Neumann-Neumann Waveform Relaxation Algorithm in Multiple subdomains for Hyperbolic Problems in 1D and 2D
This work provides a theoretical convergence guarantee for a domain decomposition method applied to wave equations, benefiting researchers in numerical methods for hyperbolic PDEs.
The paper presents a Neumann-Neumann waveform relaxation algorithm for hyperbolic problems, proving convergence in a finite number of steps for finite time intervals using Fourier-Laplace analysis, and demonstrating performance with numerical results.
We present a Waveform Relaxation (WR) version of the Neumann-Neumann algorithm for the wave equation in space-time. The method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves in space-time with corresponding interface condition, followed by a correction step. Using a Fourier-Laplace transform argument, for a particular relaxation parameter, we prove convergence of the algorithm in a finite number of steps for finite time intervals. The number of steps depends on the size of the subdomains and the time window length on which the algorithm is employed. We illustrate the performance of the algorithm with numerical results, followed by a comparison with classical and optimized Schwarz WR methods.