Numerical simulations for the chemotaxis models on surfaces via a novel characteristic finite element method

In this paper, the chemotaxis partial differential equations models which describe the movement by one community in reaction to one chemical or biological signal are given and solved numerically on surfaces. The models and the numerical method concerned are in terms of a generic form of chemotaxis models. We develop a new semi-implicit finite element scheme based on the gradient and Laplacian recoveries to linearize the equations. Meanwhile, we modify the proposed semi-implicit scheme as a characteristic form and present a novel strategy for the discretization of characteristic derivative on surface. The lumped mass modification is employed for positivity preservation. And the related analysis results are provided. We investigate the accuracy and convergence of the proposed method by numerical tests. And the simulations of blowing-up solution, pattern formulations and aggregations of bacteria demonstrate the applicability of the proposed methods. (C) 2019 Elsevier Ltd. All rights reserved.