# American Institute of Mathematical Sciences

December 2017, 12(4): 591-617. doi: 10.3934/nhm.2017024

## Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams

 1 Université Côte d'Azur, Inria, CNRS, LJAD, Parc Valrose, 06108 Nice, France 2 Institut de Mécanique des Fluides de Toulouse, CNRS UMR 5502, France 3 Université d'Orléans, MAPMO, UMR CNRS 7349, France

Received  December 2016 Revised  August 2017 Published  October 2017

We discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicle density in the situation of congestion. These models are obtained through asymptotic arguments. Hence, we are interested in the simulation of approached models that contain stiff terms and large speeds of propagation. We design schemes intended to apply with relaxed stability conditions.

Citation: Florent Berthelin, Thierry Goudon, Bastien Polizzi, Magali Ribot. Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams. Networks & Heterogeneous Media, 2017, 12 (4) : 591-617. doi: 10.3934/nhm.2017024
##### References:
 [1] A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278, doi: 10.1137/S0036139900380955. [2] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099. [3] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Disc. Cont. Dyn. Syst.-B, 19 (2014), 1869-1888. doi: 10.3934/dcdsb.2014.19.1869. [4] N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677. [5] S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications, Oxford Mathematical Monographs, Oxford University Press, 2007. [6] F. Berthelin and D. Broizat, A model for the evolution of traffic jams in multilane, Kinetic and Related Models, 5 (2012), 697-728. doi: 10.3934/krm.2012.5.697. [7] F. Berthelin, T. Goudon and S. Minjeaud, Multifluid flows: A kinetic approach, J. Sci. Comput., 66 (2016), 792-824. doi: 10.1007/s10915-015-0044-1. [8] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Rational Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9. [9] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Frontiers in math., Birkhäuser, 2004. doi: 10.1007/b93802. [10] F. Bouchut, S. Jin and X. Li, Numerical approximations of pressureless and isothermal gas dynamics, SIAM J. Numer. Anal., 41 (2003), 135-158 (electronic). doi: 10.1137/S0036142901398040. [11] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353. [12] C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551. doi: 10.4310/CMS.2007.v5.n3.a2. [13] C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interfaces Free Bound., 10 (2008), 197-221. doi: 10.4171/IFB/186. [14] P. Colella, Glimm's method for gas dynamics, SIAM J. Sci. Statist. Comput., 3 (1982), 76-110. doi: 10.1137/0903007. [15] C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B: Methodological, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z. [16] P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinet. Relat. Models, 1 (2008), 279-293. doi: 10.3934/krm.2008.1.279. [17] P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys., 10 (2011), 1-31. doi: 10.4208/cicp.210709.210610a. [18] P. Degond and J. Hua, Self-organized hydrodynamics with congestion and path formation in crowds, J. Comput. Phys., 237 (2013), 299-319. doi: 10.1016/j.jcp.2012.11.033. [19] P. Degond, J. Hua and L. Navoret, Numerical simulations of the Euler system with congestion constraint, J. Comput. Phys., 230 (2011), 8057-8088. doi: 10.1016/j.jcp.2011.07.010. [20] D. C. Gazis, R. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Research, 9 (1961), 545-567. doi: 10.1287/opre.9.4.545. [21] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Applied Math., 18 (1965), 697-715. doi: 10.1002/cpa.3160180408. [22] E. Grenier, Existence globale pour le systéme des gaz sans pression, Comptes Rendus Acad. Sci., 321 (1995), 171-174. [23] J. Jung, Schémas numériques adaptés aux accélérateurs multicoeurs pour les écoulements bifluides, PhD thesis, Univ. Strasbourg, 2014. [24] A.-Y. Le Roux, Stability for some equations of gas dynamics, Math. Comput., 37 (1981), 307-320. doi: 10.1090/S0025-5718-1981-0628697-8. [25] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [26] P. -L. Lions and N. Masmoudi, On a free boundary barotropic model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 373-410, doi: 10.1016/S0294-1449(99)80018-3. [27] T. P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys., 57 (1977), 135-148. doi: 10.1007/BF01625772. [28] B. Maury and A. Preux, Pressureless Euler equations with maximal density constraint: A time-splitting scheme, Technical report, Université Paris-Sud, 2017, 333-355, Available on https://hal.archives-ouvertes.fr/hal-01224008. doi: 10.1515/9783110430417-014. [29] P. Nelson and A. Sopasakis, The Prigogine-Herman kinetic model predicts widely scattered traffic flow data at high concentrations, Transportation Research Part B: Methodological, 32 (1998), 589-604. doi: 10.1016/S0191-2615(98)00020-4. [30] S. L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis, Transportation Research, 9 (1975), 225-235. doi: 10.1016/0041-1647(75)90063-5. [31] H. J. Payne, Freflo: A Macroscopic Simulation Model of Freeway Traffic, Transportation Research Record. [32] I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing, 1971. [33] G. Puppo, M. Semplice, A. Tosin and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Communications in Mathematical Sciences, 14 (2016), 643-669. doi: 10.4310/CMS.2016.v14.n3.a3. [34] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [35] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edition, Springer-Verlag, Berlin, 2009. doi: 10.1007/b79761. [36] R. Wegener and A. Klar, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport Theory and Stat. Phys., 25 (1996), 785-798. doi: 10.1080/00411459608203547. [37] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.

