2 ^ 2 (/i)y»)

_

1 4- Q22y>2(/t)y* + Qi2V?i(/t)l/*(l + r 2 y 2 ( / Q ) ( l + r i v » i ( f c ) ) ( l + 02av>a(ft)»*)

All t h e terms in M are clearly positive with the possible exception of the term a = 1 + r 2 y> 2 (/i). Now, it is clear t h a t a > 0 if r 2 > 0. On the other hand, if r 2 < 0, then we set a = e'T"2'fe > 0. This completes t h e proof. Third Model:

Predator-prey

x(t + h)- x(t) yi{h)

Discrete

Models

= rix(t) - anx(t)x(t

169

+ h) - a\2x{t + h)y(t)

y(t + h)- y(t) = r2V(t) - a2ix(t)y(t)
- a22y(t + h)y(t)

which yields the difference system _/_ , ii _ X[Tl -I- 1) -

»(tt)(l+riyi(ft) i+0llVl(fc)x(„)+oI:IVi(h)v(n)

'

(4.7.5) ..(„ y^n

i 1\ _ t 1) -

y(n)(l+r3
J

The matrix B in the linearized system is given by

B =

I

l+ai2yi(ft)y*

'

l+nWCO

i \

tt2lV2(ft)l/* l+022^>2(h)l/*

oi2¥>i(
\

l+r,¥>i(/i) > 1 1+022V2C>)V*

. /

We now verify conditions (i), (ii), and (iii) of the Schur-Cohn Criterion stipulated in the second model.

0) detB =

1 + ax2ipi{h)y" a\2a2\ip\(h)ip2(h)x*y' (l+ri¥>i(/i))(l + a 22V > 2 (%*)

Proving that det B < 1 amounts to proving that -ai2r2tpi(h)ip2(h)y*

< an
ana22 0 since the left-hand side becomes negative and the right-hand side is positive. Now, for r2 < 0, Inequality (7.6) holds ifwelet¥Jl(/l)=(^i)(-^r2). (ii)

l-trB

+

detB = l-(1+y^f V l+riy>i(fc)

^ * ... l ++ a22
1 + ai2
+-

.)

170

Non-standard Discretization Methods for Some Biological Models

_ a22ri
l-trB (iii)

4.8

+ detB= ^ W ^ W » V ( ° t i ° « ~ °"°") > 0 , (1 + r1
It is straightforward to show that 1 + trB + det B > 0.

Global Stability of Competitive and Cooperative Systems

In this section, we will employ the methods of monotone dynamical systems to prove global stability in the cases of competitive and cooperative systems [19]. But before doing so, we need to introduce some preliminaries and appropriate notation. Let Qi, 1 < i < 4, be the quadrants in R 2 . Then each quadrant Qi generates a partial order R2, A C R 2 , is Qj-order-preserving if F(X) N or x„+i < x n for all n > N and similarly for yn. Finally, we

Global Stability of Competitive and Cooperative

Systems

171

need the following assumption: (Q+) : if X, Y 6 A and FX
Suppose that F : A —► A satisifies the following:

(i) F isC\ (ii) det DF(X) > 0 for all X € A, (iii) DF(X) is QA— positive in A, i.e., its diagonal entries are nonnegative and its off-diagonal entries are nonpositive, (iv) F is injective, (v) A contains order intervals, i.e., if X,Y £ A and X 4CQ, Y implies that [X,Y]Ql = {Z : X
/ i+yi(ri+ai2y+nai2V>iy)

j €

-v?i<»i2(i-H-iyi)»

DF(X) = -ai2¥>2(l+r2
l+¥>2(r2+"2i »+Q2i ritp^z M5

where L = 1 + fi(anx

+ ai2y),

M = 1 + ^2(022!/ + 021a;)-

Clearly F is C 1 . By a straightforward computation, one may show that det DF > 0. Condition (iii) in Lemma 10.2, DF(X) is Q4 -positive, is selfevident. To prove Condition (iv), we assume that DF(X) = DF(X) for

172

Non-standard Discretization Methods for Some Biological Models

some points X = (

X

] ,X = ( * J € R 2 . Then x + tpiauxy = xipiauyi

(4.8.1)

y +
(4.8.2)

Equating the product of the right-hand side of Eq. (10.1) and the righthand side of Eq. (10.2) with the product of the left-hand side of Eq. (10.1) and the left-hand side of Eq. (1.02) yields xy - xy + a2iip2xx(y - y) + a12(pij/y(a; - x) = 0

(4.8.3)

Observe that Eq.(10.3) is violated if x ^ x or y ^ y. Hence x = x and y = y, which proves that F is indeed injective. Finally, it is easy to see that R2 satisfies condition (v) in Lemma 10.2. This implies that the map F in System (5.1) is competitive and Q+ holds. By Theorem 10.1, the positive equilibrium point ( I * , J / * ) is globally asymptotically stable. By our previous remarks, this also proves that (x*,y*) in the cooperative system (5.2) is globally asymptotically stable.

