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

Author

Listed:
  • Gabriella Bretti

    (Istituto per le Applicazioni del Calcolo)

  • Laurent Gosse

    (Istituto per le Applicazioni del Calcolo)

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.

Suggested Citation

  • Gabriella Bretti & Laurent Gosse, 2021. "Diffusive limit of a two-dimensional well-balanced approximation to a kinetic model of chemotaxis," Partial Differential Equations and Applications, Springer, vol. 2(2), pages 1-34, April.
    • Handle: RePEc:spr:pardea:v:2:y:2021:i:2:d:10.1007_s42985-021-00087-7
    DOI: 10.1007/s42985-021-00087-7
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