Neumann-Neumann Waveform Relaxation Algorithm in Multiple subdomains for Hyperbolic Problems in 1D and 2D

TL;DR

This paper introduces a waveform relaxation version of the Neumann-Neumann algorithm for hyperbolic wave equations in 1D and 2D, demonstrating convergence and efficiency through theoretical analysis and numerical experiments.

Contribution

It develops a novel waveform relaxation Neumann-Neumann method for hyperbolic problems, with proven finite-step convergence and performance comparisons.

Findings

Convergence in finite steps for specific parameters

Performance improvements over classical methods

Dependence of iteration count on subdomain size

Abstract

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.