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Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model

Abstract

In this paper, we apply the local discontinuous Galerkin (LDG) method to 2D Keller–Segel (KS) chemotaxis model. We improve the results upon (Epshteyn and Kurganov in SIAM J Numer Anal, 47:368–408, 2008) 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 Zhang and Shu (J Comput Phys, 229:8918–8934, 2010). 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.

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Acknowledgements

Xingjie Helen Li would like to thank the Shanghai Centre for Mathematics Science (SCMS), Fudan University, for support during her visit.

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Correspondence to Yang Yang.

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Research supported by ARO Grant W911NF-15-1-0226, NSF Grant DMS-1418750, China National Natural Science Foundation (11571367 and 11601536), Michigan Technological University, Research Excellence Fund Scholarship and Creativity Grant.

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Li, X.H., Shu, CW. & Yang, Y. Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model. J Sci Comput 73, 943–967 (2017). https://doi.org/10.1007/s10915-016-0354-y

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  • DOI: https://doi.org/10.1007/s10915-016-0354-y

Keywords

  • Local discontinuous Galerkin method
  • Keller-Segel chemotaxis model
  • Positivity preserving
  • Error estimate
  • Neumann boundary condition
  • Blow-up
  • \(L^1\) stability