Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model

In this paper, we apply the local discontinuous Galerkin (LDG) method to 2D Keller– Segel (KS) chemotaxis model. We improve the results upon (Y. Epshteyn and A. Kurganov, SIAM Journal on Numerical Analysis, 47 (2008), 368-408) and give optimal rate of convergence under special finite element spaces. Moreover, to construct physically relevant numerical approximations, we develop a positivity-preserving limiter to the scheme, extending the idea in (Y. Zhang, X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 8918-8934). With this limiter, we can prove the L-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 numerical blow-up time by using the L-norm of the L-stable numerical approximations.