August  2021, 14(4): 681-704. doi: 10.3934/krm.2021019

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

Received  December 2020 Revised  April 2021 Published  August 2021 Early access  June 2021

Fund Project: 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

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.

Citation: Jennifer Weissen, Simone Göttlich, Dieter Armbruster. Density dependent diffusion models for the interaction of particle ensembles with boundaries. Kinetic & Related Models, 2021, 14 (4) : 681-704. doi: 10.3934/krm.2021019
References:
[1]

P. Aceves-SánchezM. BostanJ.-A. Carrillo and P. Degond, Hydrodynamic limits for kinetic flocking models of Cucker-Smale type, Math. Biosci. Eng., 16 (2019), 7883-7910.  doi: 10.3934/mbe.2019396.  Google Scholar

[2]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.  Google Scholar

[3]

I. Aoki, A simulation study on the schooling mechanism in Fish, Nippon Suisan Gakkaishi, 48 (1982), 1081-1088.  doi: 10.2331/suisan.48.1081.  Google Scholar

[4]

D. ArmbrusterS. Martin and A. Thatcher, Elastic and inelastic collisions of swarms, Phys. D, 344 (2017), 45-57.  doi: 10.1016/j.physd.2016.11.008.  Google Scholar

[5]

D. ArmbrusterS. Motsch and A. Thatcher, Swarming in bounded domains, Phys. D, 344 (2017), 58-67.  doi: 10.1016/j.physd.2016.11.009.  Google Scholar

[6]

S. BerresR. BürgerK. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163.  Google Scholar

[7]

S. BoiV. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species Polyergus rufescens, Nonlinear Anal. Real World Appl., 1 (2000), 163-176.  doi: 10.1016/S0362-546X(99)00399-5.  Google Scholar

[8]

H. Brézis and M. G. Crandall, Uniqueness of solutions of the initial-value problem for $u_{t}-\Delta \varphi (u) = 0$, J. Math. Pures Appl, 58 (1979), 153-163.   Google Scholar

[9]

R. BürgerS. DiehlM. C. MartíP. MuletI. NopensE. Torfs and P. A. Vanrolleghem, Numerical solution of a multi-class model for batch settling in water resource recovery facilities, Appl. Math. Model., 49 (2017), 415-436.  doi: 10.1016/j.apm.2017.05.014.  Google Scholar

[10]

R. Bürger, P. Mulet and L. M. Villada, Regularized nonlinear solvers for IMEX methods applied to diffusively corrected multispecies kinematic flow models, SIAM J. Sci. Comput., 35 (2013), B751–B777. doi: 10.1137/120888533.  Google Scholar

[11] S. CamazineJ.-L. DeneubourgN. R. FranksJ. SneydG. Theraulaz and E. Bonabeau, Self-Organization in Biological Systems, Second printing edition, Princeton University Press, Princeton, NJ, 2003.   Google Scholar
[12]

J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361.  doi: 10.1007/s002050050152.  Google Scholar

[13]

J. A. CarrilloM. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378.  doi: 10.3934/krm.2009.2.363.  Google Scholar

[14] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, Cambridge, 1990.   Google Scholar
[15]

Y.-L. ChuangM. R. D'OrsognaD. MarthalerA. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.  doi: 10.1016/j.physd.2007.05.007.  Google Scholar

[16]

R. M. Colombo and E. Rossi, Modelling crowd movements in domains with boundaries, IMA J. Appl. Math., 84 (2019), 833-853.  doi: 10.1093/imamat/hxz017.  Google Scholar

[17]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[18]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[19]

M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Physical Review Letters, 96 (2006), 104302. doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[20]

S. GöttlichS. HoherP. SchindlerV. Schleper and A. Verl, Modeling, simulation and validation of material flow on conveyor belts, Applied Mathematical Modelling, 38 (2014), 3295-3313.  doi: 10.1016/j.apm.2013.11.039.  Google Scholar

[21]

S. GöttlichA. Klar and S. Tiwari, Complex material flow problems: A multi-scale model hierarchy and particle methods, J. Engrg. Math., 92 (2015), 15-29.  doi: 10.1007/s10665-014-9767-5.  Google Scholar

[22]

S. GöttlichS. Knapp and P. Schillen, A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches, Kinet. Relat. Models, 11 (2018), 1333-1358.  doi: 10.3934/krm.2018052.  Google Scholar

[23]

D. Grünbaum, Translating stochastic density-dependent individual behavior with sensory constraints to an Eulerian model of animal swarming, Journal of Mathematical Biology, 33 (1994), 139-161.  doi: 10.1007/BF00160177.  Google Scholar

[24]

D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286.  doi: 10.1103/PhysRevE.51.4282.  Google Scholar

