American Institute of Mathematical Sciences

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March  2021, 16(1): 31-47. doi: 10.3934/nhm.2020032

Properties of the LWR model with time delay

Simone Göttlich 1, , Elisa Iacomini 1,, and  Thomas Jung 2,

1. 

University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany
2. 

Fraunhofer Institute ITWM, 67663 Kaiserslautern, Germany

* Corresponding author: Elisa Iacomini

Received  March 2020 Revised  October 2020 Published  December 2020

In this article, we investigate theoretical and numerical properties of the first-order Lighthill-Whitham-Richards (LWR) traffic flow model with time delay. Since standard results from the literature are not directly applicable to the delayed model, we mainly focus on the numerical analysis of the proposed finite difference discretization. The simulation results also show that the delay model is able to capture Stop & Go waves.

Keywords: Macroscopic traffic flow models, hyperbolic delay partial differential equation, Lax-Friedrichs discretization, numerical simulations, stop & go waves.
Mathematics Subject Classification: 35L65, 90B20, 65M06.
Citation: Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2021, 16 (1) : 31-47. doi: 10.3934/nhm.2020032
References:
[1]

A. Aw, A. Klar, M. Rascle and T. Materne, Derivation of continuum flow traffic models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.  Google Scholar

[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.  Google Scholar

[3]

M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Physical review E, 51 (1995), 10-35.   Google Scholar

[4]

C. Bianca, M. Ferrara and L. Guerrini, The time delays' effects on the qualitative behavior of an economic growth model, Abstract and Applied Analysis, 2013 (2013). doi: 10.1155/2013/901014.  Google Scholar

[5]

S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), 217-241.  doi: 10.1007/s00211-015-0717-6.  Google Scholar

[6]

S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM Journal on Applied Mathematics, 71 (2011), 107-127.  doi: 10.1137/090754467.  Google Scholar

[7]

R. Borsche and A. Klar, A nonlinear discrete velocity relaxation model for traffic flow, SIAM Journal on Applied Mathematics, 78 (2018), 2891-2917.  doi: 10.1137/17M1152681.  Google Scholar

[8]

M. Braskstone and M. McDonald, Car following: A historical review, transportation research part f. 2., Pergamon, 2000. Google Scholar

[9]

M. Burger, S. Göttlich and T. Jung, Derivation of a first order traffic flow model of Lighthill-Whitham-Richards type, IFAC-PapersOnLine, 51 (2018), 49-54.   Google Scholar

[10]

M. Burger, S. Göttlich and T. Jung, Derivation of second order traffic flow models with time delays, Netw. Heterog. Media, 14 (2019), 265-288.  doi: 10.3934/nhm.2019011.  Google Scholar

[11]

S. Cacace, F. Camilli, R. De Maio and A. Tosin, A measure theoretic approach to traffic flow optimisation on networks, European Journal of Applied Mathematics, (2018), 1-23. doi: 10.1017/S0956792518000621.  Google Scholar

[12]

F. Camilli, R. De Maio and A. Tosin, Measure-valued solutions to nonlocal transport equations on networks, Journal of Differential Equations, 264 (2018), 7213-7241.  doi: 10.1016/j.jde.2018.02.015.  Google Scholar

[13]

R. E. Chandler, R. Herman and E. W. Montroll, Traffic dynamics: Studies in car following, Operations Research, 6 (1958), 165-184.  doi: 10.1287/opre.6.2.165.  Google Scholar

[14]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM Journal on Applied Mathematics, 63 (2003), 708-721.  doi: 10.1137/S0036139901393184.  Google Scholar

[15]

R. M. Colombo, A. Corli and M. D. Rosini, Non local balance laws in traffic models and crystal growth, ZAMM Z. Angew. Math. Mech., 87 (2007), 449-461.  doi: 10.1002/zamm.200710327.  Google Scholar

[16]

R. M. Colombo and F. Marcellini, A mixed ode-pde model for vehicular traffic, Mathematical Methods in the Applied Sciences, 38 (2015), 1292-1302.  doi: 10.1002/mma.3146.  Google Scholar

[17]

R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, Rendiconti del Seminario Matematico della Università di Padova, 131 (2014), 217-236. doi: 10.4171/RSMUP/131-13.  Google Scholar

[18]

E. Cristiani and E. Iacomini, An interface-free multi-scale multi-order model for traffic flow, Discrete & Continuous Dynamical Systems-Series B, 25 (2019). doi: 10.3934/dcdsb.2019135.  Google Scholar

[19]

E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, Netw. Heterog. Media, 11 (2016), 395-413.  doi: 10.3934/nhm.2016002.  Google Scholar

[20]

R. V. Culshaw and S. Ruan, A delay-differential equation model of hiv infection of cd4+ t-cells, Mathematical Biosciences, 165 (2000), 27-39.   Google Scholar

