A flux-vector splitting scheme for the shallow water equations extended to high-order on unstructured meshes

TL;DR

This paper introduces a simple, robust flux-vector splitting scheme for shallow water equations that achieves high-order accuracy on unstructured meshes, demonstrated through tsunami simulations.

Contribution

It extends flux-vector splitting methods to high-order schemes on unstructured meshes for shallow water equations, enabling accurate and robust simulations.

Findings

Comparable accuracy to Godunov methods

Effective on unstructured meshes in 2D

Successfully applied to tsunami wave modeling

Abstract

We present an advection-pressure flux-vector splitting method for the one and two- dimensional shallow water equations following the approach first proposed by Toro and V\'azquez for the compressible Euler equations. The resulting first-order schemes turn out to be exceedingly simple, with accuracy and robustness comparable to that of the sophisticated Godunov upwind method used in conjunction with complete non- linear Riemann solvers. The technique splits the full system into two subsystems, namely an advection system and a pressure system. The sought numerical flux results from fluxes for each of the subsystems. The basic methodology, extended on 2D unstructured meshes, constitutes the building block for the construction of numerical schemes of very high order of accuracy following the ADER approach. The presented numerical schemes are systematically assessed on a carefully selected suite of test problems with reference solutions, in one and two space dimensions.The applicabil- ity of the schemes is illustrated through simulations of tsunami wave propagation in the Pacific Ocean.