(untitled) (Advances in Applied Mathematics and Mechanics, 2024)
The Ripa model consists of the shallow water equations and terms which take the horizontal temperature fluctuations into account. The pollutant transport model describes the transport and diffusion of pollutants in shallow water flows. Both models admit hydrostatic solutions in which the gradient of the flux term is exactly balanced by the source term on the right-hand side. In this paper, we write both models in a unified form and propose a well-balanced fifth-order finite difference alternative weighted essentially non-oscillatory (AWENO) scheme, which allows using arbitrary monotone, Lipschitz continuous and consistent numerical flux. For illustration purposes, the Lax-Friedrichs flux is employed. In order to design a well-balanced AWENO scheme, reformulation of the source term and linearization of the WENO interpolation operator are made. The well-balancedness of the proposed method will be analysed theoretically in this paper. Numerical examples verify the well-balanced property, high-order accuracy and effectiveness of our approach.