Foundations of Computational Mathematics : Hong Kong 2008

We design energy preserving and energy stable schemes for the shallow water equations. A new explicit energy preserving flux is proposed and is compared with existing energy preserving fluxes. This new flux results in a considerable reduction in the computational cost. We add suitably discretized viscous terms to the energy preserving scheme in order to obtain an energy stable scheme that replicates the energy decay of the continuous problem. The addition of physical viscosity dramatically reduces the oscillations on resolved meshes. For computing on underresolved meshes, we propose a new Roe-type numerical flux that adds diffusion in terms of energy variables to an energy preserving scheme. The resulting scheme is energy stable and as accurate as the Roe scheme, at a similar computational cost. The robustness of the energy preserving