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

TL;DR

This paper introduces a high-order discontinuous finite element method for shallow water equations that effectively handles irregular bathymetry data, ensuring properties like well-balancedness, mass conservation, and positivity, with adaptive capabilities.

Contribution

The paper develops a novel high-order DG method that works with realistic, irregular bathymetry data without regularity assumptions, and demonstrates its robustness and adaptability in coastal simulations.

Findings

Method is well-balanced, mass-conserving, and positivity-preserving.

Effective in irregular bathymetries with under-resolved features.

Shows potential for adaptive coastal flow simulations.

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.