An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic systems for Chemotaxis

TL;DR

This paper develops an asymptotic preserving numerical scheme for kinetic models of chemotaxis, accurately capturing the diffusive limit and long-term behavior, including blow-up phenomena, in both one and two dimensions.

Contribution

It introduces a novel asymptotic preserving scheme for kinetic chemotaxis models that effectively approximates solutions near blow-up and extends to two-dimensional radial cases.

Findings

Scheme accurately approximates solutions before blow-up

Solutions tend to steady states over long times

Blow-up behavior is numerically characterized

Abstract

In this work we numerically study the diffusive limit of run & tumble kinetic models for cell motion due to chemotaxis by means of asymptotic preserving schemes. It is well-known that the diffusive limit of these models leads to the classical Patlak-Keller-Segel macroscopic model for chemotaxis. We will show that the proposed scheme is able to accurately approximate the solutions before blow-up time for small parameter. Moreover, the numerical results indicate that the global solutions of the kinetic models stabilize for long times to steady states for all the analyzed parameter range. We also generalize these asymptotic preserving schemes to two dimensional kinetic models in the radial case. The blow-up of solutions is numerically investigated in all these cases.