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

Cheng Wang Indiana University Jian-Guo Liu University of Maryland

Numerical Analysis and Scientific Computing mathscidoc:1702.25051

Discrete and Continuous Dynamical Systems - Series B , 3, (2), 201-228, 2003.5
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.
Conservation laws, flux splitting, limiter function, positivity.
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@inproceedings{cheng2003positivity,
  title={Positivity property of second-order flux-splitting schemes for the compressible Euler equations},
  author={Cheng Wang, and Jian-Guo Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209105002183714381},
  booktitle={Discrete and Continuous Dynamical Systems - Series B },
  volume={3},
  number={2},
  pages={201-228},
  year={2003},
}
Cheng Wang, and Jian-Guo Liu. Positivity property of second-order flux-splitting schemes for the compressible Euler equations. 2003. Vol. 3. In Discrete and Continuous Dynamical Systems - Series B . pp.201-228. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209105002183714381.
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