4.9

Leslie Predator-prey Model

The Leslie predator-prey differential model [2; 19] differs from the pre­ viously-studied predator-prey models in that the predator's response func­ tion is assumed to be basically logistic except that its carrying capacity is a function of the prey density. This model is given by the differential system ft = xWh

- anx{t)

-

auy(t)} (4.9.1)

% =

y(t)[b2-a22m]

where all the parameters b\, a n , ai2, 62, and 022 are positive real num­ bers. System (8.1) has a unique positive equilibrium point (x*,y*) with ,

ha22 aiia.22 + CI12&2

, V =

hb2 011022 + 012^2

Leslie Predator-prey Model

173

Moreover, the equilibrium point (x*,y*) is asymptotically stable for all values of the positive parameters. We propose the following discrete model x(t + h)-

x(t)

-

y(t + h)- y(t)
bix(t) - aux(t)x(t

=

b2y(t)-

+ h) - ai2x(t + h)y(t)

a22V(t)y(t + h) x(t)

which yields the difference system (4.9.2)

It can be easily verified that System (8.2) possesses the same positive equi­ librium point (x*,y*). Linearizing System (8.2) around (x*,y*) yields the coefficient matrix (

i _ QuVi(ft)x* l+6ivi(k)

niaVi (<»)»* l+6, V i(h)

\

. \

a22V2(/>)v* x' (x' +a22f7(h)y'

i i*+o22V2(ft)!/"

. /

B =

We now check the three conditions of Schur-Cohn criterion for the asymp­ totic stability of (x*,y*). (i) ana22fi{h)y* s'fl + friViW-giiyiW1*] + (1 + 6lV»! (/i))(x* + a22V2(/»)2/*) (1 + 6iVi (h))(x* + a22
det£ =

(") 1

-I- det B - i r S =

an
x' + a22
i*[l +6iy?i(/t) -aiii(/t)y2(/i)y*

(1 + b!
>0.

174

Non-standard Discretization Methods for Some Biological Models

(iii) 1

trB = 2-y^ih)x* 1 + biifi (h)

+

detB

+ =

x*[l + bifi(h) - au
-

2ampi(h)x*

+

+

x* + a22
- ana22ipi{h)ip2(h)x*y*

+ 26i
2

+

ai2a22ipi(h)
]/D

where D = (1 + 6iy?i(/i))(x* + a22tp2(h)y*). Notice that 2bnpi(h))x'3 2tpi(h)(bi - a n x * ) = 2ipi(h)x*ai2y* > 0. Also, bia22(pi(h)(p2{h)y*

-

=

ana22ipi{h)ip2(h)x'y* = a22cpi(h)(p2(h)y*(bi = a22
aux*)

> 0.

This completes the proof that (x*,j/*) is asymptotically stable.

4.10

Other Nonstandard Numerical Schemes

In this section, we consider the discretization of the following simple pre­ dator-prey model dx —

=

. ax + pxy (4.10.1)

d

y -77 at

=

, x 72/ + °xy

where x(t) represents the density of the prey at time t, and y(t) represents the density of the predator at time t. If a < 0, 0 > 0, 7 > 0, S < 0.

(4.10.2)

Then (—a/0, ~7/<5) is the only positive equilibrium point of system (8.1). All other solutions are periodic and lie on closed curves surrounding the equilibrium point (x*,y*).

Other Nonstandard Numerical

Schemes

175

If we apply Mickens discretization scheme, the resulting difference equa­ tion possesses solutions that either spiral in towards the equilibrium point (x*,y*) or spirals out of the equilibrium point (i*, y*). In [16], the author employed a discretization scheme attributed to W. Kahan that produces a difference equation whose solutions stay on closed curves. He proposed the following discretization scheme ' ( t + h 2 ~ ' ( t ) = f (x(t + h) + x(t)) + f (x(t + h)y(t) + x(t)y(t + h)) ' y(t+h y(t)

r

= 2 (V(t + h)+ y(t)) + f (x{t + h)y(t) + x(t)y(t + /,)) (4.10.3)

which yields the difference system

}

(4.10.4)

»(^')°»(")",;-t^i;".J Note that the discretization (10.3) differs from all classical discretization schemes. It replaces the nonlinear term x(t)y(t) by ±(x(t + h)y{t)+x(t)y(t