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##### References:
 [1] A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278, doi: 10.1137/S0036139900380955. [2] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099. [3] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Disc. Cont. Dyn. Syst.-B, 19 (2014), 1869-1888. doi: 10.3934/dcdsb.2014.19.1869. [4] N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677. [5] S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications, Oxford Mathematical Monographs, Oxford University Press, 2007. [6] F. Berthelin and D. Broizat, A model for the evolution of traffic jams in multilane, Kinetic and Related Models, 5 (2012), 697-728. doi: 10.3934/krm.2012.5.697. [7] F. Berthelin, T. Goudon and S. Minjeaud, Multifluid flows: A kinetic approach, J. Sci. Comput., 66 (2016), 792-824. doi: 10.1007/s10915-015-0044-1. [8] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Rational Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9. [9] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Frontiers in math., Birkhäuser, 2004. doi: 10.1007/b93802. [10] F. Bouchut, S. Jin and X. Li, Numerical approximations of pressureless and isothermal gas dynamics, SIAM J. Numer. Anal., 41 (2003), 135-158 (electronic). doi: 10.1137/S0036142901398040. [11] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353. [12] C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551. doi: 10.4310/CMS.2007.v5.n3.a2. [13] C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interfaces Free Bound., 10 (2008), 197-221. doi: 10.4171/IFB/186. [14] P. Colella, Glimm's method for gas dynamics, SIAM J. Sci. Statist. Comput., 3 (1982), 76-110. doi: 10.1137/0903007. [15] C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B: Methodological, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z. [16] P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinet. Relat. Models, 1 (2008), 279-293. doi: 10.3934/krm.2008.1.279. [17] P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys., 10 (2011), 1-31. doi: 10.4208/cicp.210709.210610a. [18] P. Degond and J. Hua, Self-organized hydrodynamics with congestion and path formation in crowds, J. Comput. Phys., 237 (2013), 299-319. doi: 10.1016/j.jcp.2012.11.033. [19] P. Degond, J. Hua and L. Navoret, Numerical simulations of the Euler system with congestion constraint, J. Comput. Phys., 230 (2011), 8057-8088. doi: 10.1016/j.jcp.2011.07.010. [20] D. C. Gazis, R. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Research, 9 (1961), 545-567. doi: 10.1287/opre.9.4.545. [21] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Applied Math., 18 (1965), 697-715. doi: 10.1002/cpa.3160180408. [22] E. Grenier, Existence globale pour le systéme des gaz sans pression, Comptes Rendus Acad. Sci., 321 (1995), 171-174. [23] J. Jung, Schémas numériques adaptés aux accélérateurs multicoeurs pour les écoulements bifluides, PhD thesis, Univ. Strasbourg, 2014. [24] A.-Y. Le Roux, Stability for some equations of gas dynamics, Math. Comput., 37 (1981), 307-320. doi: 10.1090/S0025-5718-1981-0628697-8. [25] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [26] P. -L. Lions and N. Masmoudi, On a free boundary barotropic model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 373-410, doi: 10.1016/S0294-1449(99)80018-3. [27] T. P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys., 57 (1977), 135-148. doi: 10.1007/BF01625772. [28] B. Maury and A. Preux, Pressureless Euler equations with maximal density constraint: A time-splitting scheme, Technical report, Université Paris-Sud, 2017, 333-355, Available on https://hal.archives-ouvertes.fr/hal-01224008. doi: 10.1515/9783110430417-014. [29] P. Nelson and A. Sopasakis, The Prigogine-Herman kinetic model predicts widely scattered traffic flow data at high concentrations, Transportation Research Part B: Methodological, 32 (1998), 589-604. doi: 