In this work, we develop a new hydrostatic reconstruction procedure to construct well-balanced schemes for one and multilayer shallow water flows, including wetting and drying. Initially, we derive the method for a path-conservative finite volume scheme and combine it with entropy conservative fluxes and suitable numerical dissipation to preserve an entropy inequality in the semi-discrete case. We then combine the novel hydrostatic reconstruction with a collocated nodal split-form discontinuous Galerkin spectral element method, extending the method to high-order and curvilinear meshes. The high-order method incorporates an additional positivity-limiter and is blended with a compatible subcell finite volume method to maintain well-balancedness at wet/dry fronts. We prove entropy stability, well-balancedness, and positivity-preservation for both methods. Numerical results for the high-order method validate the theoretical findings and demonstrate the robustness of the scheme.

翻译：本文提出了一种新的静水重构方法，用于构建适用于单层及多层浅水流（包括干湿界面）的保平衡格式。首先，我们在路径守恒有限体积法框架下推导该方法，并将其与熵守恒通量及适当的数值耗散相结合，以在半离散情形下保持熵不等式。随后，我们将该新型静水重构方法与配置点节点分裂形式间断伽辽金谱元法相结合，将方法扩展至高阶格式及曲线网格。高阶方法引入了额外的正性限制器，并与兼容的子单元有限体积法进行混合，以在干湿锋面处保持平衡性。我们证明了两种方法均具有熵稳定性、平衡保持性和正性保持特性。针对高阶方法的数值结果验证了理论结论，并证明了该格式的鲁棒性。