+ h))

while it is replaced by | (x(t + h)y(t + h) + x(t)y(t)) in the standard trape­ zoidal rule and by | (x(t + h) + x(t)) (y(t + h) + y(t)) in the midpoint rule. To explain why Kahan's scheme (10.3) works while most of other numer­ ical schemes produce spiraling solutions, the author in [16] observes that for systems of differential equations in the plane, the situation where all trajectories in the phase spaces are closed curves is nongeneric, i.e., atypi­ cal. Hence in this case any small pertuabation of the right-hand side may change the closed curves into spirals. The effect of numerical integration amounts to changing the system being solved into a nearby system whose solutions would typically spiral. There is, however, a class of differential equations in the plane where closed curves are typical. This is the class of the canonical Hamiltonian systems of the form dx _ dt ~

dH dy

(4.10.5) d]L _ 8H dt ~ ~5x

176

Non-standard Discretization Methods for Some Biological Models

where H = H(x,y) is a Hamiltonian function [17]. The most impor­ tant feature of System (10.5) is that is trajectories are the level curves of the Hamiltonian function H. Moreover, if all trajectories of System (10.5) are closed, then all nearby Hamiltonian systems also have closed trajecto­ ries. We also observe that the flow tph induced by System (10.5), where fh{x(t),y(t)) = (x(t + h),y(t + h)) is an area-preserving map. Hence we should look for a numerical scheme that is also area-preserving. Such nu­ merical schemes are called canonical. This precisely what Kahan's scheme (10.3) achieves. Now, according to KAM theory [17], a canonical numerical method applied to a canonical Hamiltonian system preserves the property of closed curves. It is straightforward to verify that System (10.1) is not Hamiltonian since its vector field (/,
\

(4.10.6)

$ = 7 + *e< J Observe that System (10.6) is divergent free, i.e., | £ + | | = 0, where /(£, n) = a + /3ev, g(£, rj) = 7 + 5e^. The corresponding Hamiltonian func­ tion of (10.6) is given by #(£,??) = N(e^) + M(e"), where N(z) is the antiderivative of ^-^ and M(z) is the antiderivative of ~a~"z ■ Hence JV(e*) = 7£ + Set, M{e") - -ccr) - 0e". Observe that the trajectories of System (10.6) lie on the level curvces of H in the (£, TJ) plane. This implies that the trajectories of the System (10.1) lie on the level curves of the function H(x,y) = H(lnx,lny) = 7lnx + Sx - alny - 0y.

(4.10.7)

Observe that System (10.1) can be written in the following noncanonical Hamiltonian System dx _ dt

1 a(x,y)

dH • dy

d]t _ dt

t dH cr(i,y) dx

>

where
4

(4.10.8)

A Kolmogorov Model of Cooperative

4.11

Systems

177

A Kolmogorov Model of Cooperative Systems

In [13], Robert May suggested the following differential system to model a cooperative system of two species with densities x(t) and y(t).

(4.11.1) d

l = rait) (l "

^ & l )

where r i , r2, a i , a-i, fa, fa are positive numbers. Model (11.1). It is known that if aiQ2 < 1, then System (11.1) has a globally asymp­ totically stable positive equilibrium point (x*,y*) [6]. Although the dis­ cretization used in (3.2) and (3.3) produced a dynamically inconsistent difference equation for Model (2.1), it has been effective in dealing with Model (11.1) [7]. As in (3.2), we consider a modification of System (11.1) to a system of piecewise-constant argument

(4.11.2)

£=™M(I-5T&H)

.

where [t\ denotes the greatest integer in t. Integrating both sides of Eq. (11.2) on [n, n + 1) and letting t -+ n + 1 yields the difference systems x{n + 1)

i+(Skr)*<") (4.11.3)

y(n + l) = 1

-7^m

+(^)»(») J

where n 6 Z + , and R\(n) = fa+ ot\y{n), i?2(«) = fa + ot2x(n). It can be shown that all positive Solutions of Eq.(11.3) are bounded away from zero. Moreover, if aiQ2 < 1, then all positive Solutions of Eq.(11.3) are bounded above [7]. Now, if aitt2 < 1, then there exists a positive equilibrium point (x*,y*) which satisifies the equations. -x*+any*

= -fa (4.11.4)

(*2X* -y* = -fa-

178

Non-standard Discretization Methods for Some Biological Models

The linearized system around (i*,y*) has the coefficient matrix e~ri

/

a1(l-e~ri)

\

e~r*

/

B=l

(4.11.5) \a2(l-e-^)

It is easy to verify that the matrix B satisfies the Schur-Cohn criterion. Hence the equilibrium point (x*,j/*) is (locally) asymptotically stable. Indeed, Gopalsamy and Liu [7] proved that (x*,y*) is globally asymp­ totically stable. Hence System (11.3) is dynamically consistent with the differential system (11.1). 4.12

Open Problems

We propose the following open problems: (i) Extend the nonstandard discretization schemes to differential equations with delays. One may start with a simple equation ([l]) dx

e

H

=

"xW + /(a:(*"2))'£ > °"

(ii) Find a nonstandard discretization scheme for Volterra integral equa­ tions of the form dx fl — = Ax(t) + / K(t, s)x(s)ds. dt J0 (iii) Show that the positive equilibrium in the predator-prey discrete mod­ els (7.4) and (7.5) are globally asymptotically stable.

