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arXiv · 2505.18743

High-order adaptive discontinuous finite elements for the shallow water equations with sub-grid irregular bathymetry

Luca Arpaia·Giuseppe Orlando·Christian Ferrarin·Luca Bonaventura
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Abstract

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.

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Luca Arpaia, Giuseppe Orlando, Christian Ferrarin, Luca Bonaventura. 2025-05-24. High-order adaptive discontinuous finite elements for the shallow water equations with sub-grid irregular bathymetry. https://arxiv.org/abs/2505.18743

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