Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model

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© 2017, Springer Science+Business Media New York. 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 P1 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 L1-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 L2-norm of the L1-stable numerical solution.

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Journal of Scientific Computing