[25]

H. HoldenK. H. Karlsen and K. A. Lie, Operator splitting methods for degenerate convection-diffusion equations Ⅱ: Numerical examples with emphasis on reservoir simulation and sedimentation, Computational Geosciences, 4 (2000), 287-322.  doi: 10.1023/A:1011582819188.  Google Scholar

[26]

A. L. Koch and D. White, The social lifestyle of myxobacteria, BioEssays, 20 (1998), 1030-1038.  doi: 10.1002/(SICI)1521-1878(199812)20:12<1030::AID-BIES9>3.0.CO;2-7.  Google Scholar

[27]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[28]

G. Naldi, L. Pareschi and G. Toscani, Mathematical Modeling of Collective Behaviour in Socio-Economic and Life Sciences, Birkhäuser Basel, Basel, 2010. doi: 10.1007/978-0-8176-4946-3.  Google Scholar

[29]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol. 14 of Interdisciplinary Applied Mathematics, 2nd edition, Springer-Verlag New York, New York, NY, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[30]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[31]

D. Prims, J. Kötz, S. Göttlich and A. Katterfeld, Validation of flow models as new simulation approach for parcel handling in bulk mode, Logistics Journal, 2019 (2019), 1–11. https://www.logistics-journal.de/archiv/2019/4889 Google Scholar

[32]

E. RossiJ. WeißenP. Goatin and S. Göttlich, Well-posedness of a non-local model for material flow on conveyor belts, ESAIM Math. Model. Numer. Anal., 54 (2020), 679-704.  doi: 10.1051/m2an/2019062.  Google Scholar

[33]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.  Google Scholar

[34]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[35]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[36]

A. I. Vol'pert and S. I. Hudjaev, Cauchy's problem for degenerate second order quasilinear parabolic equations, Mathematics of the USSR-Sbornik, 7 (1969), 365-387.  doi: 10.1070/SM1969v007n03ABEH001095.  Google Scholar

[37]

J. Yin, On the uniqueness and stability of $ \rm BV $ solutions for nonlinear diffusion equations, Comm. Partial Differential Equations, 15 (1990), 1671-1683.  doi: 10.1080/03605309908820743.  Google Scholar

show all references

References:
[1]

P. Aceves-SánchezM. BostanJ.-A. Carrillo and P. Degond, Hydrodynamic limits for kinetic flocking models of Cucker-Smale type, Math. Biosci. Eng., 16 (2019), 7883-7910.  doi: 10.3934/mbe.2019396.  Google Scholar

[2]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.  Google Scholar

[3]

I. Aoki, A simulation study on the schooling mechanism in Fish, Nippon Suisan Gakkaishi, 48 (1982), 1081-1088.  doi: 10.2331/suisan.48.1081.  Google Scholar

[4]

D. ArmbrusterS. Martin and A. Thatcher, Elastic and inelastic collisions of swarms, Phys. D, 344 (2017), 45-57.  doi: 10.1016/j.physd.2016.11.008.  Google Scholar

[5]

D. ArmbrusterS. Motsch and A. Thatcher, Swarming in bounded domains, Phys. D, 344 (2017), 58-67.  doi: 10.1016/j.physd.2016.11.009.  Google Scholar

[6]

S. BerresR. BürgerK. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163.  Google Scholar

[7]

S. BoiV. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species Polyergus rufescens, Nonlinear Anal. Real World Appl., 1 (2000), 163-176.  doi: 10.1016/S0362-546X(99)00399-5.  Google Scholar

[8]

H. Brézis and M. G. Crandall, Uniqueness of solutions of the initial-value problem for $u_{t}-\Delta \varphi (u) = 0$, J. Math. Pures Appl, 58 (1979), 153-163.   Google Scholar

[9]

R. BürgerS. DiehlM. C. MartíP. MuletI. NopensE. Torfs and P. A. Vanrolleghem, Numerical solution of a multi-class model for batch settling in water resource recovery facilities, Appl. Math. Model., 49 (2017), 415-436.  doi: 10.1016/j.apm.2017.05.014.  Google Scholar

[10]

R. Bürger, P. Mulet and L. M. Villada, Regularized nonlinear solvers for IMEX methods applied to diffusively corrected multispecies kinematic flow models, SIAM J. Sci. Comput., 35 (2013), B751–B777. doi: 10.1137/120888533.  Google Scholar

[11] S. CamazineJ.-L. DeneubourgN. R. FranksJ. SneydG. Theraulaz and E. Bonabeau, Self-Organization in Biological Systems, Second printing edition, Princeton University Press, Princeton, NJ, 2003.   Google Scholar
[12]

J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361.  doi: 10.1007/s002050050152.  Google Scholar

[13]

