A convergence study of multisubdomain Schwarz waveform relaxation for a class of nonlinear problems (Q1666139)

Summary: Schwarz waveform relaxation (SWR) is a new type of domain decomposition methods, which is suited for solving time-dependent PDEs in parallel manner. The number of subdomains, namely, \(N\), has a significant influence on the convergence rate. For the representative nonlinear problem \(\partial_t u = \partial_{x x} u + f(u)\), convergence behavior of the algorithm in the two-subdomain case is well-understood. However, for the multisubdomain case (i.e., \(N \geq 3\)), the existing results can only predict convergence when \(f'(u) \leq 0 (\forall u \in \mathbb{R})\). Therefore, there is a gap between \(N \geq 3\) and \(f'(u) > 0\). In this paper, we try to finish this gap. Precisely, for a specified subdomain number \(N\), we find that there exists a quantity \(d_{\max}\) such that convergence of the algorithm on unbounded time domains is guaranteed if \(f'(u) \leq d_{\max} (\forall u \in \mathbb{R})\). The quantity \(d_{\max}\) depends on \(N\) and we present concise formula to calculate it. We show that the analysis is useful to study more complicated PDEs. Numerical results are provided to support the theoretical predictions.