A well-balanced high-order scheme on van Leer-type for the shallow water equations with temperature gradient and variable bottom topography

A well-balanced high-order scheme for shallow water equations with variable topography and temperature gradient is constructed. This scheme is of van Leer-type and is based on exact Riemann solvers. The scheme is shown to be able to capture almost exactly the stationary smooth solutions as well as stationary elementary discontinuities. Numerical tests show that the scheme gives a much better accuracy than the Godunov-type scheme and can work well even in the resonant regime. Wave interaction problems are also tested where the scheme possesses a good accuracy. It turns out that the superbee limiter can provide us with more accurate approximations than van Leer’s limiter.