Bibliography

179

Bibliography

K.L. Cooke and A.F. Ivanov, On the discretization of a delay differential equation, J. Difference Eq. Appl. to appear. J. Cushing, Integrodifferential equations and delay models in population dynam­ ics, Lecture Notes in Biomathematic 20, Springier-Verlag, New York, 1977. A. Dohtani, Occurance of Chaos in higher dimensional discrete-time systems, SIAM J. Appl. Math. 52 (1992) 1702-1721. Elaydi, S., An Introduction to Difference Equations, Second Edition, SpringerVerlag, New York, 2nd ed., 1999. K. Gopalsamy, P. Liu, X. He, Global Stability in Lotka-Volterra difference equa­ tions modeling competition, preprint. K. Gopalsamy, Stability and Oscillations in Delay differential equations of popu­ lation dynamics, Kluwer Academic Pub., Dordrecht-Boston-London, 1992. K. Gopalsamy and P. Liu, On discrete model of Mutualism, New Development in difference equations and Applications, (Ed. Cheng, Elaydi, Ladas), Gordon and Breach, (199), 207-216. J. Hofbauer, V. Huston, and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, Bull. Math. Biol. 24 (1978), 553-579. H. Jian and T.D. Rogers, the discrete dynamics of symmetric competition in the plane, J. Math. Biol. 25 (1987), 573-596. W. Krawcewicz and T.D. Rogers, Perfect harmony: the discrete dynamics of cooperation, J. Math. Biol. 28, (1990), 383-410. P. Liu and S. Elaydi, Discrete Competitive and Cooperative models of LotkaVolterra type, to appear in J. Computational and Appl. Analysis. A.J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, MD. Reprinted 1956, Dover, New York, 1924. R. May, Theoretical Ecology, principles and applications, Philadelphia, 1976, 94100. R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations,

180

Bibliography

World Scientific, Singapore, 1994. R.K. Miller and A.N. Michel, Ordinary Differential Equations, Academic Press, New York, 1982. J.M. Sanz-Serna, An unconvential symphlectic integrator of W. Kahan, Applied Numerical Math. 16 (1994), 245-250. J.M. Sanz-Serna and M.P. Calvo, numerical Hamiltonian Problems, Chapman and Hall, London, 1994. H.L. Smith, Planar competitive and cooperative difference equations, J. Differ­ ence Eq. Appl., Vol.3 (1998), 335-357. J.M. Smith, Model in Ecology, Cambridge University Press, Cambridge, 1974. Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Sci­ entific, Singapore, 1996. V. Volterra, Lecons sur le theorie mathematique pour la vie, Gauthiers-Villars, Paris, 1931. J. Weiner, Generalized solutions of functional differential equations, World Sci­ entific, Singapore, 1993. A. Yakubu, The effects of planting and harvesting on endangered species in dis­ crete competition systems, Math. Biosc. 126 (1995), 1-20. S. Ushiki, Central Difference Schemes and Chaos, Physics D, 4 (1982), 407-424.

Chapter 5

An Introduction to Numerical Integrators Preserving Physical Properties

Martin J. Gander* and Rita Meyer-Spasche^ Most real life applications lead to equations which cannot be solved ana­ lytically. Numerical methods are used to find solutions and it is not evident how to choose a method from the wide variety of methods available. Classi­ cal criteria include order of accuracy, stability and ease of implementation. We emphazise the important criterion of how much of the physics of the underlying problem the numerical method can preserve. We first analyze classical numerical schemes to integrate ordinary dif­ ferential equations and show that they are capable of reproducing the exact solution for corresponding classes of problems. Thus for those problems, all the underlying physical properties are preserved by the numerical scheme. We show that such schemes are of interest in applications if the main part of the problem is in the class integrated exactly. Then we analyze how much of the underlying dynamics a scheme can preserve if it is not exact. This leads us to schemes which preserve fixed points and their stability, closed orbits by preserving the energy and symplectic schemes which preserve the area under the evolution map given by the physical problem.

'Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada, (mgand«rtmath.megill.ca). tMax-Planck-Institut fur Plasmaphysik, EURATOM-Association, D-85748 Garching, Germany (meyer-spascheeipp-garching.mpg.de). 181

182

5.1

An Introduction to Numerical Integrators Preserving Physical Properties

Introduction

There is a wide variety of methods available to integrate systems of ordinary differential equations and the choice of a suitable method is not evident. Methods are usually chosen for their (a) order of accuracy, (b) stability properties, (c) code availability or ease of implementation. Stability is the crucial issue, since systems of ordinary differential equations are often stiff, in particular when they come from discretizations of partial differential equations. Thus when choosing an explicit scheme, the step size of the time integration has to be much smaller than required by the structure of the approximated solution, because otherwise the numerical scheme becomes unstable. In addition it is common practice to solve initial boundary value problems for t -¥ oo in search of steady states. This saves computer storage and thus allows finer spatial discretizations, but uses up much computing time for the time stepping. It is thus desirable to compute with a large time step k to advance quickly to the steady state, the tran­ sient solution not being of interest. Implicit methods have the advantage that they allow larger time steps without going unstable. But they require much more computational effort per time step than explicit methods and there is the problem of superstability, where the numerical method delivers a decaying solution even though the underlying physical problem has an exponentially growing one. If the computational cost of a fully implicit method becomes too high, a mixture of explicit and implicit methods is common practice: implicit where necessary, explicit where possible. Part of the problem is that the traditional theory of difference schemes does neither treat differential equations nor difference equations as dynam­ ical systems. If the dynamical properties of the continuous equations are very different from the dynamical properties of their discretizations, we can still obtain convergence for bounded, closed time intervals and stepsize k -» 0. But on open time intervals and for long time integration it is often difficult to guarantee small global errors. Over the last few years, numerical methods for evolution problems have been investigated using dy­ namical systems theory [3; 28; 32; 35, and the references therein]. Cases are known, for which a common method (forward Euler) produces discrete chaotic trajectories even though the underlying differential equation does

Introduction

183

not have chaotic solutions. Other cases are known in which the differen­ tial equation does have chaotic trajectories which cannot be detected by a common method (backward Euler) [4], In our investigations presented here we focus on the question of how much of the physics of the underlying problem the numerical method can preserve. We show that every choice of a numerical integration scheme involves a decision if the scheme to be used

(d) is exact for a subclass of problems, (e) preserves the dynamics of the underlying problem, (f) is conserving energy or flow like the underlying problem.

At present, this decision is made unconsciously in most cases, because the knowledge necessary for this type of decision is not available yet. It was our aim to increase this knowledge so that in future it will be possible to include criteria (d), (e) and (f) in the choice of the numerical method, in addition to the classical criteria (a), (b) and (c) mentioned above. We treat these three additional criteria separately in the different sections of this chapter. We chose typical model problems to do so. We use schemes of constant order with constant step sizes to simplify the exposition. The problems and phenomena we encounter and discuss are currently avoided in practice by very small step sizes, step size control and order control. We start by analyzing in Section 5.2 classical numerical schemes to integrate ordinary differential equations and show that they are capable of reproducing the exact solution for corresponding classes of problems. Thus for those problems, all the underlying physical properties are preserved by the numerical scheme. We show that such schemes are of interest in applications if the main part of the problem is in the class of problems which are integrated exactly. Then we analyze in Sections 5.3 and 5.4 how much of the underlying dynamics a scheme which is not exact can preserve. This leads us to schemes which preserve fixed points and their stability, closed orbits by conserving the energy and symplectic schemes which preserve the area under the map given by the physical problem.

184

5.2

An Introduction to Numerical Integrators Preserving Physical Properties

Exact Difference Schemes

In this section we consider standard schemes for scalar initial value problems u = /(u),

u(0) = uo € JR.

(5.1)

We investigate if standard schemes can reproduce the exact solution for certain classes of problems, i.e. for given right hand sides f(u). Following [26] we discuss two different ways of finding exact schemes. They are related to the two questions: (1) Given a standard numerical scheme, on which differential equations is it exact? (2) Given a differential equation, which schemes are exact for it? We start by answering the first question for several classical schemes. Tra­ ditionally, error expansions are used to obtain convergence results for all differential equations. We use these error expansions to find the most gen­ eral differential equation for which a given numerical scheme is exact. The second question is the traditional one asked when dealing with exact schemes. Answering this question is essentially the same as finding explicit solutions to the differential equation. Unfortunately this is not possible in general. If, however, a low-dimensional function space is known to contain the solution to be approximated, then a variable-coefficient Runge-Kutta scheme can be contructed which is exact on this function space. We present this approach following Ozawa [31]. We conclude this section by showing how exact schemes can be used to approximate blow-up solutions. We report on work by Le Roux [19] who used exact schemes for ordinary differential equations with blow-up solutions to obtain nonstandard schemes for parabolic differential equations with blow-up solutions. 5.2.1

Standard

Numerical

Schemes

as Exact

Schemes

We start with the first question raised above: given a numerical scheme, for which differential equations is it exact ? To this end, we define the truncation error of a numerical scheme. We restrict ourselves to discussing one step schemes, although the approach we take is more general. Suppose

Exact Difference Schemes

185

we are given a numerical scheme to solve the differential equation (5.1), un+1 = A(f)(un,k),

(5.2)

where k is the time step, u n is an approximation to the exact solution u(tn) at time tn = nk, n = 0 , 1 , 2 , . . . , and A(f) denotes the evolution map given by the numerical scheme. For the forward Euler scheme, for example, the evolution map A{f) is given by u„ + i = A(f){un,

k) -un-r

kf(un).

