Positivity of flux vector splitting schemes

Abstract

Over the last ten years, robustness of schemes has raised an increasing interest among the CFD community. One mathematical aspect of scheme robustness is the positivity preserving property. At high Mach numbers, solving the conservative Euler equations can lead to negative densities or internal energy. Some schemes such as the flux vector splitting (FVS) schemes are known to avoid this drawback. In this study, a general method is detailed to analyze the positivity of FVS schemes. As an application, three classical FVS schemes (Van Leer's, Hänel's variant, and Steger and Warming's) are proved to be positively conservative under a CFL-like condition. Finally, it is proved that for any FVS scheme, there is an intrinsic incompatibility between the desirable property of positivity and the exact resolution of contact discontinuities.