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Journal of Scientific Computing

, Volume 78, Issue 3, pp 1387–1404 | Cite as

Energy Dissipative Local Discontinuous Galerkin Methods for Keller–Segel Chemotaxis Model

  • Li Guo
  • Xingjie Helen LiEmail author
  • Yang Yang
Article
  • 103 Downloads

Abstract

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.

Keywords

Energy dissipation Local discontinuous Galerkin method Keller–Segel chemotaxis model Blow-up solutions 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Data and Computer ScienceSun Yat-Sen UniversityGuangzhouPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsUNC CharlotteCharlotteUSA
  3. 3.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA

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