Godunov-type Numerical Scheme for the Shallow Water Equations with Horizontal Temperature Gradient

Abstract

We present a Godunov-type scheme for the shallow water equations with horizontal temperature gradient and variable topography. First, the exact solutions of the Riemann problem in a computational form are given, where algorithms for computing these solutions are described. Second, a Godunov-type scheme is constructed relying on exact solutions of the local Riemann problems. Computing algorithms for the scheme are given. The scheme is shown to be well-balanced and preserve the positivity of the water height. Numerical tests show that the scheme is convergent with a good accuracy, even for the resonant phenomenon, where the exact solutions contain several distinct waves propagating with the same shock speed. Furthermore, the scheme also provides us with good results for the solution of the wave interaction problem.

New Results for Riemann Solution of the Cargo-LeRoux Model by the Application of Flux-Limiter Schemes

“…A slightly different approach of changing state variables for finding the Riemann solution was then introduced by Pang et al [16]. Further developments include the solution to hyperbolic conservation laws with non-conservative terms such as ( shallow water models, horizontal temperature gradient, and pressure gradient terms, etc see in [2,23,29]. A large number of contributions has been made in this context such as the incorporation of the entropy equation for pressure gradient models by Mahmood, [24], establishing important analysis related to the existence and uniqueness of the Riemann problem for one-dimensional isentropic and non-isentropic gas dynamic equation [6,14], the modified Riemann solutions for MHD problems including magnetic field effects on elementary waves presented in [32], investigating Riemann solutions for radiating non-ideal flows [25] etc.…”