A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density
This work addresses challenges in environmental and hydraulic engineering for modeling pollutant transport in water bodies, representing an incremental improvement with specific enhancements to existing methods.
The paper tackles the problem of simulating coupled shallow water flow and solute transport with variable density by developing a numerical scheme that ensures well-balanced and positivity-preserving properties. The result is a method that accurately preserves steady-state solutions and maintains positivity of water depth and concentration, validated through numerical examples.
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.