Structure-Preserving Nonstaggered Central Difference Schemes at Wet-Dry Fronts for the Shallow Water Equations

We aim to propose a Riemann-problem-solver-free nonstaggered central difference scheme (NCDS), which is well-balanced and positivity-preserving for two-dimensional shallow water flows over nonflat bottom topography with wet-dry fronts. The well-balanced property of the NCDS for the two-dimensional shallow water equations at wet-dry fronts is still unclear. The NCDS is “generously multidimensional” and consists of three steps: a forward step, a corrector step, and a backward step. Each step needs to guarantee the water depth to be nonnegative and retain the lake at rest. The key ingredient of the NCDS is the backward step. It is very nontrivial and technical to obtain both the well-balanced and positivity-preserving properties when the computational domain contains wet-dry fronts for the NCDS. The traditional NCDS fails to retain the lake at rest when the computational domain contains wet-dry fronts. We introduce a novel backward step, which is well-balanced and positivity-preserving, to recover numerical solutions on nonstaggered cells. Finally, we show several numerical results of two-dimensional shallow water equations with wet-dry fronts to verify these properties.