July  2019, 39(7): 4001-4040. doi: 10.3934/dcds.2019161

Traveling waves for nonlocal models of traffic flow

Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA

* Corresponding author: Wen Shen

Received  August 2018 Revised  December 2018 Published  April 2019

We consider several nonlocal models for traffic flow, including both microscopic ODE models and macroscopic PDE models. The ODE models describe the movement of individual cars, where each driver adjusts the speed according to the road condition over an interval in the front of the car. These models are known as the FtLs (Follow-the-Leaders) models. The corresponding PDE models, describing the evolution for the density of cars, are conservation laws with nonlocal flux functions. For both types of models, we study stationary traveling wave profiles and stationary discrete traveling wave profiles. (See definitions 1.1 and 1.2, respectively.) We derive delay differential equations satisfied by the profiles for the FtLs models, and delay integro-differential equations for the traveling waves of the nonlocal PDE models. The existence and uniqueness (up to horizontal shifts) of the stationary traveling wave profiles are established. Furthermore, we show that the traveling wave profiles are time asymptotic limits for the corresponding Cauchy problems, under mild assumptions on the smooth initial condition.

Citation: Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161
References:
[1]

A. Aggarwal, R. M. Colombo and P. Goatin, Non-local systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963–983. doi: 10.1137/140975255.  Google Scholar

[2]

A. Aggarwal and P. Goatin, Crowd dynamics through non-local conservation laws, B. Braz. Math. Soc., 47 (2016), 37–50. doi: 10.1007/s00574-016-0120-7.  Google Scholar

[3]

D. Amadori, S. Y. Ha and J. Park, On the global well-posedness of BV weak solutions to the Kuramoto–Sakaguchi equation, J. Differential Equations, 262 (2017), 978–1022. doi: 10.1016/j.jde.2016.10.004.  Google Scholar

[4]

D. Amadori and W. Shen, Front tracking approximations for slow erosion, Dicrete Contin. Dyn. Syst., 32 (2012), 1481–1502. doi: 10.3934/dcds.2012.32.1481.  Google Scholar

[5]

P. Amorim, R. M. Colombo and A. Teixeira, On the numerical integration of scalar non-local conservation laws, EASIM: M2MAN, 49 (2015), 19–37. doi: 10.1051/m2an/2014023.  Google Scholar

[6]

B. ArgallE. CheleshkinJ. M. GreenbergC. Hinde and P. Lin, A rigorous treatment of a follow-the-leader traffic model with traffic lights present, SIAM J. Appl. Math., 63 (2002), 149-168.  doi: 10.1137/S0036139901391215.  Google Scholar

[7]

J. Aubin, Macroscopic traffic models: Shifting from densities to "celerities", Appl. Math. Comput, 217 (2010), 963-971.  doi: 10.1016/j.amc.2010.02.032.  Google Scholar

[8]

F. Betancourt, R. Bürger, K. H. Karlsen and E. M. Tory, On non-local conservation laws modelling sedimentation, Nonlinearity, 24 (2011), 855–885. doi: 10.1088/0951-7715/24/3/008.  Google Scholar

[9]

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

[10]

G.-Q. Chen and C. Christoforou, Solutions for a nonlocal conservation law with fading memory, Proc. Amer. Math. Soc., 135 (2007), 3905-3915.  doi: 10.1090/S0002-9939-07-08942-3.  Google Scholar

[11]

F. A. ChiarelloP. Goatin and E. Rossi, Stability estimates for non-local scalar conservation laws, Nonlinear Analysis: Real World Applications, 45 (2019), 668-687.  doi: 10.1016/j.nonrwa.2018.07.027.  Google Scholar

[12]

R. M. Colombo and M. Lécureux-Mercier, Nonlocal crowd dynamics models for several populations, Acta Math. Sci., 32 (2012), 177–196. doi: 10.1016/S0252-9602(12)60011-3.  Google Scholar

[13]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, Non-local crowd dynamics, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 769–772. doi: 10.1016/j.crma.2011.07.005.  Google Scholar

[14]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp. doi: 10.1142/S0218202511500230.  Google Scholar

[15]

R. M. Colombo, F. Marcellini and E. Rossi, Biological and industrial models motivating nonlocal conservation laws: A review of analytic and numerical results, Netw. Heterog. Media, 11 (2016), 49–67. doi: 10.3934/nhm.2016.11.49.  Google Scholar

