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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[37]

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

[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.

[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.

[40]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[37]

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

[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.

[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.

[40]

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

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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