Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds (Communications in Computational Physics, 2011)
We present a new Finite Volume Evolution Galerkin (FVEG) scheme for thesolution of the shallow water equations (SWE) with the bottom topography as a sourceterm. Our new scheme will be based on the FVEG methods presented in (Noelle andKraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries.The most important aspect is to preserve the positivity of the water height. We presenta general approach to ensure this for arbitrary finite volume schemes. The main idea isto limit the outgoing fluxes of a cell whenever they would create negative water height.Physically, this corresponds to the absence of fluxes in the presence of vacuum. Well-balancingis then re-established by splitting gravitational and gravity driven parts ofthe flux. Moreover, a new entropy fix is introduced that improves the reproduction ofsonic rarefaction waves.