A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations (Communications in Computational Physics, 2024)
We construct a positivity-preserving and well-balanced high order accuratefinite difference scheme for solving shallow water equations under the fourth ordercompact finite difference framework. The source term is rewritten to balance the fluxgradient 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 forwardEuler time discretization. Thus, a positivity-preserving limiter can be used to enforcethe 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 alsoused 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.
Keywords: Well-balanced; positivity-preserving; compact finite difference.