We present a robust central scheme for shallow water equations with wetting and drying that is formally second-order accurate in smooth regions. The proposed method preserves physically admissible states, particularly the non-negativity of the water height, while accurately resolving complex flows in the presence of dry areas without relying on Riemann solvers. The scheme combines a well-balanced discretization, the surface gradient formulation, with a nonlinear invariant-region-preserving limiter to ensure consistency between flux and source term discretizations, suppress spurious oscillations near wet–dry interfaces, and maintain the well-balanced property for steady states, including lake-at-rest configurations over irregular topographies. In dry and near wet–dry regions, the limiter preserves positivity, while in fully wet regions, it remains inactive so that the scheme locally recovers the original well-balanced formulation. A cutoff criterion is introduced to activate the limiter only when necessary, thereby preserving accuracy in fully wet regions. The main contribution of this work lies in the compatible coupling of the surface gradient reconstruction and invariant-region-preserving limiting within an unstaggered central framework, together with a dedicated treatment of lake-at-rest configurations involving wet–dry interfaces. Numerical experiments demonstrate second-order numerical convergence for the water height in smooth benchmark problems, as well as robustness and favorable performance compared with existing methods.
Citation: Elissa Malaeb, Toni Sayah, Rony Touma. A well-balanced and positivity-preserving central scheme for shallow water equations[J]. AIMS Mathematics, 2026, 11(8): 27529-27562. doi: 10.3934/math.20261102
We present a robust central scheme for shallow water equations with wetting and drying that is formally second-order accurate in smooth regions. The proposed method preserves physically admissible states, particularly the non-negativity of the water height, while accurately resolving complex flows in the presence of dry areas without relying on Riemann solvers. The scheme combines a well-balanced discretization, the surface gradient formulation, with a nonlinear invariant-region-preserving limiter to ensure consistency between flux and source term discretizations, suppress spurious oscillations near wet–dry interfaces, and maintain the well-balanced property for steady states, including lake-at-rest configurations over irregular topographies. In dry and near wet–dry regions, the limiter preserves positivity, while in fully wet regions, it remains inactive so that the scheme locally recovers the original well-balanced formulation. A cutoff criterion is introduced to activate the limiter only when necessary, thereby preserving accuracy in fully wet regions. The main contribution of this work lies in the compatible coupling of the surface gradient reconstruction and invariant-region-preserving limiting within an unstaggered central framework, together with a dedicated treatment of lake-at-rest configurations involving wet–dry interfaces. Numerical experiments demonstrate second-order numerical convergence for the water height in smooth benchmark problems, as well as robustness and favorable performance compared with existing methods.
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Elissa Malaeb, Toni Sayah, Rony Touma. A well-balanced and positivity-preserving central scheme for shallow water equations[J]. AIMS Mathematics, 2026, 11(8): 27529-27562. doi: 10.3934/math.20261102
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