[16]

R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, Rend. Semin. Mat. Univ. Padova, 131 (2014), 217-235.  doi: 10.4171/RSMUP/131-13.  Google Scholar

[17]

G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 523–537. doi: 10.1007/s00030-012-0164-3.  Google Scholar

[18]

C. De Filippis and P. Goatin, The initial-boundary value problem for general non-local scalar conservation laws in one space dimension, Nonlinear Anal., 161 (2017), 131-156.  doi: 10.1016/j.na.2017.05.017.  Google Scholar

[19]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871.  doi: 10.1007/s00205-015-0843-4.  Google Scholar

[20]

L. Dong and L. Tong, Shock formation in a traffic flow model with Arrhenius look-ahead dynamics, Netw. Heterog. Media, 6 (2011), 681-694.  doi: 10.3934/nhm.2011.6.681.  Google Scholar

[21]

R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, 20 Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[22]

R. D. Driver and M. D. Rosini, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal., 10 (1962), 401-426.  doi: 10.1007/BF00281203.  Google Scholar

[23]

Q. Du, J. R. Kamm, R. B. Lehoucq and M. L. Parks. A new approach for a nonlocal, nonlinear conservation law, SIAM J. Appl. Math., 72 (2012), 464–487. doi: 10.1137/110833233.  Google Scholar

[24]

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

[25]

P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: Well-posedness and micro-macro limit, Commun. Math. Sci., 15 (2017), 261-287.   Google Scholar

[26]

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

[27]

H. Holden and N. H. Risebro, Continuum limit of Follow-the-Leader models –a short proof, Discrete Contin. Dyn. Syst., 38 (2018), 715-722.  doi: 10.3934/dcds.2018031.  Google Scholar

[28]

H. Holden and N. H. Risebro, Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow, Netw. Heterog. Media, 13 (2018), 409-421.  doi: 10.3934/nhm.2018018.  Google Scholar

[29]

A. Keimer and L. Pflug, Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations, 263 (2017), 4023-4069.  doi: 10.1016/j.jde.2017.05.015.  Google Scholar

[30]

A. KeimerL. Pflug and M. Spinola, Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping, J. Math. Anal. Appl., 466 (2018), 18-55.  doi: 10.1016/j.jmaa.2018.05.013.  Google Scholar

[31]

A. KeimerL. Pflug and M. Spinola, Nonlocal Scalar Conservation Laws on Bounded Domains and Applications in Traffic Flow, SIAM J. Math. Anal., 50 (2018), 6271-6306.  doi: 10.1137/18M119817X.  Google Scholar

[32]

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.  Google Scholar

[33]

R. H. Martin, Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics, Wiley, New York, 1976.  Google Scholar

[34]

E. Rossi, A justification of a LWR model based on a follow the leader description, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 579-591.  doi: 10.3934/dcdss.2014.7.579.  Google Scholar

[35]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[36]

W. Shen, Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition, Netw. Heterog. Media, 13 (2018), 449-478.  doi: 10.3934/nhm.2018020.  Google Scholar

[37]

W. Shen, Nonlocal PDE Models for Traffic Flow on Rough Roads, Preprint, 2018. Google Scholar

[38]

W. Shen and K. Shikh-Khalil, Traveling waves for a microscopic model of traffic flow, DCDS (Discrete and Continuous Dynamical Systems), 38 (2018), 2571-2589.  doi: 10.3934/dcds.2018108.  Google Scholar

[39]

W. Shen and T. Y. Zhang, Erosion profile by a global model for granular flow, Arch. Rational Mech. Anal., 204 (2012), 837-879.  doi: 10.1007/s00205-012-0499-2.  Google Scholar

[40]

K. Zumbrun, On a nonlocal dispersive equation modeling particle suspensions, Q. Appl. Math., 57 (1999), 573–600. doi: 10.1090/qam/1704419.  Google Scholar

show all references

References:
[1]

A. Aggarwal, R. M. Colombo and P. Goatin, Non-local systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963–983. doi: 10.1137/140975255.  Google Scholar

[2]

A. Aggarwal and P. Goatin, Crowd dynamics through non-local conservation laws, B. Braz. Math. Soc., 47 (2016), 37–50. doi: 10.1007/s00574-016-0120-7.  Google Scholar

[3]

