# American Institute of Mathematical Sciences

October  2022, 17(5): 783-801. doi: 10.3934/nhm.2022026

## Splitting scheme for a macroscopic crowd motion model with congestion for a two-typed population

 LMO - Laboratoire de Mathématiques d'Orsay, France

Received  January 2022 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: The work of Félicien Bourdin is supported by the ERC grant NORIA

We study the extension of the macroscopic crowd motion model with congestion to a population divided into two types. As the set of pairs of density whose sum is bounded is not geodesically convex in the product of Wasserstein spaces, the generic splitting scheme may be ill-posed. We thus analyze precisely the projection operator on the set of admissible densities, and link it to the projection on the set of measures of bounded density in the mono-type case. We then derive a numerical scheme to adapt the one-typed population splitting scheme.

Citation: Félicien BOURDIN. Splitting scheme for a macroscopic crowd motion model with congestion for a two-typed population. Networks and Heterogeneous Media, 2022, 17 (5) : 783-801. doi: 10.3934/nhm.2022026
##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metricspaces and in the Space of Proba-bility Measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005. [2] T. M. Blackwell and P. Bentley, Don't push me! Collision-avoiding swarms, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No. 02TH8600), IEEE, 2 (2002). [3] V. J. Blue and J. L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways, Transportation Research Part B: Methodological, 35 (2001), 293-312.  doi: 10.1016/S0191-2615(99)00052-1. [4] C. E. Brennen, Fundamentals of Multiphase flow, 2005. doi: 10.1017/CBO9780511807169. [5] C. Burstedde, et al., Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A: Statistical Mechanics and its Applications, 295 (2001), 507–525. doi: 10.1016/S0378-4371(01)00141-8. [6] C. Cancés, T. O. Gallouét and L. Monsaingeon, Incompressible immiscible multiphase flows in porous media: A variational approach, Anal. PDE, 10 (2017), 1845-1876.  doi: 10.2140/apde.2017.10.1845. [7] G. Carlier and et al., Convergence of entropic schemes for optimal transport and gradient flows, SIAM J. Math. Anal., 49 (2017), 1385-1418.  doi: 10.1137/15M1050264. [8] J. A. Carrillo, M. P. Gualdani and G. Toscani, Finite speed of propagation in porous media by mass transportation methods, C. R. Math., 338 (2004), 815-818.  doi: 10.1016/j.crma.2004.03.025. [9] C. T. Crowe, Multiphase Flow Handbook, CRC press, 2005.  doi: 10.1201/9781420040470. [10] M. Cuturi, Sinkhorn distances: Lightspeed computation of optimal transport, Advances in Neural Information Processing Systems, 26 (2013), 2292-2300. [11] D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Physical review E, 51 (1995), 4282. [12] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359. [13] I. Kim and A. R. Mészáros, On nonlinear cross-diffusion systems: An optimal transport approach, Calc. Var. Partial Differential Equations, 57 (2018), 1-40.  doi: 10.1007/s00526-018-1351-9. [14] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci., 20 (2010), 1787-1821.  doi: 10.1142/S0218202510004799. [15] B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Netw. Heterog. Media, 6 (2011), 485-519.  doi: 10.3934/nhm.2011.6.485. [16] A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158. [17] D. Morale, V. Capasso and K. Oelschláger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.  doi: 10.1007/s00285-004-0279-1. [18] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7. [19] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243. [20] G. Peyré, Entropic approximation of Wasserstein gradient flows, SIAM J. Imaging Sci., 8 (2015), 2323-2351.  doi: 10.1137/15M1010087. [21] A. Roudneff-Chupin, Modélisation macroscopique de mouvements de foule, Phdthesis, PhD Thesis, Université Paris-Sud XI, 2011. [22] F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2. [23] R. H. Silsbee, Focusing in collision problems in solids, J. Appl. Physics, 28 (1957), 1246-1250. [24] V. Shvetsov and D. Helbing, Macroscopic dynamics of multilane traffic, Physical Review E, 29 (1999), 6328.

