Modeling and Computations of Shallow-Water Coastal Flows

No artificial numerical viscosity: from the 1/3 rule to energy conservative schemes for the shallow-water equations

Eitan Tadmor

University of Maryland

Entropy stability plays an important role in the dynamics of nonlinear systems of conservation laws and related convection-diffusion equations. What about the corresponding numerical framework? we present a general theory of entropy stability for difference approximations of such nonlinear equations.

We demonstrate this approach with a host of first- and second-order accurate schemes ranging from scalar examples to the shallow-water and Navier-Stokes equations. In particular, we discuss a family of energy-conservative and energy-stable schemes for the shallow-water equations, with a well-balanced description of moving equilibria states. We also present a new family of entropy stable schemes which retain the precise entropy decay of the Navier-Stokes equations. They contain no artificial numerical viscosity.