On Skew-Symmetric Splitting and Entropy Conservation Schemes for the Euler Equations

The Tadmor type of entropy conservation formulation for the Euler equations and various skew-symmetric splittings of the inviscid flux derivatives are discussed. Numerical stability of high order central and Padé type (centered compact) spatial discretization is enhanced through the application of these formulations. Numerical test on a 2-D vortex convection problem indicates that the stability and accuracy of these formulations using the same high order central spatial discretization are similar for vortex travel up to a few periods. For two to three times longer time integrations, their corresponding stability and accuracy behaviors are very different. The goal of this work is to improve treatment of nonlinear instabilities and to minimize the use of numerical dissipation in numerical simulations of shock-free compressible turbulence and turbulence with shocks.