High Order Well-Balanced Finite Difference AWENO Scheme for Ripa and Pollutant Transport Systems (Advances in Applied Mathematics and Mechanics, 2024)
The Ripa model consists of the shallow water equations and terms whichtake the horizontal temperature fluctuations into account. The pollutant transportmodel describes the transport and diffusion of pollutants in shallow water flows. Bothmodels admit hydrostatic solutions in which the gradient of the flux term is exactlybalanced by the source term on the right-hand side. In this paper, we write bothmodels in a unified form and propose a well-balanced fifth-order finite difference alternative weighted essentially non-oscillatory (AWENO) scheme, which allows usingarbitrary monotone, Lipschitz continuous and consistent numerical flux. For illustration purposes, the Lax-Friedrichs flux is employed. In order to design a well-balancedAWENO scheme, reformulation of the source term and linearization of the WENO interpolation operator are made. The well-balancedness of the proposed method willbe analysed theoretically in this paper. Numerical examples verify the well-balancedproperty, high-order accuracy and effectiveness of our approach.