(5.3)

Note that for implicit methods, the evolution map A(f) requires a non­ linear solve. In this exposition we assume that the explicit form (5.2) can always be obtained in exact arithmetic and we neglect the presence of rounding errors. We will also consider numerical methods for which the evolution map A(f) involves derivatives of / . To estimate the numerical error of a scheme, we define the truncation error T(u, k) of scheme (5.2) using the first step of the numerical method, T(u,k) := i(«(fc) - U l ) = ±(u(k)-A(f)(u0>k)).

(5.4)

For the analysis we expand T{u, k) in a Taylor series in k oo

r(u,*) = £Bi(/(u))(0)*''.

(5.5)

j=0

For forward Euler, for example, we find B

-0

B

*

dJf{u{t))

for j > 1.

(5.6)

t=o

Forward Euler is a first order method, because So = 0, B\ ^ 0 . Definition 5.1 A numerical scheme is an exact scheme for a given differential equation (5.1), if Bj(f{u))(0) = 0 for all j > 0. In the following we investigate several standard numerical schemes and derive the differential equations on which they are exact. 5.2.1.1

First Order Schemes

The two first order schemes mostly used are the forward Euler scheme and the backward Euler scheme. For the forward Euler scheme to be exact, we

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An Introduction to Numerical Integrators Preserving Physical Properties

have to obtain T{u, k) = 0. Using (5.6), we see that this is achieved if all total derivatives of f(u(t)) with respect to t vanish at t = 0. It thus suffices that the first total derivative vanishes since then all the higher derivatives will vanish as well. The first total derivative is given by

df(u(t)) dt

= f'(u(t))u(t)\t=0

= /'(«o)/(u 0 )

t=0

where we used the differential equation (5.1) in the last step. We thus find that the r.h.s. function / has to satisfy a nonlinear differential equation for the numerical scheme to be exact for any initial condition uo, / ' / = 0.

(5.7)

This differential equation has the solutions / ( u ) = 0 and /(u) = C where C is an arbitrary constant. Thus the forward Euler method is exact for any r.h.s. function which is a constant, which is not surprising, since such an equation has a linear solution and Euler is exact for linear solutions. Nevertheless this approach generalizes to arbitrary schemes and we will derive differential equations, nonlinear in general, to be satisfied by the r.h.s. function / such that the numerical method becomes exact. For backward Euler, Un+i -un

+ kf{un+i)

(5.8)

we obtain for the truncation error (5.5) after a short calculation 0

*i = < (

j =0

'j

J

\ rf 7(tx( \*fW))

j > 0

(5.9)

t=o

and thus backward Euler is also a first order method, Bo = 0, Bi ^ 0. It is exact, if all the total derivatives with respect to t of f(u(t)) at t — 0 vanish, i.e. if we obtain T(u, k) = 0. We thus obtain the same differential equation to be satisfied by / as for forward Euler,

/7 = o Backward Euler is an exact scheme for f(u) = C, C an arbitrary constant and thus for all linear solutions u(t).

Exact Difference Schemes

5.2.1.2

187

Second Order Schemes

We start with the trapezoidal rule «n+l - «n

/(«n+l) + /(«n)

U0 = U(0) (5.10) A; 2 which is a method used very often in practice. As follows from the formulas given in [26, Lemma 3.3], its truncation error is given by (5.5) and 0

U' + l)! 2jlJ

dV

t=o

j < 1, • , J>1

(5-11) '

v

and the method is second order in general, fl0 = B\ = 0, B2 # 0. Lemma 5.1 The trapezoidal rule is exact on equation (5.1) for any time step k if the right hand side function f(u) satisfies the differential equation / " / + (/') 2 = 0.

(5.12)

Proof. We have to show that (5.12) implies that Bj = 0 for all j > 0. As for the Euler methods, it suffices for the first non-zero term Z?2 in the expansion to vanish for all the others to vanish as well, since the higher order terms consist of higher total derivatives of the first non-zero term. So to make the first non-zero term Bi vanish, we need

dV(«M) dt2

= jt(f'Wt))f(u(t)))\t=0

(513)

2

2

= /"M/ K) + (/'M) /M to vanish and thus / has either to vanish itself, / ( u ) = 0, or to satisfy the differential equation (5.12) which concludes the proof. D Corollary 5.1 the form

The trapezoidal rule is exact for all r.h.s. functions f of /(«) = VCu + D,

C>0,DeR

(5.14)

and equivalently for solutions u(t) satisfying u(t) 6 span{l,t,t2}. Proof.

