Linear Combinations of Generalized Crank Nicolson Schemes

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Linear Combinations of Generalized Crank Nicolson Schemes

A . R. GOURLAY

IBM UK Scientific Centre, Athelstan House, Winchester

AND

J. LL. MORRIS

Department of Computer Science, University of Waterloo, Waterloo, Canada

[Received 16 October 1980]

The combination of two or more generalized Crank Nicolson schemes in order to obtain second, third and fourth order accurate discretizations in time is considered.

Particular attention is given to the stability properties of the methods proposed.

1. Introduction

WE WILL CONSIDER in a general setting the idea of combining together two or more members of the same dass of schemes (but with different parameter values) in order to obtain methods with a higher order of accuracy with respect to the time discretization and/or improved stability characteristics. To be specific we will consider the single evolutionary equation

where A will usually be an elliptic operator in the space variables. [For the development of the time discretizations it will not be necessary to be specific about A or the boundary or initial conditions. An alternative approach would have been to follow that of Lawson & Morris (1978), and work with the problem already discretized in the space dimensions.] The dass of schemes we wish to consider is that based on the generalized Crank Nicolson method,

U-k9A-]v^m+l = U + k(l-0)A]v^m, (1.2) which has local error in time of 0[(0 —$)Jk2+fc3] where k is the time step and vm

denotes an approximation to u(-,mk), u being the solution of Equation (1.1).

Normally we use one of three likely methods in the class (1.2) corresponding to 0 = 0 the explidt method,

0 = 1 the backward difference method, 9 = i the Crank Nicolson method.

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The latter scheme is, of course, of second order accuracy in time, one order higher than the other two. The second and third methods enjoy unconditional stability for the initial value problem, based on a simple von Neumann analysis.

It should be noted that (1.2) should be written in the "efficient" form

which is a generalization of the method discussed by Lawson & Morris (1977). For clarity we do not explicitly formulate the following methods in this form although it should be understood that in a practical implementation each member should be written in this way.

In the following we will not concern ourselves with the approximation process in the space co-ordinates, though specific examples may be quoted later. However, it should be appreciated that quite high accuracy may now be achieved in the spatial approximation process. Our motivation here is to explore the possibilities of obtaining higher order methods for the time discretization process, which are generated by linear combinations of two or more formulae of the class (1.2). As the choice 9 = $ gives us a second order method our interest will be in determining second, third and fourth order temporally accurate methods with or without stability characteristics which are an improvement on those of the Crank Nicolson method.

Alternative approaches to increasing the accuracy of time integration include defect correction (Stetter, 1978) and deferred correction (Pereyra, 1973). Saylor (1979) has made a comprehensive study of these methods.

2. Combination of Two Theta Schemes 2.1 Third Order Methods

To generate third order temporal accuracy we consider the following procedure

\l-k9A~\vv = [/+/c(l-0)/l]t>M,

U-k<t>Alv> = U+k(l-<P)A]v^m, (2.1) vm+l=xvl+(l-a)v2,

where 9, <f> and a are parameters, and u1, v2 are consistent intermediate approximations to vm+l. In operator form we may write the solution of Equation (1.1) as

(2.2) For future use we note the expansion

lI + k(l-6)A']U-k8A]~1 = I + kA + 9k2A2 + 92k3A3 + .... (2.3) Eliminating v\ v2 from (2.1) gives

vm+1 = (a[/

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To discover the temporal accuracy of (2.4), and hence (2.1), we substitute in (2.4) u(-, mk) for vm, etc., and compare the resulting expansion [in powers of kA using (2.3)] with the expansion for the exponential in (2.2). The following accuracy conditions must be satisfied in order to give us a third order method:

second order condition

«0 + ( l - « W = i, (2.5) third order condition

a6² + (l-a)<f>² = i (2.6) Equation (2.5) defines a. in terms of 9 and <f> as

<x = (i-<j>)/(6-<fi). (2.7) We note that (2.5) and (2.6) cannot both be satisfied if 9 = <f> or 9 = ±. Equations (2.6) and (2.7) now allow the determination of <f> in terms of 9 as

60), (2.8) where again the case 0 = i is excluded. This choice of 0 in terms of 9 allows us to write

(2.9) We note in passing that a can only become infinite when 9 = 4>, and that this occurs if 9 = \ ± 1A/12. At this stage therefore we have a third order in time method containing a parameter 9.

2.2 Stability of Third Order Schemes

The next step is to consider the stability of (2.4). If we assume that A is elliptic in the sense that its space Fourier transformation is a real non-positive number [as would be the case if Equation (1.1) was the scalar heat equation], then the symbol or amplification factor of the scheme (2.4) is

[l-(l-flt]

( 2 1 0 )

where t is real and non-negative. Our only interest will be in schemes which are unconditionally stable (like the Crank Nicolson method) and therefore we seek those values of the parameter 9 which will give a positive value of <f> and a non-zero value of a such that R defined by Equation (2.10) is bounded by one in modulus.

