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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 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 (Y. Zhang, X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 8918-8934). 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.

Local discontinuous Galerkin method, Keller-Segel chemotaxis model, positivity preserving, error estimate, Neumann boundary condition, blow-up, $L^1$ stability