# American Institute of Mathematical Sciences

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 and Related Models, 2021, 14 (4) : 681-704. doi: 10.3934/krm.2021019
##### References:
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Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163. [7] S. Boi, V. 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. [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. [9] R. Bürger, S. Diehl, M. C. Martí, P. Mulet, I. Nopens, E. 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. [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. [11] S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Self-Organization in Biological Systems, Second printing edition, Princeton University Press, Princeton, NJ, 2003. [12] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361.  doi: 10.1007/s002050050152. [13] J. A. Carrillo, M. 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. [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. [15] Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. 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. [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. [17] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [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. [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. [20] S. Göttlich, S. Hoher, P. Schindler, V. 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. [21] S. Göttlich, A. 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. [22] S. Göttlich, S. 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. [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. [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. [25] H. Holden, K. 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. [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. [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. [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. [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. [30] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392. [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 [32] E. Rossi, J. Weißen, P. 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. [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. [34] C. M. Topaz, A. 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. [35] T. Vicsek, A. Czirók, E. Ben-Jacob, I. 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. [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. [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.

show all references

##### References:
 [1] P. Aceves-Sánchez, M. Bostan, J.-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. [2] W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919. [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. [4] D. Armbruster, S. Martin and A. Thatcher, Elastic and inelastic collisions of swarms, Phys. D, 344 (2017), 45-57.  doi: 10.1016/j.physd.2016.11.008. [5] D. Armbruster, S. Motsch and A. Thatcher, Swarming in bounded domains, Phys. D, 344 (2017), 58-67.  doi: 10.1016/j.physd.2016.11.009. [6] S. Berres, R. Bürger, K. 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. [7] S. Boi, V. 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. [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. [9] R. Bürger, S. Diehl, M. C. Martí, P. Mulet, I. Nopens, E. 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. [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. [11] S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Self-Organization in Biological Systems, Second printing edition, Princeton University Press, Princeton, NJ, 2003. [12] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361.  doi: 10.1007/s002050050152. [13] J. A. Carrillo, M. 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. [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. [15] Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. 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. [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. [17] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [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. [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. [20] S. Göttlich, S. Hoher, P. Schindler, V. 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. [21] S. Göttlich, A. 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. [22] S. Göttlich, S. 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. [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. [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. [25] H. Holden, K. 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. [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. [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. [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. [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. [30] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392. [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 [32] E. Rossi, J. Weißen, P. 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. [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. [34] C. M. Topaz, A. 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. [35] T. Vicsek, A. Czirók, E. Ben-Jacob, I. 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. [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. [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.
Numerical approximations of the Heaviside function (27)
Experimental setup
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)
Density plots of the solutions at $t = 0.15$ for different diffusion coefficients $k(\rho)$
Real data and density plots of the solutions $t = 1.5$s
Maximum density over time
Collision of a swarm with a boundary
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
Collision of swarms
Heaviside approximations
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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