2010
DOI: 10.3934/krm.2010.3.501
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Abstract: International audienceThis paper is devoted to numerical simulations of a kinetic model describing chemotaxis. This kinetic framework has been investigated since the 80's when experimental observations have shown that the motion of bacteria is due to the alternance of 'runs and tumbles'. Since parabolic and hyperbolic models do not take into account the microscopic movement of individual cells, kinetic models have become of a great interest. Dolak and Schmeiser (2005) have then proposed a kinetic model describ… Show more

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Cited by 21 publications
(27 citation statements)
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“…Especially (see e.g. [41]) the kinetic equation, for bounded tumbling kernel, propagates in time the smoothness of its initial data. Hence, at the kinetic level, all quantities involved are well-defined and there seems to be no nonconservative product.…”
Section: A Time-splitting (Ts) Pre-processing Procedurementioning
confidence: 99%
“…Especially (see e.g. [41]) the kinetic equation, for bounded tumbling kernel, propagates in time the smoothness of its initial data. Hence, at the kinetic level, all quantities involved are well-defined and there seems to be no nonconservative product.…”
Section: A Time-splitting (Ts) Pre-processing Procedurementioning
confidence: 99%
“…In 2D, it is possible to have blow-up patterns along lines (Vauchelet 2010) Fig. 6 Fetecau and Eftimie (2010), Hasimoto (1974), Hillen and Stevens (2000), Vauchelet (2010), Leverentz et al (2009), Hillen and Levine (2003) Gradient blow-up (shocks) The density u is bounded, but its gradient ∇ x u becomes infinite at a time point t = T < ∞. Note that, in some cases, the blow-up of the gradient can cause an infinite growth in the amplitude of solutions.…”
Section: One-equation Hyperbolic Modelsmentioning
confidence: 99%
“…These methods range from existence and regularity results, to asymptotic problems and the derivation of macroscopic models. In terms of understanding the patterns, the analytical results go as far as proving the existence of bounded or blow-up solutions (Bournaveas et al 2008;Vauchelet 2010). For simpler models (e.g., models that assume an uniform distribution of velocities after the turning event) it is still possible to investigate analytically the structure of solutions.…”
Section: Velocity-jump Processesmentioning
confidence: 99%
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“…Hyperbolic models eliminate this undesirable feature, see [18,22,43]; discrete velocity kinetic models can be handled by means of similar techniques [27,28,40]. Advances in numerical analysis allow to simulate efficiently much more detailed kinetic models [2,16,23,41,42,48]: consult for instance [11,50]. Finally, we stress again the similarity between this class of distinguished numerical schemes and the ones derived for the nonlinear parabolic equations by extending the ideas of the classic Scharfetter-Gummel scheme [44] for which numerical fluxes are computed by solving a Dirichlet problem for the stationary equations, see [20].…”
Section: Outline Of the Papermentioning
confidence: 99%