We present a discontinuous finite element method for the shallow water equations which exploits high-resolution realistic bathymetry data without any regularity assumption, also in the case of high-order discretizations. We prove a number of mathematical properties specific to the proposed method that is well-balanced, mass-conserving and positivity-preserving under a mild CFL condition also in the presence of wet-dry fronts. The method includes a consistent conservative discretization for passive tracers. We use a high-order Discontinuous Galerkin (DG) method as implemented in the deal.II library. This environment provides efficient and native parallelization techniques and automatically handles non-conforming meshes to implement adaptive strategies which are tested in a coastal environment. Idealized test cases show the robustness in presence of irregular bathymetries also with under-resolved features at the grid scale. A benchmark with realistic bathymetry and a complex domain shows the potential of the proposed discretization for adaptive simulations of coastal flows.

翻译：我们提出一种适用于浅水方程的间断有限元方法，该方法无需任何正则性假设即可利用高分辨率真实地形数据，且适用于高阶离散格式。我们证明了该方法具有若干数学特性：在适度CFL条件下保持平衡性、质量守恒性及正性约束，且能处理干湿交替边界。该方法包含一种针对被动示踪物的守恒型离散格式。我们采用deal.II库中实现的高阶间断伽辽金（DG）方法进行求解。该环境提供高效的原生并行技术，可自动处理非协调网格以实现自适应策略，并在海岸环境中进行测试。理想化算例表明，该方法在处理不规则地形（包括网格尺度下欠分辨特征）时具有鲁棒性。采用真实复杂域地形的基准算例验证了所提离散格式在海岸流动自适应模拟中的潜力。