Abstract:
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 counterpartâ€”the Patlak-Keller-Segel (PKS) systemâ€”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. |