10.1016/S0191-2615(98)00020-4. [30] S. L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis, Transportation Research, 9 (1975), 225-235. doi: 10.1016/0041-1647(75)90063-5. [31] H. J. Payne, Freflo: A Macroscopic Simulation Model of Freeway Traffic, Transportation Research Record. [32] I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing, 1971. [33] G. Puppo, M. Semplice, A. Tosin and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Communications in Mathematical Sciences, 14 (2016), 643-669. doi: 10.4310/CMS.2016.v14.n3.a3. [34] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [35] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edition, Springer-Verlag, Berlin, 2009. doi: 10.1007/b79761. [36] R. Wegener and A. Klar, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport Theory and Stat. Phys., 25 (1996), 785-798. doi: 10.1080/00411459608203547. [37] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.
Numerical results in the case of transport (15) - (16). Density (left) and velocity (right) with the Glimm scheme (top) and the explicit-implicit scheme (bottom). The results are given for the three different pressures under consideration: pressure (VO1) in black, pressure (VO2) in blue and pressure (VO3) in green. The exact solution is plotted in red
Numerical results in the case of decongestion (17) - (18). Comparison of the two schemes. Density (left) and velocity (right) at final time $T=0.2$. Top: pressure (VO1) with Glimm scheme (black). On the top, the simulations are performed implicit-explicit scheme (blue) and pressure (VO2) with implicit-explicit scheme (green). Bottom: pressure (VO3) with Glimm scheme (black) and with the explicit-implicit scheme (blue). The exact solution is plotted in red. Parameters : $\gamma=2, \varepsilon =10^{-3}$ for pressures (VO1) and (VO2) and $\gamma=4$ for pressure (VO3)
Numerical results in the case of decongestion (17) - (18). Pressure (VO3) for different values of $\gamma$. Density (left) and velocity (right) at final time $T=0.2$: $\gamma=4$ (black), $\gamma=20$ (blue) and $\gamma=100$ (green). The simulations are performed with the implicit-explicit scheme and the exact solution is plotted in red
Numerical results in the case of decongestion (17) - (18). Pressure (VO2) for different values of $\varepsilon$ and $\gamma$. Density (left) and velocity (right) at final time $T=0.2$: (VO2) for $\gamma=2, \varepsilon =10^{-3}$ (black), $\gamma=2, \varepsilon =10^{-5}$ (blue), $\gamma=3, \varepsilon =10^{-5}$ (green) and $\gamma=3, \varepsilon =10^{-7}$ (orange). The simulations are performed with the implicit-explicit scheme and the exact solution is plotted in red
Numerical results in the case of congestion -Comparison of the two schemes. Density (left) and velocity (right) at final time $T=0.01$. Glimm scheme (top) and implicit-explicit scheme (bottom), with pressure (VO1) (black), (VO2) (blue) and (VO3) (green). The exact solution is plotted in red. Parameters : $\gamma=2, \varepsilon =10^{-3}$ for pressures (VO1) and (VO2) and $\gamma=4$ for pressure (VO3)
Numerical results in the case of congestion -Comparison of the two schemes -Different parameters. For (VO1) and (VO2), we use $\gamma=2$ and $\varepsilon =10^{-5}$; for (VO3), we take $\gamma=50$. Pressure (VO1) in black, pressure (VO2) in blue and pressure (VO3) in green. The exact solution is plotted in red
Numerical results in the case of congestion -Pressure (VO3) for different values of $\gamma$. Density (left) and velocity (right) at final time $T=0.01$. Pressure (VO3) for $\gamma=20$ (black), $\gamma=50$ (blue) and $\gamma=100$ (green). The simulations are performed with the Glimm scheme (top) and the mplicit-explicit scheme (bottom). The exact solution is plotted in red
