Finite-Volume Multi-Stage Scheme for Advection-Diffusion Modeling in Shallow Water Flows

This paper adopts the finite-volume multi-stage (FMUSTA) scheme to the two-dimensional coupled system combining the shallow water equations and the advection-diffusion equation. For the convection part, the numerical flux is estimated by adopting the FMUSTA scheme, where high order accuracy is achieved by the data reconstruction using the monotonic upstream schemes for conservation laws method. For the diffusion part, the evaluations of first-order derivatives are solved via the method of Jacobian transformation. The hydrostatic reconstruction method is employed for treatment of source terms. The overall accuracy of resulting scheme is second-order both in time and space. In addition, the scheme is non-oscillatory and conserves the pollutant mass during the transport process. For scheme validation, six advection and diffusion transport tests are simulated. The influences of the grid spacing and limiters on the numerical performance are also discussed. Furthermore, the scheme is employed in the simulation of suspended sediment transport in natural-irregular river topography. From the satisfactory agreements between the simulated results and the field measured data, it is demonstrated that the proposed FMUSTA scheme is practically suitable for hydraulic engineering applications.