Positivity property of second-order flux-splitting schemes for the compressible Euler equations

Abstract

A class of upwind flux splitting methods in theEuler equations of compressible flowis considered in this paper. Using the property that Euler flux $F(U)$is a homogeneous function of degree one in $U$,we reformulate the splitting fluxes with $F^{+}=A^{+} U$,$F^{-}=A^{-} U$, and the corresponding matricesare either symmetric or symmetrizable and keep onlynon-negative and non-positive eigenvalues.That leads to the conclusion that the first order schemesare positive in the sense of Lax-Liu [18],which implies that it is $L^2$-stablein some suitable sense. Moreover, the second order schemeis a stable perturbation of the first order scheme,so that the positivity of the second order schemesis also established, undera CFL-like condition. In addition, these splitting methods preservethe positivity of density and energy.