A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations

Abstract

Waveform relaxation is a technique to solve large systems of ordinary differential equations (ODEs) in parallel. The right hand side of the system is split into subsystems which are only loosely coupled. One then solves iteratively all the subsystems in parallel and exchanges information after each step of the iteration. Two classical convergence results state linear convergence on unbounded time intervals for linear systems of ODEs under some dissipation assumption and superlinear convergence on bounded time intervals for nonlinear systems under a Lipschitz condition on the splitting. To apply waveform relaxation to partial differential equations (PDEs), one traditionally discretizes the PDE in space to get a large system of ODEs, to which then the waveform relaxation algorithm is applied using a matrix splitting. There are two problems with this approach: first information about how to split the right hand side is lost during the discretization; second the convergence results derived in this fashion depend in general on the mesh parameter and convergence rates deteriorate when the mesh is refined. To avoid those problems a new waveform relaxation algorithm is formulated directly at the PDE level. The differential operator on the right hand side is split using domain decomposition. It is shown for a scalar reaction diffusion equation with variable diffusion coefficient that the new waveform relaxation algorithm converges superlinearly for bounded time intervals and linearly for unbounded time intervals, extending the two classical convergence results to this type of PDE. Interestingly the superlinear convergence rate is faster than the superlinear convergence rate obtained by the traditional matrix splitting methods. It is shown how the convergence rates depend on the overlap of the domain decomposition and a Lipschitz condition on the reaction function. The splitting of the right hand side is naturally given by the domain decomposition and the convergence rates are robust with respect to mesh refinement when the algorithm is discretized.