(5.15)

It suffices to solve (5.12) for / . Using 0og/)' = ^ ,

(log/')' =

^

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An Introduction to Numerical Integrators Preserving Physical Properties

we divide (5.12) by / / ' , integrate and find log/= d - l o g / ' ,

C,e£

Taking the exponential on both sides, we are lead to the first order differ­ ential equation /' = y ,

C2 = exp(Ci) > 0,

which gives using separation of variables \f2=C2u

+ C3

and solving for / result (5.14) follows. To get result (5.15) we have from the differential equation (5.1)
<* 3 , ,

, xx

tt

-2mt))

= ^U(t)

and thus the vanishing of the second total derivative of / corresponds to the vanishing of the third total derivative of u with respect to t which holds for polynomials of degree at most two. D The next second order method we investigate is the linearly implicit lintrap scheme " n+1 fc~ ^

=/("n)

+

\f>(«")("■*■» " " " ) '

«o = «(0).

(5.16)

Its truncation error is found after a short calculation in expanded form (5.5) with the coefficients

f 0

j < 1,

(5.17) and hence the method is second order in general, S 0 = B\ = 0, #2 # 0. Lemma 5.2 77ie lintrap scheme is exact for (5.1) if f satisfies the dif­ ferential equation 2f"f

- (/')2

=

0.

(5.18)

Exact Difference Schemes

189

Proof. We investigate first the differential equation imposed on / to have the first non-zero term Bv vanish and then show by induction, that this implies that all terms vanish in the error expansion. The first term is given by 6

dt2

t=o

= lf"(uo)f2 - ^ ( / ' M ) 2 / M

4f{uo)~dT-

t=o

and thus it vanishes if either / vanishes or / satisfies the differential equa­ tion (5.18). By induction we show now that under this condition all the error terms vanish. Using the differential equation, the terms Bj can be written as functions of u instead of / , 1 Bj =

d»+1u(t)

0' +1)! (IP+1

t=o

2 !

i «W

dtJ

t=o

and we have just shown that under condition (5.18) Bi — 0, which means in terms of u(t) 3(u(t))2 2u(t)

ii(t)

(5.19)

Hence for induction we assume for a given j > 2 that Bj = 0, which is in terms of u(t) u(t) (i+D =

[+l!iW,««>(«), 2

(5.20)

u(t)

and we have to show that Bj+\ = 0, which means in terms of u{t) the equality (5.21) But using (5.20) we have

dt

2

\

u

(u)2

and using (5.19) to simplify the fraction we find

«W0+a) = 5(; + i ) ( V + I ) + ~ t t W ) -

)

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An Introduction to Numerical Integrators Preserving Physical Properties

Now using the induction assumption (5.20) to replace u^ w

2U

2 u 2 j + lti

' V"

we obtain

y

2

u

which concludes the induction argument.

□

Thus in the case of the lintrap scheme as well, it suffices to require that the r.h.s. function / is such that the first non-zero term in the error expansion vanishes to have all terms vanish. We get immediately Corollary 5.2 The lintrap scheme is exact for all right hand side func­ tions f of the form f(u) = {Cu + D)2,

C> 0, D € R

(5.22)

or equivalently for solutions u(t) of the form

u(t) = Proof.

C>0

" § " CHiTcTy

^Z,6JB-

(523)

To show (5.22) we simply integrate (5.18), (log/)' = 2(log/')'

f = Cl(f')2,

=»

d > 0

which gives the first order differential equation

T Integrating this differential equation using separation of variables leads to result (5.22). To show (5.23) we integrate the differential equation u = f(u) = (Cu + D)2.

a

The implicit midpoint rule is the third example of second order we investigate. It is given by

£

= /(

2

>'

U = U

°

(>'

(

'

and has the error expansion (5.5) with J'<1, Bj = {

I (3

1 +1)!

#f{u{t)) dV

1 d>/((u(0-uo)/2)

J>1(5.25)

Exact Difference Schemes

191

The method is thus second order accurate in general, Bo — B\ — 0. L e m m a 5.3 The implicit midpoint rule is exact for (5.1) if f satisfies the differential equation ~ 2(f)2

f"f

(5.26)

=0

Proof. For the first non-zero term to vanish, we have to impose Bi = 0, which means 1 1 df'((u(t) 6 dt dt t=o t=o. / > o ) ( / N ) ) 2 + (/'(«o)) 2 / f"(u0)(f(u0))2 (/'(uo)) 2 /(uo)

B2

Ufa)

=

12 0.

