A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density

TL;DR

This paper introduces a novel discontinuous Galerkin method for variable-density shallow water models that preserves positivity and well-balanced steady states, validated through numerical experiments.

Contribution

A new positivity-preserving, well-balanced DG scheme for coupled variable-density shallow water and solute transport models is developed and rigorously analyzed.

Findings

Accurately preserves steady states under still water conditions

Ensures positivity of water depth and concentration

Demonstrates high accuracy and effectiveness through numerical tests

Abstract

In this paper, we present a numerical scheme designed for coupled systems of variable-topography shallow water flow and solute transport. By integrating a variable-density system with an expression for relative density of mixtures, a novel formulation of the coupled system is derived. To ensure the well-balanced property, auxiliary variables are introduced to reformulate the variable-density shallow water equations into a new form, which is then discretized using the discontinuous Galerkin (DG) method with the Lax-Friedrichs (LF) flux as the numerical flux. By selecting appropriate values for the auxiliary variables, we demonstrate that the proposed method accurately preserves steady-state solutions under still water conditions, thereby verifying its well-balanced nature. Furthermore, sufficient conditions for preserving the positivity of both water depth and concentration are proposed and rigorously proven. A positivity-preserving limiter is introduced to enforce these conditions. Finally, a series of numerical examples are conducted to validate the computational accuracy and effectiveness of the proposed method.