# American Institute of Mathematical Sciences

May  2003, 3(2): 201-228. doi: 10.3934/dcdsb.2003.3.201

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

 1 Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, United States 2 IPST and Department of Mathematics, University of Maryland, College Park, MD 20742, United States

Received  April 2002 Revised  January 2003 Published  February 2003

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.
Citation: Cheng Wang, Jian-Guo Liu. Positivity property of second-order flux-splitting schemes for the compressible Euler equations. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 201-228. doi: 10.3934/dcdsb.2003.3.201
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