Spectral method for a kinetic swarming model

Gamba, Irene M.; Haack, J.; Motsch, Sébastien

doi:10.1016/j.jcp.2015.04.033

Cited by 17 publications

(16 citation statements)

References 35 publications

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“…The 2D scheme of moment system is used to solve three Riemann problems and a vortex formation problem and will be compared to the spectral method proposed in [13]. For our computations, the computational domain of Riemann problems in x-direction is taken as [−5, 5], the CFL number is chosen as 0.5, and the value of parameter σ is set as 0.2.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Model Reduction of a Two-Dimensional Kinetic Swarming Model by Operator Projections

Duan¹,

Kuang²,

Tang³

2018

EAJAM

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This paper derives the arbitrary order globally hyperbolic moment system for a non-linear kinetic description of the Vicsek swarming model by using the operator projection. It is built on our careful study of a family of the complicate Grad type orthogonal functions depending on a parameter (angle of macroscopic velocity). We calculate their derivatives with respect to the independent variable, and projection of those derivatives, the product of velocity and basis, and collision term. The moment system is also proved to be hyperbolic, rotational invariant, and mass-conservative. The relationship between Grad type expansions in different parameter is also established. A semi-implicit numerical scheme is presented to solve a Cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method. It is also compared to the spectral method for the kinetic equation. The results show that the solutions of our hyperbolic moment system converge to the solutions of the kinetic equation for the Vicsek model as the order of the moment system increases, and the moment method can capture key features such as vortex formation and traveling waves.the notation Id is the identity operator, Ω(t, x) is the mean velocity, and J (t, x) denotes the mean flux at x and is defined by(2.3)Here K(y − x) is the characteristic function of the ball B(0, R) = {x : |x| ≤ R}, i.e. K(x) = 1 |x|

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Section: Numerical Resultsmentioning

confidence: 99%

“…In the numerical computations, J (t, x) can be considered within the approximation (2.4) because the spatial mesh stepsize may be smaller than the radius of ball B(0, R) [13]. It has been shown [14] that the non-linear kinetic equation (2.1) with (2.2) and (2.3) or (2.4) has a non-negative global weak solution in C(0, T ;…”

Section: )mentioning

confidence: 99%

Model Reduction of a Two-Dimensional Kinetic Swarming Model by Operator Projections

Duan¹,

Kuang²,

Tang³

2018

EAJAM

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“…Several schemes have already been proposed to study the kinetic equation (4) using spectral method [15], discontinuous Galerkin [13] or semi-Lagrangian [11]. However, we are now interested in the long time behavior of the solution, thus we would like to design a numerical scheme with several properties:…”

Section: Numerical Schemementioning

confidence: 99%

“…To first investigate our numerical scheme, we study the homogeneous equation, thus solving only the collision operator (15). We present our numerical experiments with the Vicsek model (4), however results are similar with the DFL dynamics except that the time step ∆t may have to be adapted (since the CFL condition depends on |j| which varies over time).…”

Section: Homogeneous Casementioning

confidence: 99%

“…Since we aim at analyzing the long-time behavior of the solution, it is essential to preserve these properties. For instance, several papers have already proposed to solve numerically the kinetic equation associated with the Vicsek model using other methods (spectral method [15], particle method [11], discontinuous Galerkin [13]). But we would rather have lower accuracy and a preserving numerical scheme to study the long-time behavior of the solution (even though our scheme is still second order accurate in the velocity-variable).…”

Section: Introductionmentioning

confidence: 99%

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Kinetic Equations and Self-organized Band Formations

Griette

Motsch

2019

Active Particles, Volume 2

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Self-organization is an ubiquitous phenomenon in nature which can be observed in a variety of different contexts and scales, with examples ranging from fish schools, swarms of birds or locusts, to flocks of bacteria. The observation of such global patterns can often be reproduced in models based on simple interactions between neighboring particles. In this paper we focus on two particular interaction dynamics closely related to the one described in the seminal paper of Vicsek and collaborators. After reviewing the current state of the art in the subject, we study a numerical scheme for the kinetic equation associated with the Vicsek models which has the specificity of reproducing many physical properties of the continuous models, like the preservation of energy and positivity and the diminution of an entropy functional. We describe a stable pattern of bands emerging in the dynamics proposed by Degond-Frouvelle-Liu dynamics and give some insights about their formation.

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Numerical Approaches for Kinetic and Hyperbolic Models

Eftimie

2018

Hyperbolic and Kinetic Models for Self-Organised Biological Aggregations

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Spectral method for a kinetic swarming model

Cited by 17 publications

References 35 publications

Model Reduction of a Two-Dimensional Kinetic Swarming Model by Operator Projections

Model Reduction of a Two-Dimensional Kinetic Swarming Model by Operator Projections

Kinetic Equations and Self-organized Band Formations

Numerical Approaches for Kinetic and Hyperbolic Models

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