Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds

摘要

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for thesolution of the shallow water equations (SWE) with the bottom topography as asource term. Our new scheme will be based on the FVEG methods presented in(Luk\'a\v{c}ov\'a, Noelle and Kraft, J. Comp. Phys. 221, 2007), but adds thepossibility to handle dry boundaries. The most important aspect is to preservethe positivity of the water height. We present a general approach to ensurethis for arbitrary finite volume schemes. The main idea is to limit theoutgoing fluxes of a cell whenever they would create negative water height.Physically, this corresponds to the absence of fluxes in the presence ofvacuum. Well-balancing is then re-established by splitting gravitational andgravity driven parts of the flux. Moreover, a new entropy fix is introducedthat improves the reproduction of sonic rarefaction waves.