A spectral collocation method for the one dimensional shallow water equations on semi-infinite domain

Abstract

We introduce a spectral collocation method for the discretization of the shallow water equations on a one dimensional semi-infinite domain, employing suitably rescaled Laguerre basis functions to obtain an accurate description of the solutions on finite regions of arbitrary size. The time discretization is based on a semi-implicit, semi-Lagrangian approach that allows to handle the highly inhomogeneous node distribution without loss of efficiency. The method is first validated on standard test cases and then applied to the implementation of absorbing open boundary conditions by coupling the semi-infinite domain to a finite size domain on which the same equations are discretized by standard finite volume methods. Numerical experiments show that the proposed approach does not produce significant spurious reflections at the interface between the finite and infinite domain, thus providing a reliable tool for absorbing boundary conditions.