A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

Abstract

We construct a positivity-preserving and well-balanced high order accurate finite difference scheme for solving shallow water equations under the fourth order compact finite difference framework. The source term is rewritten to balance the flux gradient in steady state solutions. Under a suitable CFL condition, the proposed compact difference scheme satisfies weak monotonicity, i.e., the average water height defined by the weighted average of a three-points stencil stays non-negative in forward Euler time discretization. Thus, a positivity-preserving limiter can be used to enforce the positivity of water height point values in a high order strong stability preserving Runge-Kutta method. A TVB limiter for compact finite difference scheme is also used to reduce numerical oscillations, without affecting well-balancedness and positivity. Numerical experiments verify that the proposed scheme is high-order accurate, positivity-preserving, well-balanced and free of numerical oscillations.