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 $$P^1$$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 $$L^1$$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 $$L^2$$L2-norm of the $$L^1$$L1-stable numerical solution.