High-order adaptive discontinuous finite elements for the shallow water equations with sub-grid irregular bathymetry
This work provides a robust numerical method for coastal engineers and scientists to simulate shallow water flows over complex, irregular bathymetry with high-order accuracy and adaptive meshes.
The paper presents a discontinuous finite element method for shallow water equations that handles high-resolution irregular bathymetry without regularity assumptions, achieving well-balanced, mass-conserving, and positivity-preserving properties under a mild CFL condition. The method is demonstrated on idealized and realistic benchmarks, showing robustness and potential for adaptive coastal flow simulations.
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.