A well-balanced positivity-preserving numerical scheme for shallow water models with variable density

Abstract

We develop an unstructured numerical scheme for a coupled system modeling shallow water flows and solute transport over variable topography. A novel algorithm is introduced for the reconstructions of the variables of the system with variable density. These reconstructions are used in combination with the expression of the relative density of the mixture to guarantee the positivity required for some physical parameters of the coupled model. New discretization techniques are developed to guarantee the well-balanced property of the scheme and the consistency between the transport equation and the continuity equation at the discrete level. We prove the well-balanced property of the proposed method as well as the positivity preserving property of the scheme for both the water depth and concentrations. The performance of the proposed scheme is tested on a number of numerical examples, among which we consider nontrivial analytical solutions for the equilibrium with mixture constituents, parabolic wave with pollutant transport, and dam-break problem for modeling solute transport in rapidly varying flow. The numerical results confirm stability and well-balanced property of the scheme, consistency between the discretizations of the continuity and transport equations, positivity preserving property required for some physical parameters, and accuracy of the proposed method in modeling the dynamics of water flow and scalar transport.