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In this paper, we construct a well-balanced, positivity preserving finite volume scheme for the shallow water equations based on a continuous, piecewise linear discretization of the bottom topography. The main new technique is a special reconstruction of the flow variables in wet-dry cells, which is presented in this paper for the one dimensional case. We realize the new reconstruction in the framework of the second-order semi-discrete central-upwind scheme from (A. Kurganov and G. Petrova, Commun. Math. Sci., 2007). The positivity of the computed water height is ensured following (A. Bollermann, S. Noelle and M. Lukáčová, Commun. Comput. Phys., 2010): The outgoing fluxes are limited in case of draining cells.Key words. Hyperbolic systems of conservation and balance laws, Saint-Venant system of shallow water equations, finite volume methods, well-balanced schemes, positivity preserving schemes, wet/dry fronts.

Abstract. 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.

We give a further examination of the stabilized Residual Distribution schemes for the solution of the shallow water equations proposed in (Ricchiuto and Bollermann, J.Comp.Phys., available online 25 October 2008). Based on a non-linear variant of a Lax-Friedrichs scheme, the scheme is wellbalanced, able to handle dry areas and, for smooth regions of the solution, obtains second order of accuracy. We will analyze the accuracy when dry areas are included in the domain of computation.1991 Mathematics Subject Classification. 35L65, 65M12, 76M25, 76B15.

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