Analysis of an asymptotic preserving scheme for the Euler-Poisson system in the quasineutral limit

Pierre Degond Institut de Math´ematiques de Toulouse Jian-Guo Liu University of Maryland Marie-Hélène Vignal Institut de Math´ematiques de Toulouse

Numerical Analysis and Scientific Computing mathscidoc:1702.25037

SIAM Journal on Numerical Analysis, 46, (3), 1298–1322, 2008.6
In a previous work [P. Crispel, P. Degond, and M.-H. Vignal, J. Comput. Phys., 223 (2007), pp. 208–234], a new numerical discretization of the Euler–Poisson system was proposed. This scheme is “asymptotic preserving” in the quasineutral limit (i.e., when the Debye length $\varepsilon$ tends to zero), which means that it becomes consistent with the limit model when $\varepsilon \to 0$. In the present work, we show that the stability domain of the present scheme is independent of $\varepsilon$. This stability analysis is performed on the Fourier transformed (with respect to the space variable) linearized system. We show that the stability property is more robust when a space-decentered scheme is used (which brings in some numerical dissipation) rather than a space-centered scheme. The linearization is first performed about a zero mean velocity and then about a nonzero mean velocity. At the various stages of the analysis, our scheme is compared with more classical schemes and its improved stability property is outlined. The analysis of a fully discrete (in space and time) version of the scheme is also given. Finally, some considerations about a model nonlinear problem, the Burgers–Poisson problem, are also discussed.
stiffness, Debye length, electron plasma period, Burgers–Poisson, sheath problem, Klein–Gordon
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@inproceedings{pierre2008analysis,
  title={Analysis of an asymptotic preserving scheme for the Euler-Poisson system in the quasineutral limit},
  author={Pierre Degond, Jian-Guo Liu, and Marie-Hélène Vignal},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209005538986705351},
  booktitle={SIAM Journal on Numerical Analysis},
  volume={46},
  number={3},
  pages={1298–1322},
  year={2008},
}
Pierre Degond, Jian-Guo Liu, and Marie-Hélène Vignal. Analysis of an asymptotic preserving scheme for the Euler-Poisson system in the quasineutral limit. 2008. Vol. 46. In SIAM Journal on Numerical Analysis. pp.1298–1322. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209005538986705351.
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