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

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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.

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Authors and Affiliations

  1. School of Data and Computer Science, Sun Yat-Sen University, Guangzhou, 510006, Guang Dong, People’s Republic of China

    Li Guo

  2. Department of Mathematics and Statistics, UNC Charlotte, Charlotte, NC, 28223, USA

    Xingjie Helen Li

  3. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Yang Yang

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  1. Li Guo
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Correspondence to Xingjie Helen Li.

Additional information

The research of Li Guo was supported by the National Natural Science Foundation of China under the Grant 11601536. The research of Xingjie Helen Li was supported by the Simons Foundation Collaboration Grant: Award ID: 426935. The research of Yang Yang was supported by the NSF Grant DMS-1818467.

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Guo, L., Li, X.H. & Yang, Y. Energy Dissipative Local Discontinuous Galerkin Methods for Keller–Segel Chemotaxis Model. J Sci Comput 78, 1387–1404 (2019). https://doi.org/10.1007/s10915-018-0813-8

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