Numerical simulation of rotation dominated linear shallow water flows using finite volume methods and fourth order Adams scheme

Abstract

In this paper, we study the performance of some finite volume schemes for linear shallow water equations on a rotating frame. It is shown here that some well-known upwind schemes, which perform well for gravity waves, lead to a high level of damping or numerical oscillation for Rossby waves. We present a modified five-point upwind finite volume scheme which leads to a low level of numerical diffusion and oscillation for Rossby waves. The method uses a high-order upwind method for the calculation of the numerical flux and a fourth-order Adams method for time integration of the equations and is considerably more efficient than the fourth-order Runge–Kutta method that is usually used for temporal integration of shallow water equations in the presence of the Coriolis term. In the method proposed here, the Coriolis term is treated analytically in two stages: before and after calculation of computational fluxes. It is shown that the energy dissipation of the proposed method is considerably less than other upwind methods that are widely used, such as the third-order upwind method.