A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes

TL;DR

This paper presents a discontinuous Galerkin finite element method for simulating free surface flows using a simplified class of Green-Naghdi equations on unstructured meshes, enabling accurate and stable modeling of nonlinear wave phenomena.

Contribution

It introduces a novel discontinuous Galerkin formulation for a new class of Green-Naghdi equations, allowing high-order approximation, mesh flexibility, and preservation of physical properties.

Findings

Validated through benchmarks with nonlinear wave transformations.

Handles arbitrary unstructured meshes effectively.

Ensures positivity and steady state preservation.

Abstract

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies.