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

Gabriella Bretti () and Laurent Gosse ()
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Gabriella Bretti: Istituto per le Applicazioni del Calcolo
Laurent Gosse: Istituto per le Applicazioni del Calcolo

Partial Differential Equations and Applications, 2021, vol. 2, issue 2, 1-34

Abstract: Abstract A $$(2+2)$$ ( 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.

Keywords: Diffusive limit; Kinetic well-balanced scheme; Run-and-Tumble; 65M06; 35J15; 76M45; 92B05 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00087-7

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