Schwarz Waveform Relaxation Algorithms for the Linear Viscous Equatorial Shallow Water Equations

Abstract

We are interested in numerically solving the viscous shallow water equations with a small Coriolis force on a large domain. Specifically, we develop and analyze Schwarz waveform relaxation algorithms: we split the domain of computation into two subdomains and with appropriate transmission conditions an iterative procedure leads to the global solution. We first analyze the Dirichlet-type transmission conditions—computing the convergence rate in Fourier–Laplace variables and proving it is less than 1 yields the convergence of the algorithm. The algorithm requires an overlap between subdomains, and the convergence is slow. We propose a better algorithm for which the overlap is unnecessary: the transmission conditions are now an approximation of the absorbing boundary conditions. We also prove convergence; if the domains overlap, the arguments are similar to the Dirichlet-type problem, if not, we use variational arguments. A numerical scheme is then proposed and numerical results are shown which highlight the...