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

TLDR: In this article, the diffusive limit of run-and-tumble kinetic models for cell motion due to chemotaxis was studied by means of asymptotic preserving schemes.

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 demonstrate an aggregative behavior from small to a large unique aggregate for the kinetic solutions after the blow-up time in the Patlak--Keller--Segel model. 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.