
-
Previous Article
Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $
- NHM Home
- This Issue
-
Next Article
A 2-dimensional shape optimization problem for tree branches
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 |
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.
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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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 |








[1] |
Michael Herty, Reinhard Illner. On Stop-and-Go waves in dense traffic. Kinetic & Related Models, 2008, 1 (3) : 437-452. doi: 10.3934/krm.2008.1.437 |
[2] |
Michael Herty, Reinhard Illner. Analytical and numerical investigations of refined macroscopic traffic flow models. Kinetic & Related Models, 2010, 3 (2) : 311-333. doi: 10.3934/krm.2010.3.311 |
[3] |
John D. Towers. The Lax-Friedrichs scheme for interaction between the inviscid Burgers equation and multiple particles. Networks & Heterogeneous Media, 2020, 15 (1) : 143-169. doi: 10.3934/nhm.2020007 |
[4] |
Paolo Baiti, Alberto Bressan, Helge Kristian Jenssen. Instability of travelling wave profiles for the Lax-Friedrichs scheme. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 877-899. doi: 10.3934/dcds.2005.13.877 |
[5] |
Hermano Frid. Invariant regions under Lax-Friedrichs scheme for multidimensional systems of conservation laws. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 585-593. doi: 10.3934/dcds.1995.1.585 |
[6] |
Paola Goatin, Elena Rossi. Comparative study of macroscopic traffic flow models at road junctions. Networks & Heterogeneous Media, 2020, 15 (2) : 261-279. doi: 10.3934/nhm.2020012 |
[7] |
Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161 |
[8] |
Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571 |
[9] |
Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351 |
[10] |
Marte Godvik, Harald Hanche-Olsen. Car-following and the macroscopic Aw-Rascle traffic flow model. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 279-303. doi: 10.3934/dcdsb.2010.13.279 |
[11] |
Mohamed Benyahia, Massimiliano D. Rosini. A macroscopic traffic model with phase transitions and local point constraints on the flow. Networks & Heterogeneous Media, 2017, 12 (2) : 297-317. doi: 10.3934/nhm.2017013 |
[12] |
Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503 |
[13] |
Helge Holden, Nils Henrik Risebro. Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Networks & Heterogeneous Media, 2018, 13 (3) : 409-421. doi: 10.3934/nhm.2018018 |
[14] |
John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501 |
[15] |
Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9 |
[16] |
Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Numerical approximations of a traffic flow model on networks. Networks & Heterogeneous Media, 2006, 1 (1) : 57-84. doi: 10.3934/nhm.2006.1.57 |
[17] |
Wen Shen, Karim Shikh-Khalil. Traveling waves for a microscopic model of traffic flow. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2571-2589. doi: 10.3934/dcds.2018108 |
[18] |
Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control & Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121 |
[19] |
Rolf Rannacher. A short course on numerical simulation of viscous flow: Discretization, optimization and stability analysis. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1147-1194. doi: 10.3934/dcdss.2012.5.1147 |
[20] |
Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255 |
2019 Impact Factor: 1.053
Tools
Article outline
Figures and Tables
[Back to Top]