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

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(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.Wellbalancing 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.

IMEX Large Time Step Finite Volume Methods for Low Froude Number Shallow Water Flows

We present new large time step methods for the shallow water flows in the lowFroude number limit.In order to take into accountmultiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection.We propose to approximate fast linear waves implicitly in time and in space bymeans of a genuinely multidimensional evolution operator.On the other hand,we approximate nonlinear advection part explicitly in time and in space bymeans of themethod of characteristics or some standard numerical flux function.Time integration is realized by the implicit-explicit(IMEX)method.We apply the IMEX Euler scheme,two step Runge Kutta Cranck Nicolson scheme,as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit.Numerical experiments demonstrate stability,accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

A Novel Full-Euler Low Mach Number IMEX Splitting

In this paper,we introduce an extension of a splitting method for singularly perturbed equations,the socalled RS-IMEX splitting[Kaiser et al.,Journal of Scientific Computing,70(3),1390–1407],to deal with the fully compressible Euler equations.The straightforward application of the splitting yields sub-equations that are,due to the occurrence of complex eigenvalues,not hyperbolic.A modification,slightly changing the convective flux,is introduced that overcomes this issue.It is shown that the splitting gives rise to a discretization that respects the low-Mach number limit of the Euler equations;numerical results using finite volume and discontinuous Galerkin schemes show the potential of the discretization.