Numerical results in the case of congestion -Pressure (VO2) for different values of $\gamma$ and $\varepsilon$. Density (left) and velocity (right) at final time $T=0.01$. We compare the pressure (VO2) for $\gamma=2, \varepsilon =10^{-5}$ (black), $\gamma=3, \varepsilon =10^{-5}$ (blue) and $\gamma=3, \varepsilon =10^{-7}$ (green). The simulations are performed with the Glimm scheme (top) and the implicit-explicit scheme (bottom). The exact solution is plotted in red
Numerical results in the case of a shock between two blocks -Pressure (VO1) for $\gamma=2$ and different values of $\varepsilon$: $\varepsilon =10^{-3}$ (black), $\varepsilon =10^{-4}$ (blue), $\varepsilon =10^{-5}$ (green) and $\varepsilon =10^{-6}$ (orange). Figure 9a (on the left) represents the densities whereas figure 9b (on the right) represents the velocities. The exact solution is plotted in red
Numerical results in the case of a shock between two blocks -Pressure VO2 for $\gamma=2$ and different values of $\varepsilon$: $\varepsilon =10^{-3}$ (black), $\varepsilon =10^{-4}$ (blue), $\varepsilon =10^{-5}$ (green) and $\varepsilon =10^{-6}$ (orange). On top, simulations are performed with the Glimm scheme and on the bottom, with the implicit-explicit scheme. We display the densities on the left and the velocities on the right. The exact solution is plotted in red
Numerical results in the case of a shock between two blocks -Pressure VO3 for different values of $\gamma$: $\gamma=16$ (black), $\gamma=32$ (blue), $\gamma=64$ (green) and $\gamma=128$ (orange). On top, simulations are performed with the Glimm scheme and on the bottom, with the implicit-explicit scheme. We display the densities on the left and the velocities on the right. The exact solution is plotted in red
Comparaison of the numerical scheme and velocity offset for the initial data (AI). We display the densities on the left and the velocities on the right, at $t=0.2$ (top), $t=0.4$ (middle) and $t=0.6$ (bottom). Glimm scheme with pressures (VO1) (in black), (VO2) (in green) and (VO3) (in purple); implicit-explicit scheme with pressures (VO1) (in blue), (VO2) (in orange) and (VO3) (in grey). The exact solution is plotted in red
Comparaison of the numerical scheme and velocity offset for the initial data (AⅢ). We display the densities on the left and the velocities on the right, at $t=0.27$ (top), $t=0.53$ (middle) and $t=0.8$ (bottom). Glimm scheme with pressures (VO1) (in black), (VO2) (in green) and (VO3) (in purple); implicit-explicit scheme with pressures (VO1) (in blue), (VO2) (in orange) and (VO3) (in grey). The exact solution is plotted in red
Time steps -comparison between Glimm scheme and the explicit-implicit scheme for the congestion case. Pressure (VO3) for different values of γ and pressure (VO2) for γ = 2 and different values of ". The time step is the smallest time step used during the simulation and the factor is the ratio of the time step for the explicit-implicit scheme over the time step for the Glimm scheme
 Pressure & param Time step Glimm scheme Time step explicit-implicit scheme Factor Pressure (VO2), ε = 10−4 ∆t = 2·10−6 ∆t = 2·10−6 1 Pressure (VO2), ε = 10−5 ∆t = 7·10−7 ∆t = 10−6 1.39 Pressure (VO2), ε = 10−6 ∆t = 2·10−7 ∆t = 7:7·10−7 3.22 Pressure (VO2), ε = 10−7 ∆t = 7:5·10−8 ∆t = 6:2·10−7 8.18 Pressure (VO3), γ = 50 ∆t = 9·10−6 ∆t = 10−5 1.12 Pressure (VO3), γ = 100 ∆t = 4:8·10−6 ∆t = 6:4·10−6 1.36 Pressure (VO3), γ = 200 ∆t = 2:4·10−6 ∆t = 5:6·10−6 2.33 Pressure (VO3), γ = 500 ∆t = 9:5·10−7 ∆t = 2:7·10−5 27.95
 Pressure & param Time step Glimm scheme Time step explicit-implicit scheme Factor Pressure (VO2), ε = 10−4 ∆t = 2·10−6 ∆t = 2·10−6 1 Pressure (VO2), ε = 10−5 ∆t = 7·10−7 ∆t = 10−6 1.39 Pressure (VO2), ε = 10−6 ∆t = 2·10−7 ∆t = 7:7·10−7 3.22 Pressure (VO2), ε = 10−7 ∆t = 7:5·10−8 ∆t = 6:2·10−7 8.18 Pressure (VO3), γ = 50 ∆t = 9·10−6 ∆t = 10−5 1.12 Pressure (VO3), γ = 100 ∆t = 4:8·10−6 ∆t = 6:4·10−6 1.36 Pressure (VO3), γ = 200 ∆t = 2:4·10−6 ∆t = 5:6·10−6 2.33 Pressure (VO3), γ = 500 ∆t = 9:5·10−7 ∆t = 2:7·10−5 27.95
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