Abstract In this work, we present a semi-implicit structure-preserving nonstaggered central scheme (SP-NCS) that is both well-balanced and positivity-preserving, designed to solve the two-dimensional shallow water Exner equations with time-dependent bottom topography. The SP-NCS can be viewed as a two-dimensional extension of the numerical method presented in D. Li and J. Dong, A robust hybrid unstaggered central and Godunov-type scheme for Saint-Venant–Exner equations with wet/dry fronts, Comput. & Fluids 235 2022, Article ID 105284, which is uncoupled for the shallow water Exner equations. The SP-NCS is also inspired by the nonstaggered central scheme presented in J. Dong and X. Qian, Structure-preserving nonstaggered central difference schemes at wet-dry fronts for the shallow water equations, Commun. Appl. Math. Comput. 8 2026, 1, 366–410, but their method is significantly adapted to preserve the still-water steady state when the computational domain contains wetting and drying transitions. Retaining the stationary solution in the two-dimensional case, especially when the domain include wetting and drying transitions, presents a significant challenge. This is because the backward step and the discretization of the source term in the corrector step become more complex when dealing with such fronts. A key innovation is the introduction of a structure-preserving parameter that helps retain the stationary solution even in the presence of wet-dry fronts. This task is nontrivial due to the robust requirements involved in ensuring the stability and accuracy of the solution. In particular, the SP-NCS demonstrates robustness in handling the relatively strong interactions between the two coupled models even with a large time step. This is particularly relevant when the rate of change in the bed topography is much slower than the speed of the water surface waves. We rigorously prove the positivity-preserving and well-balanced properties of the SP-NCS. Finally, we conduct several numerical experiments to validate the theoretical results and demonstrate the effectiveness of the proposed method in preserving the stationary solution and in scenarios involving wet-dry fronts, especially for the relatively strong interactions between the two coupled models.
Wei et al. (Sat,) studied this question.