(/"(uo)(/(«o)) 2 - 2 ( / ' M ) 2 / M )

Again / either has to vanish, f(u) = 0, or it has to satisfy a differential equation (5.26). Integrating this differential equation, 2(log/)' = (log/')'

/2 = Ci/',

=>

C!>0

we find the first order differential equation f2 /' = — which integrated using separation of variables leads to the r.h.s function 1

/(«) = D-Cu

(5.27)

C>Q,D£R

for which the first non-zero term vanishes. Integrating the differential equa­ tion with this right hand side, ii = f(u) =

1 D-Cu

,

u(0) = u 0

(5.28)

we find the solutions u(t) u(t) = ^(D±

y/D2 + C2u2 - 2C(Cuo + t)\ ,

C> 0, D G R.

(5.29)

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An Introduction to Numerical Integrators Preserving Physical Properties

To show that the midpoint rule is exact, we apply the midpoint rule (5.24) to (5.28) and simplify, u(k)-u0 _ k

u2(fc)-2gti(fc) + S*+2gtio-u3

1 D- C"W + °°

(

*(«(*) + "(0) - 2D/C)

'

and show that the discrete scheme has the same solutions in this case as the underlying differential equation (5.28). Solving (5.30) for u(k) we find indeed u(k) = ±;(D± ^ / D 2 + C2u20 - 2C(Du0 + k)\ which means that u(k) lies in the solution space (5.29). Hence the implicit midpoint rule is exact for all r.h.s. functions of the form (5.27) which is equivalent to (5.26). □ Corollary 5.3 f of the form

The implicit midpoint rule is exact for all right hand sides

f{u) =

D~^ai'

C> D R

^ ^-

and equivalently for all solutions u(t) of the form u(t) = ^[D±

y/D2 - 2C(Ci + «)) ,

C>0, d,

DeM.

Proof. The proof is already contained in equations (5.27) and (5.29) of Lemma 5.3. □ 5.2.1.3

Higher Order Schemes

We consider the family of Taylor methods, see for instance [1, p. 215ff]. The higher-order Taylor methods are obtained by adding more and more terms of the Taylor expansion to the numerical method, so the Taylor method of order m is given by

A i d>7(«(0) 3=0

J

£=0

where the total derivatives with respect to time are evaluated to lead to a numerical scheme. For example the first order Taylor method is identical to

Exact Difference Schemes

193

the forward Euler method, while the second order Taylor method is given by k? " n + l = « n + kf{un)

+ y/'(u„)/(«„).

The error expansion (5.5) for the Taylor method of order m contains the coefficients

(

0

j < m, •^

(5-31)

t=o

1 *f(u(t)) and hence the method is in general m-th order accurate, (j + 1)! dti and hence the method is in general m-th order accurate, Bo = B\ — ... — Bm-i = 0, Bm The ? 0. Taylor method of order 2 is exact for all f satisfying L e m m a 5.4 The Taylor method of order 2 is exact for all f satisfying the differential equation f'f + (/') 2 = 0. (5.32) Proof. It suffices for the Taylor methods that the first non-zero term of the error expansion vanishes for the method to become exact, because all the higher order terms are derivatives of the previous ones. For the second order Taylor method, the first non-zero error term is given by

1
= J(/>o)(/(«o))2 + (/'(«o))2/M) t=0

6

and thus the Taylor method of second order is exact if either / vanishes or if / satisfies the differential equation (5.32). D Note that this is the same differential equation that we found for the trape­ zoidal rule and thus the second order Taylor method is exact on the same problems the trapezoidal rule is. Corollary 5.4 side functions

The second order Taylor method is exact for right hand f(u) = VCu + D,

C>0,Dem.

and equivalently for solutions u(t) satisfying u(t) e Proof.

span{l,t,t2}.

The proof is identical to the proof of Corollary 5.1.

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An Introduction to Numerical Integrators Preserving Physical Properties

Next we consider the third order Taylor method. Lemma 5.5 The third order Taylor method is exact for f which satisfy the differential equation f'"f2 Proof. by

+ f3 = 0.

(5.33)

The first non-zero term in the error expansion is S3 which is given

„ <E>3

+ Af'f'f

=

1 24

;rr

d3f{u{t)) dt3 t=o

r(/"'M(/(uo))3 + 4/"( Uo )/'M(/M) 2 + ( / ' M ) 3 / M .

24'

Since all higher order terms vanish if S3 = 0 the result (5.33) follows.

□

Thus for each Taylor method a higher and higher order non-linear differ­ ential equation can be defined and if / satisfies this differential equation, the Taylor method will be exact for this type of problems. However it be­ comes difficult to solve the nonlinear differential equations for / for higher order Taylor methods. Looking at the solution space however, we find the following, intuitive Lemma 5.6 isfying

The Taylor methods of order m are exact for solutions sat­ span{l,t,t2,...,tm}.

u(t) e

Proof. It is convenient to transform the error terms into derivatives of the solution u using the differential equation, dj+1 dP+1

w