Prevention of Expansion Shock by the Control of Numerical Dissipation of Upwind Schemes

This paper is concerned with the prevention of expansion shock given by some upwind schemes. Upwind schemes for the Euler equations of gas dynamics, be it flux vector splitting (FVS) or flux difference splitting (FDS), have many attractive shock-capturing properties. However, some of these schemes are known to give expansion shock, thus violating the so-called entropy condition. It is not very easy to say whether a particular upwind scheme violates the entropy condition, as lack of evidence cannot be taken as a rigorous proof for satisfaction of the condition. For some schemes that are known to violate the entropy condition, a number of entropy fixes based on the flux function have been suggested in the literature. Here we propose a more general method of fixing the problem by controlling numerical dissipation of the schemes.