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##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metricspaces and in the Space of Proba-bility Measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005. [2] T. M. Blackwell and P. Bentley, Don't push me! Collision-avoiding swarms, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No. 02TH8600), IEEE, 2 (2002). [3] V. J. Blue and J. L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways, Transportation Research Part B: Methodological, 35 (2001), 293-312.  doi: 10.1016/S0191-2615(99)00052-1. [4] C. E. Brennen, Fundamentals of Multiphase flow, 2005. doi: 10.1017/CBO9780511807169. [5] C. Burstedde, et al., Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A: Statistical Mechanics and its Applications, 295 (2001), 507–525. doi: 10.1016/S0378-4371(01)00141-8. [6] C. Cancés, T. O. Gallouét and L. Monsaingeon, Incompressible immiscible multiphase flows in porous media: A variational approach, Anal. PDE, 10 (2017), 1845-1876.  doi: 10.2140/apde.2017.10.1845. [7] G. Carlier and et al., Convergence of entropic schemes for optimal transport and gradient flows, SIAM J. Math. Anal., 49 (2017), 1385-1418.  doi: 10.1137/15M1050264. [8] J. A. Carrillo, M. P. Gualdani and G. Toscani, Finite speed of propagation in porous media by mass transportation methods, C. R. Math., 338 (2004), 815-818.  doi: 10.1016/j.crma.2004.03.025. [9] C. T. Crowe, Multiphase Flow Handbook, CRC press, 2005.  doi: 10.1201/9781420040470. [10] M. Cuturi, Sinkhorn distances: Lightspeed computation of optimal transport, Advances in Neural Information Processing Systems, 26 (2013), 2292-2300. [11] D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Physical review E, 51 (1995), 4282. [12] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359. [13] I. Kim and A. R. Mészáros, On nonlinear cross-diffusion systems: An optimal transport approach, Calc. Var. Partial Differential Equations, 57 (2018), 1-40.  doi: 10.1007/s00526-018-1351-9. [14] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci., 20 (2010), 1787-1821.  doi: 10.1142/S0218202510004799. [15] B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Netw. Heterog. Media, 6 (2011), 485-519.  doi: 10.3934/nhm.2011.6.485. [16] A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158. [17] D. Morale, V. Capasso and K. Oelschláger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.  doi: 10.1007/s00285-004-0279-1. [18] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7. [19] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243. [20] G. Peyré, Entropic approximation of Wasserstein gradient flows, SIAM J. Imaging Sci., 8 (2015), 2323-2351.  doi: 10.1137/15M1010087. [21] A. Roudneff-Chupin, Modélisation macroscopique de mouvements de foule, Phdthesis, PhD Thesis, Université Paris-Sud XI, 2011. [22] F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2. [23] R. H. Silsbee, Focusing in collision problems in solids, J. Appl. Physics, 28 (1957), 1246-1250. [24] V. Shvetsov and D. Helbing, Macroscopic dynamics of multilane traffic, Physical Review E, 29 (1999), 6328.
The interpolation between two opposite configurations of spheres along generalized geodesics. In particular, for $t = 0.45$ the pair $(\alpha_1^t, \alpha_2^t)$ is not in $K_2$
The cell of the mesh in position $(k, l)$. The densities are defined inside the cell, whereas the velocities are defined at the edges
The distribution of the image of the cell $(2, 2)$ by T. We first compute the image of the center, then draw a box of size $h$. The lower left part is lifted on the cell $(3, 4)$, the upper left part on $(3, 5)$, the upper right part on $(4, 5)$ and the lower right part on $(4, 4)$
Distribution of the 2-Wasserstein distances between pairs of estimated projections of the density in example 3
The motion of two crossing discs. The first column represents the sum of the two densities, and the other two the separated densities. The total time of the simulation is $T = 0.5$ for a timestep $dt = 0.01$ and a mesh size $N = 100$. The random projection step is averaged on $50$ experiments
The motion of two crossing discs in the presence of chemoattraction. We chose the same parameters that in the previous simulation, with $\kappa = 10$
Aggregation of a composite crowd driven by chemoattraction and short-range interactions. For $t = 0.50$, we see small numerical diffusion due to the stochastic projection on $K_2$. The parameters here are $N = 150$, $\kappa = 8$, $\alpha = 0.2$, $\eta = 0.1$, $R = 0.04$. The random projection step is averaged on $100$ runs and the mesh size is $N = 150$. The time parameters are still $T = 0.5$ and $dt = 0.01$
In solid line, the function $g_5$. In dashed, its approximation by a strictly convex smooth function $f_5$. In dotted line is displayed the common limit to $f_n$ and $g_n$
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