Solving the depth-integrated solute transport equation with a TVD-MacCormack scheme

Abstract

Details are given of a numerical model for predicting solute transport in shallow waters. The alternating operator-splitting technique is used to separate the spatial discretisation of the advection and diffusion terms in each step of the time marching procedure. In the diffusion stage, a second-order accurate central scheme is used. In the advection stage, a five-point Total Variation Diminishing (TVD) modification is made to the standard MacCormack scheme. The model performance is examined by comparing numerical predictions with analytical solutions and some other numerical model results for idealised test cases. The model is also used to simulate the motion of a contaminant cloud in the River Thames estuary, where extensive wetting and drying occurs with the tidal oscillation. The present model displays sufficiently high efficiency and accuracy in solving the solute transport problems in a natural aquatic environment. It is free of fictitious oscillations close to sharp concentration gradients, while retaining second-order accuracy in smooth regions. Attention has also been paid to ensure the accurate treatment of the cross-diffusion terms and mass balance in complex flow situations.