Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity

Hui Yu Tsinghua University Hailiang Liu Iowa State University

Numerical Analysis and Scientific Computing mathscidoc:1906.25001

Journal of Computational Physics, 391, 14-36, 2019.4
For a class of convection-diffusion equations with variable diffusivity, we construct third order accurate discontinuous Galerkin (DG) schemes on both one and two dimensional rectangular meshes. The DG method with an explicit time stepping can well be applied to nonlinear convection–diffusion equations. It is shown that under suitable time step restrictions, the scaling limiter proposed in Liu and Yu (2014) [23]when coupled with the present DG schemes preserves the solution bounds indicated by the initial data, i.e., the maximum principle, while maintaining uniform third order accuracy. These schemes can be extended to rectangular meshes in three dimension. The crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions. This is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages. Numerical results are presented to validate the numerical methods and demonstrate their effectiveness.
Diffusion. Discontinuous Galerkin method. Maximum principle. High order accuracy.
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@inproceedings{hui2019third,
  title={Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity},
  author={Hui Yu, and Hailiang Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190626133203171283367},
  booktitle={Journal of Computational Physics},
  volume={391},
  pages={14-36},
  year={2019},
}
Hui Yu, and Hailiang Liu. Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity. 2019. Vol. 391. In Journal of Computational Physics. pp.14-36. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190626133203171283367.
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