A well-defined moving steady states capturing Godunov-type scheme for Shallow-water model

Abstract :

The present work concerns the derivation of a well-balanced scheme to approximate the weak solutions of the shallow-water model. Here, the numerical scheme exactly captures all the smooth steady solutions with nonvanishing velocities. To address such an issue, a Godunov-type scheme is adopted. A particular attention is paid on the derivation of the intermediate states within the approximate Riemann solver. Indeed, because of the moving steady states, the intermediate states may be ill-defined. Here, we introduce a suitable correction in order to get a fully well-defined finite volume scheme. In addition, the numerical method is established to be positive preserving and to satisfy a discrete entropy inequality up to small perturbations. Several numerical experiments, including wet/dry transition, illustrate the relevance of the designed scheme.