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

A class of upwind flux splitting methods in the Euler equations of compressible flow is 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 matrices are either symmetric or symmetrizable and keep only non-negative and non-positive eigenvalues. That leads to the conclusion that the first order schemes are positive in the sense of Lax-Liu [18], which implies that it is $L^2$-stable in some suitable sense. Moreover, the second order scheme is a stable perturbation of the first order scheme, so that the positivity of the second order schemes is also established, under a CFL-like condition. In addition, these splitting methods preserve the positivity of density and energy.