The first observation that we can make is that neither 9 nor <f> can be allowed to assume a negative value: otherwise there would exist some value of t such that one of the denominators in (2.10) would become zero. As 0 is defined in terms of 9 the condition for ^ to be non-negative, together with the non-degeneracy requirement, becomes

0 > } or OsSfl^f

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f

To continue our analysis of (2.10) we rewrite it in the following two different ways

Using the definition of <£ in (2.8) we see that

Since the numerator in the second expression is always positive, it follows that unconditional stability occurs only when 9 > \. Those schemes employing a value of 0 < 9 < ^ enjoy only conditional stability. However, as we observed above it is possible for a to become infinite at certain points. Therefore we must not allow 9 to assume a value which would cause a to be infinite. The only such value in the region of unconditional stability is

= 0-788675, and therefore a realistic stability region would be, for example,

" [(i, oo)-(0-75,0-83)].

The exclusion of the small interval in fact ensures that a never exceeds four in magnitude. It should be noted that a is only positive for 9 in the lower of the two intervals forming the stability region. It is somewhat curious to note that the critical value of 9 does not in fact lead to an unstable process but causes degeneracy to a first order method to occur with 9 = <j>. The allowable range of values for a excludes the interval between 0 and 1 and therefore (2.1) is never a simple convex combination of two unconditionally stable methods. This is presumably a result of the third order accuracy condition. It is of interest to consider the asymptotic value of R as t tends to infinity. This is

which has a maximum value of — 0-732 at the excluded value of 9. The values at 0-75 and 0-83 are, however, both roughly —0-733. We can see that there is therefore a certain amount of asymptotic damping and oscillation present, but that this is favourable in comparison with the Crank Nicolson scheme. As the method contains a free parameter 9 it might be hoped that we could choose it in order to make the method fourth order. The coefficient of the fourth order term is proportional to

This is non-zero for real 9 but its magnitude is minimized when 9 = $ + lA/l2> the excluded point.

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2.3 Controlled Second Order Schemes

We now consider (2.1) as a process for improving the stability characteristics of the Crank Nicolson method. This we know is unconditionally stable but not L0-stablc, to use a phrase from the numerical analysis of ordinary differentia] equations. In practice the Crank Nicolson method behaves badly when high frequency components are present (see Lawson & Morris, 1978). In contrast, the backward difference method [0 = 1 in (1.2)] has L0-stability but only first order accuracy in time. In Fig. 1 two curves are shown; the Crank Nicolson method and the third order method

- I -

we developed above have the characteristics of the curve A, namely that of decreasing (usually monotonically) to a negative number RL. For the Crank Nicolson method RL = -l, whereas for the third order method - 1 < RL < 1 - ^ 3 = -0-732.

Curve B shows the behaviour of the backward difference method, namely that of tending to an asymptotic value of zero (the L0-stability limit value). This second curve has a better stability profile than that of curve A. The qualitative behaviour we seek is that of stability everywhere \R\ < 1, a small, usually zero asymptotic value RL, and generally a rapid decrease to this asymptotic value with small negative values of R (if it assumes any). In the Lawson & Morris (1978), and Gourlay & Morris (1980) papers, an extrapolation process was used to construct schemes with this pattern of behaviour. In this section we explore the potential of (2.1) in the construction of schemes with improved stability profiles.

We return to scheme (2.1) and now only impose the second order accuracy condition (2.5), leading to the definition of a in (2.7). In place of the third order accuracy condition we impose the requirement that RL = 0. (We could, in fact, chose any value that we felt would lead to the required stability profile.) Using the definition of a in (2.10) and letting r-tend to infinity, we find that

(2.12)

(2.13) and that for RL = 0, we must satisfy the condition

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This, together with (2.7), gives the representation for a in terms of 0 as

a = 0/(40 - 2 ^ - 1 ) , (2.14) and the curve of the function (2.14) is shown in Fig. 2. This, together with the positivity requirement for <p defined by (2.13) and the stability conditions

- i - 0(0 ~ i ) / ( 0 - 1 ) ^ 0 ,

shows that there are two allowable ranges for 0, namely 0 < 0 < }, excluding the point 0 = 1 - 1 / ^ / 2 , and 0 > 1 with the exclusion of the point 0 = 1 +1/^/2. In both of these ranges the graph of R decreases monotonically to zero. The parameter a is negative for 0 < 0 < 1 —1/^/2. and positive in the ranges 1 -1/^/2 < 0 < i and 1 < 0 < 1 + 1A/2. These combination schemes therefore have as good, if not better, stability profiles than those described in Gourlay & Morris (1980). A further advantage is that they involve only two generalized theta schemes in order to achieve second order accuracy. It is only possible to force third order accuracy if we are prepared to accept a value of the limit RL < 1—

3. Combinations of Three Theta Schemes 3.1 Fourth Order Methods

The natural extension of method (2.1) to a combination of three theta methods, in order to try to obtain a fourth order in time method is given by

., (3-1)

where 0, <p, \p, a and /) are all parameters.