D. Amadori, S. Y. Ha and J. Park, On the global well-posedness of BV weak solutions to the Kuramoto–Sakaguchi equation, J. Differential Equations, 262 (2017), 978–1022. doi: 10.1016/j.jde.2016.10.004.  Google Scholar

[4]

D. Amadori and W. Shen, Front tracking approximations for slow erosion, Dicrete Contin. Dyn. Syst., 32 (2012), 1481–1502. doi: 10.3934/dcds.2012.32.1481.  Google Scholar

[5]

P. Amorim, R. M. Colombo and A. Teixeira, On the numerical integration of scalar non-local conservation laws, EASIM: M2MAN, 49 (2015), 19–37. doi: 10.1051/m2an/2014023.  Google Scholar

[6]

B. ArgallE. CheleshkinJ. M. GreenbergC. Hinde and P. Lin, A rigorous treatment of a follow-the-leader traffic model with traffic lights present, SIAM J. Appl. Math., 63 (2002), 149-168.  doi: 10.1137/S0036139901391215.  Google Scholar

[7]

J. Aubin, Macroscopic traffic models: Shifting from densities to "celerities", Appl. Math. Comput, 217 (2010), 963-971.  doi: 10.1016/j.amc.2010.02.032.  Google Scholar

[8]

F. Betancourt, R. Bürger, K. H. Karlsen and E. M. Tory, On non-local conservation laws modelling sedimentation, Nonlinearity, 24 (2011), 855–885. doi: 10.1088/0951-7715/24/3/008.  Google Scholar

[9]

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

[10]

G.-Q. Chen and C. Christoforou, Solutions for a nonlocal conservation law with fading memory, Proc. Amer. Math. Soc., 135 (2007), 3905-3915.  doi: 10.1090/S0002-9939-07-08942-3.  Google Scholar

[11]

F. A. ChiarelloP. Goatin and E. Rossi, Stability estimates for non-local scalar conservation laws, Nonlinear Analysis: Real World Applications, 45 (2019), 668-687.  doi: 10.1016/j.nonrwa.2018.07.027.  Google Scholar

[12]

R. M. Colombo and M. Lécureux-Mercier, Nonlocal crowd dynamics models for several populations, Acta Math. Sci., 32 (2012), 177–196. doi: 10.1016/S0252-9602(12)60011-3.  Google Scholar

[13]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, Non-local crowd dynamics, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 769–772. doi: 10.1016/j.crma.2011.07.005.  Google Scholar

[14]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp. doi: 10.1142/S0218202511500230.  Google Scholar

[15]

R. M. Colombo, F. Marcellini and E. Rossi, Biological and industrial models motivating nonlocal conservation laws: A review of analytic and numerical results, Netw. Heterog. Media, 11 (2016), 49–67. doi: 10.3934/nhm.2016.11.49.  Google Scholar

[16]

R. M. Colombo and E. Rossi, On the micro-macro limit in traffic flow, Rend. Semin. Mat. Univ. Padova, 131 (2014), 217-235.  doi: 10.4171/RSMUP/131-13.  Google Scholar

[17]

G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 523–537. doi: 10.1007/s00030-012-0164-3.  Google Scholar

[18]

C. De Filippis and P. Goatin, The initial-boundary value problem for general non-local scalar conservation laws in one space dimension, Nonlinear Anal., 161 (2017), 131-156.  doi: 10.1016/j.na.2017.05.017.  Google Scholar

[19]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871.  doi: 10.1007/s00205-015-0843-4.  Google Scholar

[20]

L. Dong and L. Tong, Shock formation in a traffic flow model with Arrhenius look-ahead dynamics, Netw. Heterog. Media, 6 (2011), 681-694.  doi: 10.3934/nhm.2011.6.681.  Google Scholar

[21]

R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, 20 Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[22]

R. D. Driver and M. D. Rosini, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal., 10 (1962), 401-426.  doi: 10.1007/BF00281203.  Google Scholar

[23]

Q. Du, J. R. Kamm, R. B. Lehoucq and M. L. Parks. A new approach for a nonlocal, nonlinear conservation law, SIAM J. Appl. Math., 72 (2012), 464–487. doi: 10.1137/110833233.  Google Scholar

[24]

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

[25]

P. Goatin and F. Rossi, A traffic flow model with non-smooth metric interaction: Well-posedness and micro-macro limit, Commun. Math. Sci., 15 (2017), 261-287.   Google Scholar

[26]

