Wang, C; Liu, J-G, *Positivity property of second-order flux-splitting schemes for the compressible Euler equations*,
Discrete and Continuous Dynamical Systems - Series B, vol. 3 no. 2
(2003),
pp. 201-228 .
**Abstract:**

*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 L2- 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.*