High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory (WENO) scheme. The exact C-property is investigated, and comparison with the standard finite difference WENO scheme is made. Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems. The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems, indicating smaller errors compared with the Lax-Friedrichs solver. In addition, we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.