An asymptotic preserving scheme for kinetic models for chemotaxis phenomena

What is it about?

In this paper, we extend a method of micro-macro decomposition in order to construct asymptotic preserving schemes (AP) for kinetic equations describing chemotaxis phenomena. Our strategy consists in rewriting the kinetic equation as a coupled system of kinetic part and macroscopic one, by using the micro-macro decomposition of the distribution function. By using a classical projection technique, we obtain an evolution equation for the macroscopic parameters of the equilibrium coupled to a kinetic equation for the non-equilibrium part. This method is validated by various test cases and compared to other standard methods.

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Why is it important?

This paper has developed a computational approach to a class of pattern formation models derived from the celebrated Keller-Segel model obtained by the underlying description delivered by generalized kinetic theory methods. The derivation is based on a decomposition with two scales, namely the microscopic and the macroscopic one technically related, as we have seen, by suitable small parameters accounting for the time and space dynamics. The novelty of our paper is that the computational scheme which follows precisely the derivation hallmarks by using the same decomposition and parameters. This idea improves the stability properties of the solutions with respect to classical approaches known in the literature.

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