A family of functions for mass and energy flux splitting of the Euler equations

Flux vector splitting algorithms for the Euler equations are based on dividing the mass, momentum and energy fluxes into a “forward directed flux” F⁺$F^{+}$ and a “backward directed flux” F^-$F^{-}$ (with F^-=0$F^{-}=0$ for Mach numbers M>1$M>1$ and F⁺=0$F^{+}=0$ for M<-1$M<-1$). van Leer (1979, 1982) 4 and 5 proposed using polynomials of the Mach number for computing F⁺$F^{+}$ and F^-$F^{-}$ in the subsonic regime, and derived the lowest order polynomials that satisfy a set of chosen criteria. In this paper, we explore the possibility of increasing the order of these polynomials, with the purpose of reducing the diffusion across slow moving contact discontinuities of the flux vector splitting algorithm. We find that a moderate reduction of the diffusion, resulting in sharper shocks and contact discontinuities, can indeed be obtained with the higher order polynomials for the split fluxes.