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

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 timemarching 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.
Tipo pubblicazione
Altri Autori
Gabriella Bretti; Laurent Gosse
SN Partial Differential Equations and Applications