Entropy splitting and numerical dissipation

A rigorous stability estimate for arbitrary order of accuracy of spatial ce ntral difference schemes for initial boundary value problems of nonlinear s ymmetrizable systems of hyperbolic conservation laws was established recent ly by Olsson and Oliger (1994, "Energy and Maximum Norm Estimates for Nonli near Conservation Laws," RIACS Report, NASA Ames Research Center) and Olsso n (1995, Math. Comput. 64, 212) and was applied to the two-dimensional comp ressible Euler equations fur a perfect gas by Gerritsen and Olsson (1996, J . Comput. Phys. 129, 245) and Gerritsen (1996, "Designing an Efficient Solu tion Strategy for Fluid Flows, Ph. D. Thesis, Stanford). The basic building block in developing the stability estimate is a generalized energy approac h based on a special splitting of the flux derivative via a convex entropy function and certain homogeneous properties. Due to some of the unique prop erties of the compressible Euler equations for a perfect gas, the splitting resulted in the sum of a conservative portion and a non-conservative porti on of the flux derivative, hereafter referred to as the "entropy splitting. " There art: several potentially desirable attributes and side benefits of the entropy splitting for the compressible Euler equations that were not fu lly explored in Gerritsen and Olsson. This paper has several objectives, Th e first is to investigate the choice of the arbitrary parameter that determ ines the amount of splitting and its dependence on the type of physics of c urrent interest to computational fluid dynamics. The second is to investiga te in what manner the splitting affects the nonlinear stability of the cent ral schemes fur long time integrations of unsteady Rows such as in nonlinea r aeroacoustics and turbulence dynamics. If numerical dissipation indeed is needed to stabilize the central scheme, can the splitting help minimize th e numerical dissipation compared to its un-split cousin? Extensive numerica l study on the vortex preservation capability of the splitting in conjuncti on with central schemes furlong time integrations will be presented. The th ird is to study the effect of the non-conservative proportion of splitting in obtaining the correct shock location for high speed complex shock-turbul ence interactions. The fourth is to determine if this method can be extende d to other physical equations of state and other evolutionary equation sets . Tf numerical dissipation is needed, the Yee, Sandham, and Djomehri (1999, J. Comput. Phys. 150, 199) numerical dissipation is employed. The Yee er a l. schemes fit in the Olsson and Oliger framework.