[21]

M. Di Francesco, S. Fagioli and M. D. Rosini, Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic, Math. Biosci. Eng., 14 (2017), 127-141.  doi: 10.3934/mbe.2017009.  Google Scholar

[22]

M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales and B. Seibold, Self-sustained nonlinear waves in traffic flow, Phys. Rev. E, 79 (2009), 56-113.  doi: 10.1103/PhysRevE.79.056113.  Google Scholar

[23]

J. Friedrich, O. Kolb and S. Göttlich, A Godunov type scheme for a class of LWR traffic flow models with non-local flux, Netw. Heterog. Media, 13 (2018), 531-547.  doi: 10.3934/nhm.2018024.  Google Scholar

[24]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 2006.  Google Scholar

[25]

P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107-121.  doi: 10.3934/nhm.2016.11.107.  Google Scholar

[26]

D. Green Jr. and H. W. Stech, Diffusion and hereditary effects in a class of population models, in Differential Equations and Applications in Ecology, Epidemics, and Population Problems, Elsevier, 1981, 19-28.  Google Scholar

[27]

B. D. Greenshields, A study in highway capacity, Highway Research Board Proc., 1935, (1935), 448-477. Google Scholar

[28]

M. Herty, G. Puppo, S. Roncoroni and G. Visconti, The bgk approximation of kinetic models for traffic, Kinetic & Related Models, 13 (2020), 279-307.  doi: 10.3934/krm.2020010.  Google Scholar

[29]

A. Keimer and L. Pflug, Nonlocal conservation laws with time delay, Nonlinear Differential Equations and Applications NoDEA, 26 (2019), 54. doi: 10.1007/s00030-019-0597-z.  Google Scholar

[30]

W. Kwon and A. Pearson, Feedback stabilization of linear systems with delayed control, IEEE Transactions on Automatic control, 25 (1980), 266-269.  doi: 10.1109/TAC.1980.1102288.  Google Scholar

[31]

M. J. Lighthill and G. B. Whitham, On kinematic waves Ⅱ. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[32]

G. F. Newell, Nonlinear effects in the dynamics of car following, Operations Research, 9 (1961), 209-229.  doi: 10.1287/opre.9.2.209.  Google Scholar

[33]

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Encyclopedia of Complexity and Systems Science, Springer, New York, NY, 2009, 9727-9749. doi: 10.1007/978-1-4614-1806-1_112.  Google Scholar

[34]

G. Puppo, M. Semplice, A. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinetic & Related Models, 10 (2016), 823. doi: 10.3934/krm.2017033.  Google Scholar

[35]

A. D. Rey and C. Mackey, Multistability and boundary layer development in a transport equation, Canadian Applied Mathematics Quarterly, (1993), 61-81. Google Scholar

[36]

P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

[37]

B. G. Ros, V. L. Knoop, B. van Arem and S. P. Hoogendoorn, Empirical analysis of the causes of stop-and-go waves at sags, IET Intell. Transp. Syst., 8 (2014), 499-506.   Google Scholar

[38]

M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer International Publishing, 2013. doi: 10.1007/978-3-319-00155-5.  Google Scholar

[39]

H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, 57, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[40]

J. Song and S. Karni, A second order traffic flow model with lane changing, Journal of Scientific Computing, 81 (2019), 1429-1445.  doi: 10.1007/s10915-019-01023-z.  Google Scholar

[41]

R. E. Stern, et al., Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transportation Res. Part C, 89 (2018), 205-221. Google Scholar

[42]

A. Tordeux, G. Costeseque, M. Herty and A. Seyfried, From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models, SIAM Journal on Applied Mathematics, 78 (2018), 63-79.  doi: 10.1137/16M110695X.  Google Scholar

[43]

M. Treiber, A. Hennecke and D. Helbing, Derivation, properties, and simulation of a gas-kinetic-based, nonlocal traffic model, Physical Review E, 59 (1999), 239. Google Scholar

[44]

M. Treiber and A. Kesting, Traffic flow dynamics, in Traffic Flow Dynamics: Data, Models and Simulation, Springer-Verlag Berlin Heidelberg, 2013. doi: 10.1007/978-3-642-32460-4.  Google Scholar

[45]

G. Visconti, M. Herty, G. Puppo and A. Tosin, Multivalued fundamental diagrams of traffic flow in the kinetic Fokker-Planck limit, Multiscale Model. Simul., 15 (2017), 1267-1293.  doi: 10.1137/16M1087035.  Google Scholar

[46]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.   Google Scholar

[47]

Y. Zhao and H. M. Zhang, A unified follow-the-leader model for vehicle, bicycle and pedestrian traffic, Transportation Res. Part B, 105 (2017), 315-327.   Google Scholar

show all references

References:
[1]

