Entropy Stable Approximations of Navier-Stokes equations with no Artificial Numerical Viscosity
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. Our approach is based on comparing numerical viscosities relative to certain entropy conservative schemes.It yields precise characterizations of entropy stability which is enforced in rare factions while keeping sharp resolution of shocks. We demonstrate this approach with a host of first- and second-order accurate schemes ranging
from scalar examples to Euler and Navier-Stokes equations. In particular, we construct a new family of entropy stable schemes which retain the precise entropy decay of the Navier-Stokes equations. They contain no artificial numerical viscosity. Numerical experiments provide a remarkable evidence for the different roles of viscosity and heat conduction in forming sharp monotone profiles in the immediate neighborhoods of shocks and contacts.