Waveform Relaxation for Functional Differential Equations

Barbara Zubik-Kowal and Stefan Vandewalle

Katholieke Universiteit Leuven
Department of Computer Science
Celestijnenlaan 200A
B-3001 Heverlee, Belgium


Abstract

Waveform relaxation, or dynamic iteration, is an iterative method for solving large-scale systems of differential equations. The method extends classical iterative methods (such as Jacobi, Gauss-Seidel, SOR, multigrid, etc.) to function spaces. That is, the iterates are a sets of continuous functions, instead of sets of scalar variables. So far, the method has been applied primarily to solve ordinary differential equations, possibly obtained by semi-discretisition from a time-dependent partial differential equation, and differential-algebraic equations.

In this talk we study the application of the waveform relaxation technique for solving functional differential equations. Such problems arise for example in population dynamics, in the modelling of visco-elastic materials and in the study of nonlinear materials with memory. These equations include differential integral equations, delay differential equations as well as ordinary differential equations as special cases. By using the waveform iteration such problems can be solved iteratively by computing the solution as the limit of a series of solutions to classical differential equations without functional terms.

We will first define the type of equation that we consider, and illustrate the application of three different waveform relaxation schemes by means of three examples. Convergence of the iterative scheme under very general and nonlinear conditions on the equation's right-hand side is studied next, and a very general error estimate is derived. We continue by analysing the method's convergence under a more specialised, time-dependent Lipschitz condition for the equation's right-hand side, and construct precise error bounds for various special cases. These iteration error estimates are compared to the classical iteration error estimates that have been obtained previously under constant Lipschitz conditions.

We also present results of actual computations, and validate the quality of the theoretical estimates by extensive numerical data. This is done by several examples from partial functional-differential equations of parabolic type. Besides offering the potential for good parallel performance, the methods considered in this talk also proved to be very easy to implement. Hence, it is our belief that the waveform relaxation method could become quite an effective numerical method for solving functional differential equations, if the convergence could be improved further by using more sophisticated iteration schemes.