New Trends in Quantum and Classical Kinetic Equations and Related PDEs

An Asymptotic Preserving Scheme for Kinetic Chemotaxis Models in Two Space Dimensions

Alina Chertock

North Carolina State University

We study a two-dimensional multiscale chemotaxis model based on a combination of the macroscopic evolution equation for chemoattractant and the microscopic model for cell evolution. The latter is governed by a Boltzmann-type kinetic equation with a local turning kernel operator which describes the velocity change of the cells. The parabolic scaling yields a nondimensional kinetic model with a small parameter, which represents the mean free path of the cells. We propose a new asymptotic preserving (AP) numerical scheme that reflects the convergence of the studied micro-macro model to its macroscopic counterpart—the Patlak-Keller-Segel (PKS) system—in the singular limit. The AP property of our numerical approach is achieved by implementing an operator splitting technique combined with the idea of the even-odd formulation and we prove the the resulting scheme yields a consistent approximation of the PKS system as the mean-free path tends to 0. The performance of the proposed numerical method is illustrated on a number of numerical experiments.