On the K-consistence property of difference schemes of gas dynamics. I - An analysis of first-order schemes. II - An analysis of second-order schemes

It is shown that it is possible to introduce the K-consistence notion directly for finite difference shock-capturing schemes approximating the Euler equation system for an inviscid, compressible gas. Methods for increasing the accuracy of the numerical solutions in the vicinity of a contact discontinuity are proposed and theoretically justified. These methods are applicable in the cases when the equation of state employed does not satisfy the so-called K-consistence condition and when the finite difference second-order schemes are used for gas dynamic computations in Eulerian variables. The construction of corresponding algorithms is accomplished for the Lax-Wendroff scheme and the MacCormack scheme. Theoretical conclusions are illustrated by computational examples.