Convergence Analysis of Waveform Relaxation Method to Compute Coupled Advection-Diffusion-Reaction Equations

TL;DR

This paper analyzes the convergence of the Schwarz waveform relaxation method for coupled advection-diffusion-reaction equations, proposing an optimized algorithm that accelerates convergence for both linear and nonlinear cases, including viscous Burgers equations.

Contribution

It introduces an optimized algorithm for Dirichlet conditions that significantly speeds up convergence in waveform relaxation methods for coupled PDEs, extending effectiveness to nonlinear equations.

Findings

Optimized algorithm achieves substantial convergence speedup.

Method effective for nonlinear equations like viscous Burgers.

Numerical examples confirm improved efficiency.

Abstract

We study the computation of coupled advection-diffusion-reaction equations by the Schwarz waveform relaxation method. The study starts with linear equations, and it analyzes the convergence of the computation with a Dirichlet condition, a Robin condition, and a combination of them as the transmission conditions. Then, an optimized algorithm for the Dirichlet condition is presented to accelerate the convergence, and numerical examples show a substantial speedup in the convergence. Furthermore, the optimized algorithm is extended to the computation of nonlinear equations, including the viscous Burgers equation, and numerical experiments indicate the algorithm may largely remain effective in the speedup of convergence.