An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic systems for Chemotaxis
Provides a numerical method for studying the diffusive limit of kinetic chemotaxis models, relevant for researchers in mathematical biology and numerical analysis.
This work develops an asymptotic preserving scheme for the diffusive limit of run & tumble kinetic models for chemotaxis, accurately approximating solutions before blow-up for small parameters and showing long-time stabilization to steady states.
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.