Diffusive limit of a two-dimensional well-balanced approximation to a kinetic model of chemotaxis

Abstract

A $$(2+2)$$ -dimensional kinetic equation, directly inspired by the run-and-tumble modeling of chemotaxis dynamics is studied so as to derive a both “2D well-balanced” and “asymptotic-preserving” numerical approximation. To this end, exact stationary regimes are expressed by means of Laplace transforms of Fourier–Bessel solutions of associated elliptic equations. This yields a scattering S-matrix which permits to formulate a time-marching scheme in the form of a convex combination in kinetic scaling. Then, in the diffusive scaling, an IMEX-type discretization follows, for which the “2D well-balanced property” still holds, while the consistency with the asymptotic drift-diffusion equation is checked. Numerical benchmarks, involving “nonlocal gradients” (or finite sampling radius), carried out in both scalings, assess theoretical findings. Nonlocal gradients appear to inhibit blowup phenomena.