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

Properties of the LWR model with time delay

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  March 2021 Early access  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.

Citation: Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks and Heterogeneous Media, 2021, 16 (1) : 31-47. doi: 10.3934/nhm.2020032
References:
[1]

A. AwA. KlarM. 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.

[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]

M. BandoK. HasebeA. NakayamaA. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Physical review E, 51 (1995), 10-35. 

[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.

[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.

[6]

S. BlandinD. WorkP. GoatinB. 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.

[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.

[8]

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

[9]

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

[10]

M. BurgerS. 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.

[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.

[12]

F. CamilliR. 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.

[13]

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

[14]

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

[15]

R. M. ColomboA. 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.

[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.

[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.

[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.

[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.

[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. 

[21]

M. Di FrancescoS. 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.

[22]

M. R. FlynnA. R. KasimovJ.-C. NaveR. 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.

[23]

J. FriedrichO. 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.

[24]

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

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[37]

B. G. RosV. L. KnoopB. 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. 

[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.

[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.

[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.

[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.

[42]

A. TordeuxG. CostesequeM. 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.

[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.

[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.

[45]

G. ViscontiM. HertyG. 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.

[46]

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

[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. 

show all references

References:
[1]

A. AwA. KlarM. 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.

[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]

M. BandoK. HasebeA. NakayamaA. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Physical review E, 51 (1995), 10-35. 

[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.

[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.

[6]

S. BlandinD. WorkP. GoatinB. 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.

[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.

[8]

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

[9]

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

[10]

M. BurgerS. 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.

[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.

[12]

F. CamilliR. 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.

[13]

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

[14]

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

[15]

R. M. ColomboA. 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.

[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.

[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.

[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.

[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.

[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. 

[21]

M. Di FrancescoS. 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.

[22]

M. R. FlynnA. R. KasimovJ.-C. NaveR. 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.

[23]

J. FriedrichO. 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.

[24]

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

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[37]

B. G. RosV. L. KnoopB. 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. 

[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.

[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.

[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.

[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.

[42]

A. TordeuxG. CostesequeM. 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.

[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.

[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.

[45]

G. ViscontiM. HertyG. 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.

[46]

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

[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. 

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