Telescopic projective methods for parabolic differential equations

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Abstract

Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components.

Introduction

In [1] we introduced the projective method which is based on the following very simple idea: any stable method (the inner integrator) is used to integrate a problem over a number of small steps and then a projective2 step uses polynomial extrapolation to compute an approximation to the solution far ahead of the inner integration steps. The first k steps of the inner integrator serve to damp the fast components in the solution. The projective step then uses the result of the last step and the results from the next q inner steps to extrapolate forward. This is shown in Fig. 1 with k=2 and q=1. It is clear that the slope of the chord through y2 and y3 in Fig. 1 is a first-order approximation to the derivative, so we call this example a Projective Forward Euler (PFE) method.

The combination of the inner steps and the projective step effectively constitutes another integrator which we call the outer integrator. It was shown that this integrator could be constructed so that it was absolutely stable (hereafter referred to as “stable”) if the eigenvalues of the problem Jacobian matrix were in one of two regions, one corresponding to rapidly decaying components handled very stably by the inner step and one corresponding to an approximation to the stability region of the outer integrator (the Forward Euler in the example of Fig. 1). Thus, for problems with a large gap between the time constants of the fast inactive components and the time constants of the slow active components (those still causing changes in the solution) one is able to project forward over large steps, commensurate with the slow components, and gain speed (i.e., increase average step size). In the example shown in Fig. 1 we use three inner integration steps of length h0 to cover a distance h1=(3+M)h0. If we assume that the inner integration step represents the bulk of the work (because, for example, evaluations of the derivatives are very expensive) and that it is not possible to use a larger step size in the inner integrator, then we can define the speedup of the projective method as the number of inner integration steps needed to integrate over the interval directly divided by the number used when combined with the projective step. Thus the PFE method in Fig. 1 has a speedup ofS=(3+M)/3.

We are particularly interested in the application of these methods when the inner integrator is a legacy code that performs one time step. Often it is extremely difficult to modify such codes because they represent many years of development, employ numerous devices such as split steps, and the original developers may have long since left.

In this paper we consider an obvious extension of the projective method: Since the outer, or projective integrator can be viewed as just another integrator, why not use it as the inner integrator in yet a further projective integrator, and so on, ad infinitum? This is illustrated in Fig. 2 which has two projective levels. Note that in this illustration the speedup of the 2nd level method is [(3+M)/3]2 since we cover a distance (3+M)2h with nine inner integrations.

The methods resulting from this iteration of the projective step can have two quite different sets of properties. In one case they can handle problems that have multiple gaps – that is, whose eigenvalues lie in one of a number of well-separated regions of the complex plane. In the second case we can choose the method parameters so that its stability region includes a large section of the negative real axis and neighboring points in the complex plane – methods we will call “[0,1] stable” for reasons that will be apparent later. This is the case that is applicable to parabolic equations and will be discussed in this paper. The multiple gap case is discussed in [2].

In the next section we will briefly review the stability analysis of projective methods and their important properties so that we can discuss [0,1] stable methods in the third section.

Section snippets

Stability analysis

The usual linear stability analysis of time-stepping methods discusses stability in the -plane, where h is the time step and λ is an eigenvalue of the local linearization of the problem. We are applying a projective step to any inner integrator (we may have, for example, a legacy code that performs one time step in a manner we do not fully understand). Since the nature of the inner integrator affects the stability and since we wish to analyze the stability of the projective process

Stability and speedup of TP methods

One way to compute the stability region is to map the unit circle under the inverse of the mapping (4) for a number of iterations. Actually, then we have the boundary of the stability region of only that many applications of the TP method. Since the region shrinks at each iteration (and starts from the finite unit disk) it must converge and in practice we quickly get a reasonable approximation to the infinitely iterated region. Fig. 5 shows the stability region for 10 iterations of the P2–1–3

Conclusion

We have shown how the projective method can be iterated to achieve larger regions of absolute stability. Some simple numerical examples are given in [2] where it is also shown how to handle problems with multiple gaps in their eigenspectrum. A combination of these two objectives could be used to have fewer gaps and larger stability regions where needed. While the methods discussed do not appear to be as efficient as Runge–Kutta methods designed to have extended stability regions, telescopic

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Supported in part by AFOSR (Dynamics and Control program, Dr. Mark Jacobs and Dr. Belinda King).

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