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Self-alignment driven by jump processes: macroscopic limit and numerical investigation. (English) Zbl 1341.35170

Summary: In this paper, we are interested in studying self-alignment mechanisms described as jump processes. In the dynamics proposed, active particles are moving at a constant speed and align with their neighbors at random times following a Poisson process. This dynamics can be viewed as an asynchronous version of the so-called Vicsek model. Starting from this particle dynamics, we introduce the related kinetic description and then derive a continuum hydrodynamic model. We then introduce different discretization strategies for the hierarchy of proposed models, we numerically study the convergence of the schemes and compare the behaviors of the different systems for several test cases.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35L40 First-order hyperbolic systems
35L67 Shocks and singularities for hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
92D25 Population dynamics (general)
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics

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