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 efficiency of the new transmission conditions. The influence of the time window lengths is discussed. Key words. Schwarz waveform relaxation, shallow water equations, domain decompositio