Commun. Comput. Phys., 7 (2010), pp. 1049-1075.

A Well-Balanced and Non-Negative Numerical Scheme for Solving the Integrated Shallow Water and Solute Transport Equations

Qiuhua Liang 1*

1 School of Civil Engineering and Geosciences, Newcastle University, Newcastle upon Tyne, NE1 7RU, England, UK.

Received 14 September 2009; Accepted (in revised version) 23 November 2009
Available online 6 January 2010


Based on the recent development in shallow flow modelling, this paper presents a finite volume Godunov-type model for solving a 4x4 hyperbolic matrix system of conservation laws that comprise the shallow water and depth-averaged solute transport equations. The adopted governing equations are derived to preserve exactly the solution of lake at rest so that no special numerical technique is necessary in order to construct a well-balanced scheme. The HLLC approximate Riemann solver is used to evaluate the interface fluxes. Second-order accuracy is achieved using the MUSCL slope limited linear reconstruction together with a Runge-Kutta time integration method. The model is validated against several benchmark tests and the results are in excellent agreement with analytical solutions or other published numerical predictions.

AMS subject classifications: 76R99, 65E99

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Key words: Solute transport, shallow water equations, advection-diffusion equation, well-balanced scheme, wetting and drying, source terms, Riemann solver.

*Corresponding author.
Email: (Q. Liang)

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