Following the same procedure as before to determine the accuracy conditions leads to the following:

second order condition

<x0+/?</> + ( l - « - W = i (3.2) third order condition

a02 + /?02 + ( l - a - ^ J = i (3.3) fourth order condition

3 3 3 = i . (3.4)

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10

- O - 7 0-3 1-0 1-3

FlO. 2.

These equations are satisfied if

(3.5) (3.6) (3.7) provided no two of 6, <f> and ip are equal, and the denominator in (3.5) is nonzero.

Therefore the set of excluded points will include those where 6 = <f>, <j> = ip, \j/ = 9,

3.2 Stability of Fourth Order Schemes The symbol of the scheme (3.1) is given by

[ 1 + 0 0 ' This may be written in the two ways:

+ {i-a-py- (3.8)

R= 1 -

= - 1 where

B =

D = - i ) = 294»j/-B.

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For ty defined in (3.5) these reduce further to

D = A/6.

Using these expressions it is straightforward to verify that \R\ < 1 if 9, </>, \ji > 0, and - } > 0 ,

Considering the two cases of 9 < \ and 9 > \ it is straightforward to verify that only the latter case allows all these conditions to be satisfied. Also we observe that if the above conditions are met this will ensure that ip > } since

Therefore if we assume that 9 is greater than \ the condition for unconditional stability becomes

In 9 — <f> parameter space the excluded points where degeneracy occurs are defined by the equations

In Fig. 3 we present the stability region for the method (3.1) with the excluded points marked by broken lines. It should be noted that this only gives the region of unconditional stability. There is a region of conditional stability when 0 < 6 < $.

3.3 Controlled Third Order Schemes

We now repeat the analysis carried out above only requiring third order accuracy [conditions (3.2) and (3.3)] but now imposing the extra constraint that the method should have a symbol which has the asymptotic value of zero (RL = 0). From (3.8) we see that

and since our third order accuracy requirement will mean that definitions of a and /?

will be as in (3.6) and (3.7) it follows that RL

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0-5

a Fio. 3.

and therefore the value of \j/ giving RL = 0 is

(3.9) The requirements for stability are the same as before with the new definition of \j/.

Using (3.9) the quantities B, C, D defined above now have the simple values

B = 6<t>ip, C = 2e<t>ip + i D = 6(t>4i,

and the numerator [2+2At + Ct²+Dt³'] may be written in the form 2(6 + <f> + il/)t + d<t>ip(2t² + t³) + 2-t + t²/6,

which is positive for positive 6, <p, \j/. Therefore stability only requires that 0,

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for all non-negative t. This restriction may be written in the form

Using the definition of \ji the stability region therefore consists of three parts (i) 9,4> > I with

(ii) the region < 0,

)(

including the point (^, 1),

(iii) the region 9, <£ < \ where 9<fi-(9 + 4>)/2+$ > 0 but excluding that part not satisfying (9 + <p + 4i-{)² < 49(f>4i.

It seems easier simply to check whether this condition is satisfied for any values of 9, <p < i that are selected. For the values of9,4>> 1 it appears that the stability curve goes monotonically to zero with increasing t, but this behaviour does not characterize those schemes with 9, <f> < {. Once again certain points must be excluded from the region of stability, namely those where any denominator becomes zero that is when 9 = <(>, (f> = i/> or 9 = &. The stability regions are displayed in Fig. 4.

FIG. 4.

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4. Conclusion

We have shown that there exist families of second, third and fourth order in time methods based on linear combinations of generalized Crank Nicolson methods.

There are methods which have unconditional stability and third (fourth) order temporal accuracy based on a combination of two (three) schemes. The imposition of a requirement that the amplification factor tends asymptotically to zero reduces the order achievable in each case by one. For each case there is a considerable freedom in parameter choice.

REFERENCES

GOURLAY, A. R. & MORRIS, J. LL. 1980 The extrapolation of first order methods for parabolic partial differential equations II. SIAM J. Num. Analysis 17 (5), 641-655.

LAWSON, J. D. & MORRIS, J. LL. 1977 A note on theefficient implementation of splitting methods in two space variables. BIT 17, 492-493.

LAWSON, J. D. & MORRIS, J. LL. 1978 The extrapolation of first order methods for parabolic partial differential equations I. SIAM J. Num. Analysis 15, 1212-1224.

PEREYRA, V. 1973 High order finite difference solution of differential equations. Stanford Rep.

S/an-GS-73-348, Stanford, California.

SAYLOR, A. 1979 Extrapolation, deferred correction, and defect correction on discrete-time Galerkin methods for linear parabolic problems. Ph.D. Thesis, University of Kentucky.

STETTER, H. 1978 The defect correction principle and discretization methods. Num. Math. 29, 425-443.