Density dependent diffusion models for the interaction of particle ensembles with boundaries

Jennifer Weissen; Simone Göttlich; Dieter Armbruster

doi:10.3934/krm.2021019

Article Contents

2021, Volume 14, Issue 4: 681-704. Doi: 10.3934/krm.2021019

This issue Previous Article Inelastic Boltzmann equation driven by a particle thermal bath Next Article Lower bound for the Boltzmann equation whose regularity grows tempered with time

Density dependent diffusion models for the interaction of particle ensembles with boundaries

1.
University of Mannheim, Department of Mathematics, 68131, Mannheim, Germany
2.
Arizona State University, School of Mathematical and Statistical Sciences, Tempe, AZ 85287-1804, USA

^* Corresponding author
^* Corresponding author

Received: December 2020

Revised: April 2021

Early access: June 2021

Published: August 2021

The authors are grateful for the support of their joint research by the DAAD (Project-ID 57444394). J. Weissen and S. Göttlich are supported by the DFG project GO 1920/7-1

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Abstract

The transition from a microscopic model for the movement of many particles to a macroscopic continuum model for a density flow is studied. The microscopic model for the free flow is completely deterministic, described by an interaction potential that leads to a coherent motion where all particles move in the same direction with the same speed known as a flock. Interaction of the flock with boundaries, obstacles and other flocks leads to a temporary destruction of the coherent motion that macroscopically can be modeled through density dependent diffusion. The resulting macroscopic model is an advection-diffusion equation for the particle density whose diffusion coefficient is density dependent. Examples describing ⅰ) the interaction of material flow on a conveyor belt with an obstacle that redirects or restricts the material flow and ⅱ) the interaction of flocks (of fish or birds) with boundaries and ⅲ) the scattering of two flocks as they bounce off each other are discussed. In each case, the advection-diffusion equation is strictly hyperbolic before and after the interaction while the interaction phase is described by a parabolic equation. A numerical algorithm to solve the advection-diffusion equation through the transition is presented.

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Mathematics Subject Classification: Primary: 35M10, 35K65; Secondary: 35L65.

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Figure 1. Numerical approximations of the Heaviside function (27)

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Figure 2. Experimental setup

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Figure 3. Maximum density as a function of time for Eq.(14) with diffusion coefficient $ k_1 $ (solid line) and $ k_2 $ (dashed line), and for Eq.(18) with diffusion coefficient $ k_3 $ (dashed dotted) and $ k_4 $ (dotted line)

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Figure 4. Density plots of the solutions at $ t = 0.15 $ for different diffusion coefficients $ k(\rho) $

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Figure 5. Real data and density plots of the solutions $ t = 1.5 $s

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Figure 6. Maximum density over time

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Figure 7. Collision of a swarm with a boundary

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Figure 8. Reflection angle for $ {\theta ^0} = $ 30, 45 and 60 deg. The asterisk marks the diffusion constant for which the flock as a whole reflects specularly, i.e. $ {\theta ^0} = {\theta ^r} $. The results of Table 1 are marked with a circle

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Figure 9. Collision of swarms

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Figure 10. Heaviside approximations

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Table 1. Reflection angle of the experiments in Figure 7c

$ \delta $ 1 2 3

$ {\theta ^r} $ 52.81 24.93 16.15

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