A. Aw, A. Klar, M. Rascle and T. Materne, Derivation of continuum flow traffic models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.  Google Scholar

[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.  Google Scholar

[3]

M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Physical review E, 51 (1995), 10-35.   Google Scholar

[4]

C. Bianca, M. Ferrara and L. Guerrini, The time delays' effects on the qualitative behavior of an economic growth model, Abstract and Applied Analysis, 2013 (2013). doi: 10.1155/2013/901014.  Google Scholar

[5]

S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), 217-241.  doi: 10.1007/s00211-015-0717-6.  Google Scholar

[6]

S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM Journal on Applied Mathematics, 71 (2011), 107-127.  doi: 10.1137/090754467.  Google Scholar

[7]

R. Borsche and A. Klar, A nonlinear discrete velocity relaxation model for traffic flow, SIAM Journal on Applied Mathematics, 78 (2018), 2891-2917.  doi: 10.1137/17M1152681.  Google Scholar

[8]

M. Braskstone and M. McDonald, Car following: A historical review, transportation research part f. 2., Pergamon, 2000. Google Scholar

[9]

M. Burger, S. Göttlich and T. Jung, Derivation of a first order traffic flow model of Lighthill-Whitham-Richards type, IFAC-PapersOnLine, 51 (2018), 49-54.   Google Scholar

[10]

M. Burger, S. Göttlich and T. Jung, Derivation of second order traffic flow models with time delays, Netw. Heterog. Media, 14 (2019), 265-288.  doi: 10.3934/nhm.2019011.  Google Scholar

[11]

S. Cacace, F. Camilli, R. De Maio and A. Tosin, A measure theoretic approach to traffic flow optimisation on networks, European Journal of Applied Mathematics, (2018), 1-23. doi: 10.1017/S0956792518000621.  Google Scholar

[12]

F. Camilli, R. De Maio and A. Tosin, Measure-valued solutions to nonlocal transport equations on networks, Journal of Differential Equations, 264 (2018), 7213-7241.  doi: 10.1016/j.jde.2018.02.015.  Google Scholar

[13]

R. E. Chandler, R. Herman and E. W. Montroll, Traffic dynamics: Studies in car following, Operations Research, 6 (1958), 165-184.  doi: 10.1287/opre.6.2.165.  Google Scholar

[14]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM Journal on Applied Mathematics, 63 (2003), 708-721.  doi: 10.1137/S0036139901393184.  Google Scholar

[15]

R. M. Colombo, A. Corli and M. D. Rosini, Non local balance laws in traffic models and crystal growth, ZAMM Z. Angew. Math. Mech., 87 (2007), 449-461.  doi: 10.1002/zamm.200710327.  Google Scholar

[16]

R. M. Colombo and F. Marcellini, A mixed ode-pde model for vehicular traffic, Mathematical Methods in the Applied Sciences, 38 (2015), 1292-1302.  doi: 10.1002/mma.3146.  Google Scholar

[17]

R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, Rendiconti del Seminario Matematico della Università di Padova, 131 (2014), 217-236. doi: 10.4171/RSMUP/131-13.  Google Scholar

[18]

E. Cristiani and E. Iacomini, An interface-free multi-scale multi-order model for traffic flow, Discrete & Continuous Dynamical Systems-Series B, 25 (2019). doi: 10.3934/dcdsb.2019135.  Google Scholar

[19]

E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, Netw. Heterog. Media, 11 (2016), 395-413.  doi: 10.3934/nhm.2016002.  Google Scholar

[20]

R. V. Culshaw and S. Ruan, A delay-differential equation model of hiv infection of cd4+ t-cells, Mathematical Biosciences, 165 (2000), 27-39.   Google Scholar

[21]

M. Di Francesco, S. Fagioli and M. D. Rosini, Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic, Math. Biosci. Eng., 14 (2017), 127-141.  doi: 10.3934/mbe.2017009.  Google Scholar

[22]

M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales and B. Seibold, Self-sustained nonlinear waves in traffic flow, Phys. Rev. E, 79 (2009), 56-113.  doi: 10.1103/PhysRevE.79.056113.  Google Scholar

[23]

J. Friedrich, O. Kolb and S. Göttlich, A Godunov type scheme for a class of LWR traffic flow models with non-local flux, Netw. Heterog. Media, 13 (2018), 531-547.  doi: 10.3934/nhm.2018024.  Google Scholar

[24]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 2006.  Google Scholar

[25]

P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107-121.  doi: 10.3934/nhm.2016.11.107.  Google Scholar

[26]

D. Green Jr. and H. W. Stech, Diffusion and hereditary effects in a class of population models, in Differential Equations and Applications in Ecology, Epidemics, and Population Problems, Elsevier, 1981, 19-28.  Google Scholar

[27]

