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A second-order asymptotic-preserving and positive-preserving discontinuous Galerkin scheme for the Kerr-Debye model

Juntao Huang Tsinghua University Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1804.25003

Mathematical Models and Methods in Applied Sciences, 27, 549–579, 2017
In this paper, we develop a second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) scheme for the Kerr-Debye model. By using the approach first introduced by Zhang and Shu in Ref.~\refcite{ZhSh:04} with an energy estimate and Taylor expansion, the asymptotic-preserving property of the semi-discrete DG methods is proved rigorously. In addition, we propose a class of unconditional positivity-preserving implicit-explicit (IMEX) Runge-Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr-Debye model. The new IMEX Runge-Kutta methods are based on the modification of the strong-stability-preserving (SSP) implicit Runge-Kutta method and have second-order accuracy. The numerical results validate our analysis.
Discontinuous Galerkin; positivity-preserving; asymptotic-preserving; stiff systems; Runge-Kutta methods; implicit-explicit; Kerr-Debye model
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@inproceedings{juntao2017a,
  title={A second-order asymptotic-preserving and positive-preserving discontinuous Galerkin scheme for the Kerr-Debye model},
  author={Juntao Huang, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416091946068133038},
  booktitle={Mathematical Models and Methods in Applied Sciences},
  volume={27},
  pages={549–579},
  year={2017},
}
Juntao Huang, and Chi-Wang Shu. A second-order asymptotic-preserving and positive-preserving discontinuous Galerkin scheme for the Kerr-Debye model. 2017. Vol. 27. In Mathematical Models and Methods in Applied Sciences. pp.549–579. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416091946068133038.
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