J. A. CarrilloM. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378.  doi: 10.3934/krm.2009.2.363.  Google Scholar

[14] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, Cambridge, 1990.   Google Scholar
[15]

Y.-L. ChuangM. R. D'OrsognaD. MarthalerA. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.  doi: 10.1016/j.physd.2007.05.007.  Google Scholar

[16]

R. M. Colombo and E. Rossi, Modelling crowd movements in domains with boundaries, IMA J. Appl. Math., 84 (2019), 833-853.  doi: 10.1093/imamat/hxz017.  Google Scholar

[17]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[18]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[19]

M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Physical Review Letters, 96 (2006), 104302. doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[20]

S. GöttlichS. HoherP. SchindlerV. Schleper and A. Verl, Modeling, simulation and validation of material flow on conveyor belts, Applied Mathematical Modelling, 38 (2014), 3295-3313.  doi: 10.1016/j.apm.2013.11.039.  Google Scholar

[21]

S. GöttlichA. Klar and S. Tiwari, Complex material flow problems: A multi-scale model hierarchy and particle methods, J. Engrg. Math., 92 (2015), 15-29.  doi: 10.1007/s10665-014-9767-5.  Google Scholar

[22]

S. GöttlichS. Knapp and P. Schillen, A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches, Kinet. Relat. Models, 11 (2018), 1333-1358.  doi: 10.3934/krm.2018052.  Google Scholar

[23]

D. Grünbaum, Translating stochastic density-dependent individual behavior with sensory constraints to an Eulerian model of animal swarming, Journal of Mathematical Biology, 33 (1994), 139-161.  doi: 10.1007/BF00160177.  Google Scholar

[24]

D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286.  doi: 10.1103/PhysRevE.51.4282.  Google Scholar

[25]

H. HoldenK. H. Karlsen and K. A. Lie, Operator splitting methods for degenerate convection-diffusion equations Ⅱ: Numerical examples with emphasis on reservoir simulation and sedimentation, Computational Geosciences, 4 (2000), 287-322.  doi: 10.1023/A:1011582819188.  Google Scholar

[26]

A. L. Koch and D. White, The social lifestyle of myxobacteria, BioEssays, 20 (1998), 1030-1038.  doi: 10.1002/(SICI)1521-1878(199812)20:12<1030::AID-BIES9>3.0.CO;2-7.  Google Scholar

[27]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[28]

G. Naldi, L. Pareschi and G. Toscani, Mathematical Modeling of Collective Behaviour in Socio-Economic and Life Sciences, Birkhäuser Basel, Basel, 2010. doi: 10.1007/978-0-8176-4946-3.  Google Scholar

[29]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol. 14 of Interdisciplinary Applied Mathematics, 2nd edition, Springer-Verlag New York, New York, NY, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[30]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[31]

D. Prims, J. Kötz, S. Göttlich and A. Katterfeld, Validation of flow models as new simulation approach for parcel handling in bulk mode, Logistics Journal, 2019 (2019), 1–11. https://www.logistics-journal.de/archiv/2019/4889 Google Scholar

[32]

E. RossiJ. WeißenP. Goatin and S. Göttlich, Well-posedness of a non-local model for material flow on conveyor belts, ESAIM Math. Model. Numer. Anal., 54 (2020), 679-704.  doi: 10.1051/m2an/2019062.  Google Scholar

[33]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.  Google Scholar

[34]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[35]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[36]

A. I. Vol'pert and S. I. Hudjaev, Cauchy's problem for degenerate second order quasilinear parabolic equations, Mathematics of the USSR-Sbornik, 7 (1969), 365-387.  doi: 10.1070/SM1969v007n03ABEH001095.  Google Scholar

[37]

J. Yin, On the uniqueness and stability of $ \rm BV $ solutions for nonlinear diffusion equations, Comm. Partial Differential Equations, 15 (1990), 1671-1683.  doi: 10.1080/03605309908820743.  Google Scholar

Figure 1.  Numerical approximations of the Heaviside function (27)
Figure 2.  Experimental setup
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)
Figure 4.  Density plots of the solutions at $ t = 0.15 $ for different diffusion coefficients $ k(\rho) $
Figure 5.  Real data and density plots of the solutions $ t = 1.5 $s
Figure 6.  Maximum density over time
Figure 7.  Collision of a swarm with a boundary
Table 1 are marked with a circle">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
Figure 9.  Collision of swarms
Figure 10.  Heaviside approximations
Table 1.  Reflection angle of the experiments in Figure 7c
$ \delta $ 1 2 3
$ {\theta ^r} $ 52.81 24.93 16.15
$ \delta $ 1 2 3
$ {\theta ^r} $ 52.81 24.93 16.15
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