This paper proposes a robust structure-preserving dissipation-reducing nonstaggered central scheme for the multi-dimensional two-layer shallow water equations with wet-dry fronts. Since the two-layer shallow water equations are conditionally hyperbolic, computing Riemann solutions for numerical fluxes across cell interfaces is challenging. This issue is circumvented by the proposed scheme, which is genuinely Riemann-problem-solver-free, meaning that the numerical fluxes are independent of characteristic speed information. The source term and non-conservative product arising from the momentum exchange, are discretized using auxiliary variables. One key advantage of the scheme is its ability to preserve stationary solutions, even in the presence of wet-dry fronts in the computational domain. The auxiliary variables are interpreted as point values along paths in the phase space, which are used to discretize the non-conservative product, reinterpreted as a Borel measure. We prove that the scheme is well-balanced and positivity-preserving. Specifically, we establish a Lax-Wendroff type convergence theorem, in which the dissipation-reducing parameter plays a crucial and nontrivial role. Finally, we apply the scheme to several classical problems of the two-layer shallow water equations, with numerical results confirming its efficiency and robustness.
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Funding
National Natural Science Foundation of China(12371374, 12271523)
RIGHTS & PERMISSIONS
Shanghai University
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