Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients

Abstract

In this paper, we investigate the effect of the space and time discretisation on the convergence properties of Schwarz Waveform Relaxation (SWR) algorithms. We consider a reaction-diffusion problem with discontinuous coefficients discretised on two non-overlapping domains with several numerical schemes (in space and time). A methodology to determine the rate of convergence of the classical SWR method with standard interface conditions (Dirichlet-Neumann or Robin-Robin) accounting for discretisation errors is presented. We discuss how such convergence rates differ from the ones derived at a continuous level (i.e. assuming an exact discrete representation of the continuous problem). In this work we consider a second-order finite difference scheme and a finite volume scheme based on quadratic spline reconstruction in space, combined with either a simple backward Euler scheme or a twostep "Padé" scheme (resembling a Diagonally Implicit Runge Kutta scheme) in time. We prove those combinations of space-time schemes to be unconditionally stable on bounded domains. We illustrate the relevance of our analysis with specifically designed numerical experiments.

Optimized Schwarz waveform relaxation method for the incompressible Stokes problem

“…This has indeed recently been observed for the heat equation in [2]. As far as time discretization is concerned, the inclusion of its effect in the convergence analysis of OSWR methods is a current topic of research, and is for example addressed in [15] where a Z− transform is used and in [2], where a discrete-time analysis of the OSWR method is proposed. This issue is also addressed in Section 9.2.…”

“…We first follow this standard approach in this work, but in a second step modify it to also include the effect of the discretization in the time direction; the Robin parameters obtained with such a modification improve the convergence rate over the standard choice in our numerical tests. Note that studying the influence of the numerical scheme over the OSWR convergence rate is a recent approach, pursued for example in [15,26,2].…”

Discrete-time analysis of optimized Schwarz waveform relaxation with Robin parameters depending on the targeted iteration count

“…This may be due to the fact that the Fourier transform supposes an infinite time interval, while the actual simulation is necessarily performed on a finite one; switching to Fourier series does not solve this issue since the error does not vanish at the final time as it does at the initial time. Another approach, based on discrete-time analysis is proposed in [9,27]; for simple schemes, it is based on the so-called one-sided 𝒵 transform, which is a discrete equivalent of the Laplace transform. However, this also requires either to consider infinite intervals in time or to neglect the error at the final time.…”