Local discontinuous Galerkin method for the Keller-Segel chemotaxis model

Xingjie Helen Li Chi-Wang Shu Brown University Yang Yang

Numerical Analysis and Scientific Computing mathscidoc:1804.25004

Journal of Scientific Computing, 73, 943-967, 2017
In this paper, we apply the local discontinuous Galerkin (LDG) method to 2D Keller--Segel (KS) chemotaxis model. We improve the results upon (Y. Epshteyn and A. Kurganov, SIAM Journal on Numerical Analysis, 47 (2008), 368-408) and give optimal rate of convergence under special finite element spaces before the blow-up occurs (the exact solutions are smooth). Moreover, to construct physically relevant numerical approximations, we consider $P^1$ LDG scheme and develop a positivity-preserving limiter to the scheme, extending the idea in (Y. Zhang, X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 8918-8934). With this limiter, we can prove the $L^1$-stability of the numerical scheme. Numerical experiments are performed to demonstrate the good performance of the positivity-preserving LDG scheme. Moreover, it is known that the chemotaxis model will yield blow-up solutions under certain initial conditions. We numerically demonstrate how to find the approximate blow-up time by using the $L^2$-norm of the $L^1$-stable numerical solution.
Local discontinuous Galerkin method, Keller-Segel chemotaxis model, positivity preserving, error estimate, Neumann boundary condition, blow-up, $L^1$ stability
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@inproceedings{xingjie2017local,
  title={Local discontinuous Galerkin method for the Keller-Segel chemotaxis model},
  author={Xingjie Helen Li, Chi-Wang Shu, and Yang Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416092310788301039},
  booktitle={Journal of Scientific Computing},
  volume={73},
  pages={943-967},
  year={2017},
}
Xingjie Helen Li, Chi-Wang Shu, and Yang Yang. Local discontinuous Galerkin method for the Keller-Segel chemotaxis model. 2017. Vol. 73. In Journal of Scientific Computing. pp.943-967. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416092310788301039.
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