A well-balanced scheme for kinetic models of chemotaxis derived from one-dimensional local forward–backward problems

Gosse, Laurent

doi:10.1016/j.mbs.2012.12.009

Cited by 12 publications

(9 citation statements)

References 46 publications

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“…So, it appears logical to try to extend our present results toward continuous (at least, larger) velocity models for which strong aggregation occurs, see e.g. [13,17,20,39]. Relying either on "Caseology" methods [1] or on approximations [21, §10.3-4], it is still possible to derive scattering matrices S n j− 1 2 and to set up a scheme of the form (3.5).…”

Section: A Time-splitting (Ts) Pre-processing Procedurementioning

confidence: 99%

Numerical High-Field Limits in Two-Stream Kinetic Models and 1D Aggregation Equations

Gosse¹,

Vauchelet²

2016

SIAM J. Sci. Comput.

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Abstract. Numerical resolution of two-stream kinetic models in strong aggregative setting is considered. To illustrate our approach, we consider an 1D kinetic model for chemotaxis in hydrodynamic scaling and the high field limit of the Vlasov-Poisson-Fokker-Planck system. A difficulty is that, in this aggregative setting, weak solutions of the limiting model blow up in finite time, therefore the scheme should be able to handle Dirac measures. It is overcome thanks to a careful discretization of the macroscopic velocity resulting of Vol'pert calculus: accordingly, a new well-balanced (WB) and asymptotic preserving (AP) numerical scheme is provided. Numerical simulations confirm a good behavior of solutions.

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Section: A Time-splitting (Ts) Pre-processing Procedurementioning

confidence: 99%

Numerical High-Field Limits in Two-Stream Kinetic Models and 1D Aggregation Equations

Gosse¹,

Vauchelet²

2016

SIAM J. Sci. Comput.

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“…Except from the Keller-Segel model (2.11), the schemes are based on the relaxation methods proposed in [26] and [25]. For more sophisticated relaxation schemes for kinetic and Cattaneo model we refer to [18,19,6,31]. The main advantage of this scheme is that they are asymptotic preserving, i.e.…”

Section: Numerical Schemes On An Intervalmentioning

confidence: 99%

Kinetic and related macroscopic models for chemotaxis on networks

Borsche

Kall

Klar

et al. 2016

Math. Models Methods Appl. Sci.

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In this paper we consider kinetic and associated macroscopic models for chemotaxis on a network. Coupling conditions at the nodes of the network for the kinetic problem are presented and used to derive coupling conditions for the macroscopic approximations. The results of the different models are compared and relations to a Keller-Segel model on networks are discussed. For a numerical approximation of the governing equations asymptotic preserving relaxation schemes are extended to directed graphs. Kinetic and macroscopic equations are investigated numerically and their solutions are compared for tripod and more general networks

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“…Recently designing schemes that are either AP or WB, or have both properties attracts a lot of interests, both for kinetic chemotaxi model [13,14,27] and grey radiative transfer equation [12,33]. Generally, it is very hard to have a scheme that have both properties and this is the goal of our present paper.…”

Section: Introductionmentioning

confidence: 99%

“…We take N v = 32 points for the velocity and N x = 500 points for the space discretization. Parameters in (1.1) are chosen as in[14] such that χ S = 1, D = 15, β = 60, α = 3.…”

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confidence: 99%

Well-balanced and asymptotic preserving schemes for kinetic models

Emako,

Tang

2016

Preprint

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In this paper, we propose a general framework for designing numerical schemes that have both well-balanced (WB) and asymptotic preserving (AP) properties, for various kinds of kinetic models. We are interested in two different parameter regimes, 1) When the ratio between the mean free path and the characteristic macroscopic length ε tends to zero, the density can be described by (advection) diffusion type (linear or nonlinear) macroscopic models; 2) When ε = O(1), the models behave like hyperbolic equations with source terms and we are interested in their steady states. We apply the framework to three different kinetic models: neutron transport equation and its diffusion limit, the transport equation for chemotaxis and its Keller-Segel limit, and grey radiative transfer equation and its nonlinear diffusion limit. Numerical examples are given to demonstrate the properties of the schemes.

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A well-balanced scheme for kinetic models of chemotaxis derived from one-dimensional local forward–backward problems

Cited by 12 publications

References 46 publications

Numerical High-Field Limits in Two-Stream Kinetic Models and 1D Aggregation Equations

Numerical High-Field Limits in Two-Stream Kinetic Models and 1D Aggregation Equations

Kinetic and related macroscopic models for chemotaxis on networks

Well-balanced and asymptotic preserving schemes for kinetic models

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