Optimal Schwarz Waveform Relaxation for the One Dimensional Wave Equation

We introduce a nonoverlapping variant of the Schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension. We derive transmission conditions which lead to convergence of the algorithm in a number of iterations equal to the number of subdomains, independently of the length of the time interval. These optimal transmission conditions are in general nonlocal, but we show that the nonlocality depends on the time interval under consideration, and we introduce time windows to obtain optimal performance of the algorithm with local transmission conditions in the case of piecewise constant wave speed. We show that convergence in two iterations can be achieved independently of the number of subdomains in that case. The algorithm thus scales optimally with the number of subdomains, provided the time windows are chosen appropriately. For continuously varying coefficients we prove convergence of the algorithm with local transmission conditions using energy estimates. We then introduce a finite volume discretization which permits computations on nonmatching grids, and we prove convergence of the fully discrete Schwarz waveform relaxation algorithm. We finally illustrate our analysis with numerical experiments.

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