We present a systematic study of novel entropy stable approximations for a variety of nonlinear conservation laws, from the scalar Burgers equation to one dimensional Navier-Stokes and two dimensional shallow water equations.

To this end, we construct a new family of second-order entropy stable difference schemes which retain the precise entropy decay of the original partial differential equations. Here we employ the entropy conservative differences of Tadmor's 2004 paper to discretize the convective fluxes, and center differences to discretize the dissipative fluxes. This resulting family of difference schemes are free of artificial numerical viscosity in the sense that their entropy dissipation is then dictated solely by physical dissipation terms. The numerical results of 1D compressible Navier-Stokes equations equations provide us 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. Further implementation in 2D shallow water equations is realized dimension by dimension.