An investigation of overlapping domain decomposition methods for one-dimensional dispersive long wave equations

Abstract

This paper contains a thorough investigation of the numerical accuracy and efficiency of an overlapping domain decomposition (Schwarz iteration) method for weakly dispersive and non-linear water waves. The investigation is restricted to one-dimensional wave propagation. In our tests, a global domain is divided into two overlapping subdomains. A local boundary value problem (at a time level) is solved in each subdomain, with boundary values extracted from the neighboring domains. The size of the overlap and number of iterations over the domains are the principal parameters that influence the convergence speed and numerical accuracy. We also investigate different finite difference and finite element formulations. The finite signal speed in wave problems makes domain decomposition methods particularly efficient, and the proposed method converges satisfactorily already for an overlap of some typical water depths and a few (2–3) iterations. Numerical artifacts and instabilities may develop at the subdomain boundaries for certain choices of overlap, wave lengths, grid sizes, and (discrete) wave velocities. A special filtering technique is designed to make the method more robust with respect to such instabilities.