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Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 2-3/2017

05.01.2017

Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model

verfasst von: Xingjie Helen Li, Chi-Wang Shu, Yang Yang

Erschienen in: Journal of Scientific Computing | Ausgabe 2-3/2017

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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.
Vorheriger Artikel An Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Hamilton–Jacobi Equations
Nächster Artikel An Augmented Method for 4th Order PDEs with Discontinuous Coefficients

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Metadaten
Titel
Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model
verfasst von
Xingjie Helen Li
Chi-Wang Shu
Yang Yang
Publikationsdatum
05.01.2017
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 2-3/2017
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-016-0354-y

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