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Positivity-preserving third order DG schemes for Poisson-Nernst-Planck equations

Stephan Gerster Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Germany Michael Herty Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Germany Hui Yu Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

Analysis of PDEs mathscidoc:2205.03010

Communications in Mathematical Sciences, 19, (3), 787-806, 2021.5
We consider the stabilization of linear kinetic equations with a random relaxation term. The well-known framework of hypocoercivity by J. Dolbeault, C. Mouhot and C. Schmeiser (2015) ensures the stability in the deterministic case. This framework, however, cannot be applied directly for arbitrarily small random relaxation parameters. Therefore, we introduce a Galerkin formulation, which reformulates the stochastic system as a sequence of deterministic ones. We prove for the γ-distribution that the hypocoercivity framework ensures the stability of this series and hence the stochastic stability of the underlying random kinetic equation. The presented approach also yields a convergent numerical approximation.
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@inproceedings{stephan2021positivity-preserving,
  title={Positivity-preserving third order DG schemes for Poisson-Nernst-Planck equations},
  author={Stephan Gerster, Michael Herty, and Hui Yu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220520141647608189303},
  booktitle={Communications in Mathematical Sciences},
  volume={19},
  number={3},
  pages={787-806},
  year={2021},
}
Stephan Gerster, Michael Herty, and Hui Yu. Positivity-preserving third order DG schemes for Poisson-Nernst-Planck equations. 2021. Vol. 19. In Communications in Mathematical Sciences. pp.787-806. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220520141647608189303.
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