A Well-Balanced Wet–Dry Front Reconstruction for Two-Layer Shallow Water Flows

In this paper, a well-balanced and positivity-preserving scheme for the nonconservative two-layer shallow water equations is developed in the framework of the finite volume method. To address the challenges posed by wet–dry fronts, the focus of our study is on reconstructing them for each layer to ensure a well-balanced property. To this end, a new numerical discretization and a special wet–dry front reconstruction are proposed. In addition, the draining time method is employed to ensure the positivity of the water depth. We prove that the proposed scheme is both well-balanced in steady-state solutions and positivity-preserving. Finally, numerical experiments demonstrate the robustness of the scheme.