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

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Abstract

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.

Keywords

Surface chemotaxis models
Surface finite element method
Surface gradient and Laplacian recoveries
Modified method of surface characteristic
Positivity preservation

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This work is in parts supported by the Excellent Doctor Innovation Program of Xinjiang University (No. XJUBSCX-2016006), the Graduate Student Research Innovation Program of Xinjiang (No. XJGRI2017013), the Research Fund from Key Laboratory of Xinjiang Province (No. 2017D04030), the Xinjiang Provincial University Research Foundation of China (No. XJEDU2018I002), and the NSF of China (No. 11671345, No. 11362021, No. U1703262).