High Order Well-Balanced and Positivity-Preserving AWENO Scheme for Rotating Shallow Water Equations with Coriolis Force

Abstract

The rotating shallow water equations (RSWEs), incorporating the Coriolis force, induce the geostrophic equilibrium with a nonzero velocity field, which poses more challenges in designing well-balanced numerical schemes than solving the classical SWEs. To tackle this issue, we leverage the Coriolis force’s primitive function, specifically, the approach of apparent topography. Additionally, we introduce a positivity-preserving and well-balanced finite difference AWENO scheme for solving the RSWEs. Our approach applies characteristic-wise WENO interpolation directly to conservative variables, eliminating the need for the commonly used WENO linearization technique. To realize the purpose of a well-balanced property, the Lax-Friedrichs numerical flux is modified by appropriately adjusting the viscosity coefficients to avoid the dissipation from non-zero momentum in equilibrium-state simulations. The corresponding theoretical proof is provided. Furthermore, a positivity-preserving limiter is introduced for numerical simulations involving dry topography to avoid the negative water depth. Numerical results demonstrate that the proposed scheme preserves the equilibria exactly with the optimal convergence order and outperforms non-well-balanced methods in resolving sharp gradients, and accurately simulates the evolution of geophysical flows with the impact of the Coriolis force.