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Energy dissipative local discontinuous Galerkin methods for Keller-Segel chemotaxis model

Li Guo Sun Yat-Sen University XIngjie Li UNC Charlotte Yang Yang Michigan Technological University

Numerical Analysis and Scientific Computing mathscidoc:1907.25006

Journal of Scientific Computing, 78, (3), 1387–1404, 2019.3
In this paper, we apply the local discontinuous Galerkin (LDG) method to solve the classical Keller-Segel (KS) chemotaxis model. The exact solution of the KS chemotaxis model may exhibit blow-up patterns with certain initial conditions, and is not easy to approximate numerically. Moreover, it has been proved that there exists a definition of free energy of the KS system which dissipates with respect to time. We will construct a consistent numerical energy and prove the energy dissipation with the LDG discretization. Several numerical experiments in one and two space dimensions will be given. Especially, for solutions with blow-up (converge to Dirac delta functions), the densities of KS model are computed to be strictly positive in the numerical experiments and the energies are also numerically observed to be strictly positive and decreasing as are seen in the figures. Therefore, the scheme is stable for the KS model with blow-up solutions.
Energy dissipation, local discontinuous Galerkin method, Keller-Segel chemotaxis model, blow-up solutions
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@inproceedings{li2019energy,
  title={Energy dissipative local discontinuous Galerkin methods for Keller-Segel chemotaxis model},
  author={Li Guo, XIngjie Li, and Yang Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190701095154736978386},
  booktitle={Journal of Scientific Computing},
  volume={78},
  number={3},
  pages={1387–1404},
  year={2019},
}
Li Guo, XIngjie Li, and Yang Yang. Energy dissipative local discontinuous Galerkin methods for Keller-Segel chemotaxis model. 2019. Vol. 78. In Journal of Scientific Computing. pp.1387–1404. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190701095154736978386.
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