A finite-difference representation of the Coriolis force in numerical models of Godunov type for rotating shallow water flows

Abstract

A finite-difference representation describing the Coriolis force in numerical methods of the Godunov type for flows of rotating shallow water is proposed in the paper. Finite-difference schemes are proposed for modelling flows both on a flat bed and on a bed of an arbitrary profile. The influence of the Coriolis force is simulated by introducing a fictitious unsteady boundary. The numerical algorithms developed here are based on representing an arbitrary bed surface and the Coriolis force by a complex unsteady step boundary. Due to inhomogeneity of the bed surface and the influence of the Coriolis force, a quasi-two-layer model of a liquid flow over a step boundary is used for numerical approximation of the source terms. The model takes into account the hydrodynamic peculiarities. This allows us to give a visual interpretation of nonlinear processes caused by inhomogeneity of the bottom and by the fictitious unsteady boundary describing the Coriolis force. The use of the quasi-two-layer model provides a better approximation to the original Euler equations in the presence of source terms. Comparative analysis of the well-known finite-difference schemes describing the rotation and inhomogeneity of the bottom surface profile is performed. Numerical calculations confirming the efficiency of the proposed method are also performed.