Overlapping Schwarz Waveform Relaxation for Convection-Dominated Nonlinear Conservation Laws

We analyze the convergence of the overlapping Schwarz waveform relaxation algorithm applied to convection-dominated nonlinear conservation laws in one spatial dimension. For two subdomains and bounded time intervals we prove superlinear asymptotic convergence of the algorithm in the parabolic case and convergence in a finite number of steps in the hyperbolic limit. The convergence results depend on the overlap, the viscosity, and the length of the time interval under consideration, but they are independent of the number of subdomains, as a generalization of the results to many subdomains shows. To investigate the behavior of the algorithm for a long time, we apply it to the Burgers equation and use a steady state argument to prove that the algorithm converges linearly over long time intervals. This result reveals an interesting paradox: while for the superlinear convergence rate on bounded time intervals decreasing the viscosity improves the performance, in the linear convergence regime decreasing the viscosity slows down the convergence rate and the algorithm can converge arbitrarily slowly, if there is a standing shock wave in the overlap. We illustrate our theoretical results with numerical experiments.