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

[27]

H. Holden and N. H. Risebro, Continuum limit of Follow-the-Leader models –a short proof, Discrete Contin. Dyn. Syst., 38 (2018), 715-722.  doi: 10.3934/dcds.2018031.  Google Scholar

[28]

H. Holden and N. H. Risebro, Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow, Netw. Heterog. Media, 13 (2018), 409-421.  doi: 10.3934/nhm.2018018.  Google Scholar

[29]

A. Keimer and L. Pflug, Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations, 263 (2017), 4023-4069.  doi: 10.1016/j.jde.2017.05.015.  Google Scholar

[30]

A. KeimerL. Pflug and M. Spinola, Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping, J. Math. Anal. Appl., 466 (2018), 18-55.  doi: 10.1016/j.jmaa.2018.05.013.  Google Scholar

[31]

A. KeimerL. Pflug and M. Spinola, Nonlocal Scalar Conservation Laws on Bounded Domains and Applications in Traffic Flow, SIAM J. Math. Anal., 50 (2018), 6271-6306.  doi: 10.1137/18M119817X.  Google Scholar

[32]

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.  Google Scholar

[33]

R. H. Martin, Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics, Wiley, New York, 1976.  Google Scholar

[34]

E. Rossi, A justification of a LWR model based on a follow the leader description, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 579-591.  doi: 10.3934/dcdss.2014.7.579.  Google Scholar

[35]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[36]

W. Shen, Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition, Netw. Heterog. Media, 13 (2018), 449-478.  doi: 10.3934/nhm.2018020.  Google Scholar

[37]

W. Shen, Nonlocal PDE Models for Traffic Flow on Rough Roads, Preprint, 2018. Google Scholar

[38]

W. Shen and K. Shikh-Khalil, Traveling waves for a microscopic model of traffic flow, DCDS (Discrete and Continuous Dynamical Systems), 38 (2018), 2571-2589.  doi: 10.3934/dcds.2018108.  Google Scholar

[39]

W. Shen and T. Y. Zhang, Erosion profile by a global model for granular flow, Arch. Rational Mech. Anal., 204 (2012), 837-879.  doi: 10.1007/s00205-012-0499-2.  Google Scholar

[40]

K. Zumbrun, On a nonlocal dispersive equation modeling particle suspensions, Q. Appl. Math., 57 (1999), 573–600. doi: 10.1090/qam/1704419.  Google Scholar

Figure 1.  Typical profiles $P(\cdot)$ with $v(\rho) = 1-\rho, h = 0.2, \ell = 0.01$ and various $[\rho^-, \rho^+]$ values given in the legends. In the left we use the weight function $w(x) = \frac{2}{h}-\frac{2x}{h^2}$ on $(0, h)$ where $w'<0$, while in right plot we use $w(x) = \frac{2x}{h^2}$ on $(0, h)$ where $w'>0$
Figure 2.  Typical solutions of the FtLs model $(z_i(t), \rho_i(t))$ at various time $t$, with oscillatory initial condition. Above: $w(x) = \frac{2}{h}-\frac{2x}{h^2}$ and the solution approaches some profile as $t$ grows. Below: $w(x) = \frac{2x}{h^2}$ and the solution oscillates more as $t$ grows
Figure 3.  Left: Graphs of $Q(x)$ and $\widehat Q(x)$ on $[x_1, x_1+h]$. Right: Graphs of shifted functions $Q(s+x_1)$ and $\widehat Q(s+x_2)$
Figure 4.  Typical profiles $Q(\cdot)$ with $v(\rho) = 1-\rho, h = 0.2$, and various asymptotic values $[\rho^-, \rho^+]$ given in the legends. For the left plot we use $w(x) = \frac{2}{h}-\frac{2x}{h^2}$ for $x\in(0, h)$, and for right plot we use $w(x) = \frac{2x}{h^2}$ for $x\in(0, h)$
Figure 5.  Solution for the nonlocal conservation law (1.1) with oscillatory initial condition, at $t = 0, 0.4, 0.8$. Top: With $w(x) = \frac{2}{h}-\frac{2x}{h^2}$, i.e., $w'<0$, the solution quickly approaches the traveling wave profile $Q(x)$ as $t$ grows. Bottom: With $w(x) = \frac{2x}{h^2}$, i.e., $w'>0$, the solution becomes more oscillatory as $t$ grows
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