Optimized Schwarz Waveform Relaxation for Advection Reaction Diffusion Equations in Two Dimensions

TL;DR

This paper develops and analyzes optimized Schwarz waveform relaxation methods with effective transmission conditions for solving advection reaction diffusion equations in two dimensions, ensuring rapid convergence and high performance.

Contribution

It provides a detailed mathematical analysis and closed-form asymptotic parameters for Robin and Ventcel transmission conditions, enhancing the efficiency of waveform relaxation methods.

Findings

Proves best approximation results for Robin and Ventcel conditions.

Derives asymptotic parameter values for optimal performance.

Validates analysis with extensive numerical experiments.

Abstract

Optimized Schwarz Waveform Relaxation methods have been developed over the last decade for the parallel solution of evolution problems. They are based on a decomposition in space and an iteration, where only subproblems in space-time need to be solved. Each subproblem can be simulated using an adapted numerical method, for example with local time stepping, or one can even use a different model in different subdomains, which makes these methods very suitable also from a modeling point of view. For rapid convergence however, it is important to use effective transmission conditions between the space-time subdomains, and for best performance, these transmission conditions need to take the physics of the underlying evolution problem into account. The optimization of these transmission conditions leads to a mathematically hard best approximation problem of homographic type. We study in this paper in detail this problem for the case of linear advection reaction diffusion equations in two spatial dimensions. We prove comprehensively best approximation results for transmission conditions of Robin and Ventcel type. We give for each case closed form asymptotic values for the parameters, which guarantee asymptotically best performance of the iterative methods. We finally show extensive numerical experiments, and we measure performance corresponding to our analysis