Implicit positivity-preserving high order discontinuous Galerkin methods for conservation laws

Tong Qin Brown University Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1804.25023

SIAM Journal on Scientific Computing, 40, A81-A107, 2018
Positivity-preserving discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws have been extensively studied in the last several years. But nearly all the developed schemes are coupled with explicit time discretizations. Explicit discretizations suffer from the constraint for the Courant-Friedrichs-Levis (CFL) number. This makes explicit methods impractical for problems involving unstructured and extremely varying meshes or long-time simulations. Instead, implicit DG schemes are often popular in practice, especially in the computational fluid dynamics (CFD) community. In this paper we develop a high-order positivity-preserving DG method with the backward Euler time discretization for conservation laws. We focus on one spatial dimension, however the result easily generalizes to multidimensional tensor product meshes and polynomial spaces. This work is based on a generalization of the positivity-preserving limiters in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), pp.~3091--3120) and (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), pp.~8918--8934) to implicit time discretizations. Both the analysis and numerical experiments indicate that a lower bound for the CFL number is required to obtain the positivity-preserving property. The proposed scheme not only preserves the positivity of the numerical approximation without compromising the designed high-order accuracy, but also helps accelerate the convergence towards the steady-state solution and add robustness to the nonlinear solver. Numerical experiments are provided to support these conclusions.
Positivity-preserving; Discontinuous Galerkin method; Backward Euler
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@inproceedings{tong2018implicit,
  title={Implicit positivity-preserving high order discontinuous Galerkin methods for conservation laws},
  author={Tong Qin, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416104744633801058},
  booktitle={SIAM Journal on Scientific Computing},
  volume={40},
  pages={A81-A107},
  year={2018},
}
Tong Qin, and Chi-Wang Shu. Implicit positivity-preserving high order discontinuous Galerkin methods for conservation laws. 2018. Vol. 40. In SIAM Journal on Scientific Computing. pp.A81-A107. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416104744633801058.
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