B. D. Greenshields, A study in highway capacity, Highway Research Board Proc., 1935, (1935), 448-477. Google Scholar

[28]

M. Herty, G. Puppo, S. Roncoroni and G. Visconti, The bgk approximation of kinetic models for traffic, Kinetic & Related Models, 13 (2020), 279-307.  doi: 10.3934/krm.2020010.  Google Scholar

[29]

A. Keimer and L. Pflug, Nonlocal conservation laws with time delay, Nonlinear Differential Equations and Applications NoDEA, 26 (2019), 54. doi: 10.1007/s00030-019-0597-z.  Google Scholar

[30]

W. Kwon and A. Pearson, Feedback stabilization of linear systems with delayed control, IEEE Transactions on Automatic control, 25 (1980), 266-269.  doi: 10.1109/TAC.1980.1102288.  Google Scholar

[31]

M. J. Lighthill and G. B. Whitham, On kinematic waves Ⅱ. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[32]

G. F. Newell, Nonlinear effects in the dynamics of car following, Operations Research, 9 (1961), 209-229.  doi: 10.1287/opre.9.2.209.  Google Scholar

[33]

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Encyclopedia of Complexity and Systems Science, Springer, New York, NY, 2009, 9727-9749. doi: 10.1007/978-1-4614-1806-1_112.  Google Scholar

[34]

G. Puppo, M. Semplice, A. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinetic & Related Models, 10 (2016), 823. doi: 10.3934/krm.2017033.  Google Scholar

[35]

A. D. Rey and C. Mackey, Multistability and boundary layer development in a transport equation, Canadian Applied Mathematics Quarterly, (1993), 61-81. Google Scholar

[36]

P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

[37]

B. G. Ros, V. L. Knoop, B. van Arem and S. P. Hoogendoorn, Empirical analysis of the causes of stop-and-go waves at sags, IET Intell. Transp. Syst., 8 (2014), 499-506.   Google Scholar

[38]

M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer International Publishing, 2013. doi: 10.1007/978-3-319-00155-5.  Google Scholar

[39]

H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, 57, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[40]

J. Song and S. Karni, A second order traffic flow model with lane changing, Journal of Scientific Computing, 81 (2019), 1429-1445.  doi: 10.1007/s10915-019-01023-z.  Google Scholar

[41]

R. E. Stern, et al., Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transportation Res. Part C, 89 (2018), 205-221. Google Scholar

[42]

A. Tordeux, G. Costeseque, M. Herty and A. Seyfried, From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models, SIAM Journal on Applied Mathematics, 78 (2018), 63-79.  doi: 10.1137/16M110695X.  Google Scholar

[43]

M. Treiber, A. Hennecke and D. Helbing, Derivation, properties, and simulation of a gas-kinetic-based, nonlocal traffic model, Physical Review E, 59 (1999), 239. Google Scholar

[44]

M. Treiber and A. Kesting, Traffic flow dynamics, in Traffic Flow Dynamics: Data, Models and Simulation, Springer-Verlag Berlin Heidelberg, 2013. doi: 10.1007/978-3-642-32460-4.  Google Scholar

[45]

G. Visconti, M. Herty, G. Puppo and A. Tosin, Multivalued fundamental diagrams of traffic flow in the kinetic Fokker-Planck limit, Multiscale Model. Simul., 15 (2017), 1267-1293.  doi: 10.1137/16M1087035.  Google Scholar

[46]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.   Google Scholar

[47]

Y. Zhao and H. M. Zhang, A unified follow-the-leader model for vehicle, bicycle and pedestrian traffic, Transportation Res. Part B, 105 (2017), 315-327.   Google Scholar

Figure 1.  Comparison between density profiles computed with (right) and without(left) CFL condition in case of a rarefaction wave
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Figure 2.  Comparison between density profiles computed with (right) and without (left) CFL condition in a shock framework
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Figure 3.  Test 0: Comparing the density evolution computed by the delayed model (left) and the LWR model (right)
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Figure 4.  Test 0: Comparison between density profiles corresponding to different grid steps size
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Figure 5.  Test 0: Density evolution and profile, at time $ T = \frac{1}{3}T_f $, in case of a too high delay
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7], with $ \rho^0(x,1) $ on the left and $ \rho^0(x,2) $ on the right">Figure 6.  Test 1: Reproducing the simulation presented in [7], with $ \rho^0(x,1) $ on the left and $ \rho^0(x,2) $ on the right
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Figure 7.  Test 2: Density values in the $ (x,t) $-plane (left) and density profile at time $ T = T_f $ (right)
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Figure 8.  Test 2: Density values in the $ (x,t) $-plane with low delay term, $ T_\Delta = 4 \Delta t $, (left) and density profile at $ T = T_f $ (right)
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Figure 9.  Triggering of Stop & Go waves: Density values in the $ (x,t) $-plane
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