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

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.