An asymptotic preserving scheme for the two-dimensional shallow water equations with Coriolis forces

Abstract

We consider the two-dimensional Saint-Venant system of shallow water equations with Coriolis forces. We focus on the case of a low Froude number, in which the system is stiff and conventional explicit numerical methods are extremely inefficient and often impractical. Our goal is to design an asymptotic preserving (AP) scheme, which is uniformly asymptotically consistent and stable for a broad range of (low) Froude numbers. The goal is achieved using the flux splitting proposed in [Haack et al., Commun. Comput. Phys., 12 (2012), pp. 955–980] in the context of isentropic Euler and Navier-Stokes equations. We split the flux into the stiff and nonstiff parts and then use an implicit-explicit approach: apply an explicit hyperbolic solver (we use the second-order central-upwind scheme) to the nonstiff part of the system while treating the stiff part of it implicitly. Moreover, the stiff part of the flux is linear and therefore we reduce the implicit stage of the proposed method to solving a Poisson-type elliptic equation, which is discretized using a standard second-order central difference scheme.We conduct a series of numerical experiments, which demonstrate that the developed AP scheme achieves the theoretical second-order rate of convergence and the time-step stability restriction is independent of the Froude number. This makes the proposed AP scheme an efficient and robust alternative to fully explicit numerical methods.