Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds
For computational fluid dynamics researchers, this work incrementally extends an existing FVEG method to handle dry boundaries, a known challenge in shallow water simulations.
The paper presents a Finite Volume Evolution Galerkin scheme for shallow water equations that handles dry beds and preserves positivity of water height. The method limits outgoing fluxes to avoid negative water height and includes a new entropy fix for sonic rarefaction waves.
We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Lukáčová, Noelle and Kraft, 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 present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to 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-balancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.