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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 ∂ t u = ∂ 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 ≥ 3 ), the existing results can only predict convergence when f ′ ( u ) ≤ 0 ( ∀ u ∈ R ) . Therefore, there is a gap between N ≥ 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 ) ≤ d max ( ∀ u ∈ 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.

A Convergence Study of Multisubdomain Schwarz Waveform Relaxation for a Class of Nonlinear Problems