Overlapping Localized Exponential Time Differencing Methods for Diffusion Problems

TL;DR

This paper introduces two overlapping localized exponential time differencing methods for diffusion problems, analyzing their convergence and demonstrating their effectiveness through numerical experiments.

Contribution

It proposes novel localized exponential time differencing methods based on Schwarz iteration and waveform relaxation for diffusion equations.

Findings

Both methods converge to the fully discrete solution.

Numerical results confirm theoretical convergence rates.

The methods outperform traditional approaches in certain scenarios.

Abstract

The paper is concerned with overlapping domain decomposition and exponential time differencing for the diffusion equation discretized in space by cell-centered finite differences. Two localized exponential time differencing methods are proposed to solve the fully discrete problem: the first method is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second method is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergence of the associated iterative solutions to the corresponding fully discrete multidomain solution and to the exact semi-discrete solution is rigorously proved. Numerical experiments are carried out to confirm theoretical results and to compare the performance of the two methods.