Generalized conservative approximations of split convective derivative operators

We propose a strategy to design locally conservative finite-difference approximations of convective derivatives for shock-free compressible flows with arbitrary order of accuracy, that generalizes the approach of Ducros et al. (2000) [1], and that can be applied as a building block of low-dissipative, hybrid shock-capturing methods. The approximations stem from application of standard central difference formulas to split forms of the convective terms in the compressible Euler equations, which guarantee strong numerical stability and (near) energy preservation in the inviscid limit. A convenient implementation of the high-order fluxes is suggested, which guarantees improved computational efficiency over existing methods. Numerical tests performed for isotropic turbulence at zero viscosity show stability of schemes with order of accuracy up to ten, and effectiveness of convective splitting of Kennedy and Gruber (2008) [2] in providing extra stability in the presence of strong density variations. Numerical simulations of compressible turbulent boundary layer flow indicate suitability of the method for non-uniform grids, and overall support superior computational efficiency of high-order schemes.