The FVC scheme on unstructured meshes for the two-dimensional Shallow Water Equations

TL;DR

This paper introduces a novel finite volume Eulerian-Lagrangian scheme called FVC for solving two-dimensional shallow water equations on unstructured meshes, combining accuracy, conservation, and non-oscillatory properties without Riemann problem solutions.

Contribution

The paper presents the FVC scheme, a new predictor-corrector finite volume method using characteristics for shallow water equations on unstructured meshes, avoiding Riemann problems and enhancing accuracy.

Findings

The FVC scheme is accurate and conservative.

It is non-oscillatory and avoids Riemann problem solutions.

Test results outperform traditional Roe methods.

Abstract

The fluid flow transport and hydrodynamic problems often take the form of hyperbolic systems of conservation laws. In this work we will present a new scheme of finite volume methods for solving these evolution equations. It is a family of finite volume Eulerian-Lagrangian methods for the solution of non-linear problems in two space dimensions on unstructured triangular meshes. The proposed approach belongs to the class of predictor-corrector procedures where the numerical fluxes are reconstructed using the method of characteristics, while an Eulerian method is used to discretize the conservation equation in a finite volume framework. The scheme is accurate, conservative and it combines advantages of the modified method of characteristics to accurately solve the non-linear conservation laws with a finite volume method to discretize the equations. The proposed Finite Volume Characteristics (FVC) scheme is also non-oscillatory and avoids the need to solve a Riemann problem. Several test examples will be presented for the shallow water equations. The results will be compared to those obtained with the Roe.