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September  2018, 13(3): 449-478. doi: 10.3934/nhm.2018020

Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition

Wen Shen ,

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

* Corresponding author: Wen Shen

Received  November 2017 Revised  January 2018 Published  July 2018

We study a Follow-the-Leader (FtL) ODE model for traffic flow with rough road condition, and analyze stationary traveling wave profiles where the solutions of the FtL model trace along, near the jump in the road condition. We derive a discontinuous delay differential equation (DDDE) for these profiles. For various cases, we obtain results on existence, uniqueness and local stability of the profiles. The results here offer an alternative approximation, possibly more realistic than the classical vanishing viscosity approach, to the conservation law with discontinuous flux for traffic flow.

Keywords: Traffic flow, rough coefficient, traveling waves, microscopic models, delay differential equation, local stability.
Mathematics Subject Classification: Primary: 65M20, 35L02, 35L65; Secondary: 34B99, 35Q99.
Citation: Wen Shen. Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition. Networks & Heterogeneous Media, 2018, 13 (3) : 449-478. doi: 10.3934/nhm.2018020
References:
[1]

B. Andreianov, New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux, CANUM 2014—42e Congrès National d'Analyse Numérique, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 50 (2015), 40–65. doi: 10.1051/proc/201550003.  Google Scholar

[2]

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

[3]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1869-1888.  doi: 10.3934/dcdsb.2014.19.1869.  Google Scholar

[4]

A. Bressan, Unique solutions for a class of discontinuous differential equations, Proc. Amer. Math. Soc., 104 (1988), 772-778.  doi: 10.1090/S0002-9939-1988-0964856-0.  Google Scholar

[5]

A. Bressan and W. Shen, Uniqueness for discontinuous ODE and conservation laws, Nonlinear Anal, 34 (1998), 637-652.  doi: 10.1016/S0362-546X(97)00590-7.  Google Scholar

[6]

A. Bressan and W. Shen, Unique solutions of discontinuous O.D.E.'s in Banach spaces, Anal. Appl. (Singap.), 4 (2006), 247-262.  doi: 10.1142/S0219530506000772.  Google Scholar

[7]

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

[8]

A. Corli, L. di Ruvo, L. Malaguti and M. D. Rosini, Traveling waves for degenerate diffusive equations on networks, Netw. Heterog. Media, 12 (2017), 339-370.  doi: 10.3934/nhm.2017015.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[14]

T. Gimse and N. H. Risebro, Riemann problems with a discontinuous flux function, Third International Conference on Hyperbolic Problems, Vol. I, II, 488–502, Studentlitteratur, Lund, 1990.  Google Scholar

[15]

G. Guerra and W. Shen, Vanishing viscosity solutions of riemann problems for models in polymer flooding, Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, (2017), 261-285.  doi: 10.4171/186-1/12.  Google Scholar

[16]

H. Holden and N. H. Risebro, Continuum limit of Follow-the-Leader models - a short proof, To appear in DCDS, 38 (2018), 715-722.  doi: 10.3934/dcds.2018031.  Google Scholar

[17]

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, Networks & Heterogeneous Media, 13 (2018). Google Scholar

[18]

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

[19]

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

[20]

W. Shen, On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 37, 25pp. doi: 10.1007/s00030-017-0461-y.  Google Scholar

[21]

W. Shen, Scilab codes for simulations and plots used in this paper www.personal.psu.edu/wxs27/SIM/Traffic-DDDE, 2017. Google Scholar

[22]

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

show all references

References:
[1]

B. Andreianov, New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux, CANUM 2014—42e Congrès National d'Analyse Numérique, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 50 (2015), 40–65. doi: 10.1051/proc/201550003.  Google Scholar

[2]

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

[3]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1869-1888.  doi: 10.3934/dcdsb.2014.19.1869.  Google Scholar

[4]

A. Bressan, Unique solutions for a class of discontinuous differential equations, Proc. Amer. Math. Soc., 104 (1988), 772-778.  doi: 10.1090/S0002-9939-1988-0964856-0.  Google Scholar

[5]

A. Bressan and W. Shen, Uniqueness for discontinuous ODE and conservation laws, Nonlinear Anal, 34 (1998), 637-652.  doi: 10.1016/S0362-546X(97)00590-7.  Google Scholar

[6]

A. Bressan and W. Shen, Unique solutions of discontinuous O.D.E.'s in Banach spaces, Anal. Appl. (Singap.), 4 (2006), 247-262.  doi: 10.1142/S0219530506000772.  Google Scholar

[7]

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

[8]

A. Corli, L. di Ruvo, L. Malaguti and M. D. Rosini, Traveling waves for degenerate diffusive equations on networks, Netw. Heterog. Media, 12 (2017), 339-370.  doi: 10.3934/nhm.2017015.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[14]

T. Gimse and N. H. Risebro, Riemann problems with a discontinuous flux function, Third International Conference on Hyperbolic Problems, Vol. I, II, 488–502, Studentlitteratur, Lund, 1990.  Google Scholar

[15]

G. Guerra and W. Shen, Vanishing viscosity solutions of riemann problems for models in polymer flooding, Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, (2017), 261-285.  doi: 10.4171/186-1/12.  Google Scholar

[16]

H. Holden and N. H. Risebro, Continuum limit of Follow-the-Leader models - a short proof, To appear in DCDS, 38 (2018), 715-722.  doi: 10.3934/dcds.2018031.  Google Scholar

[17]

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, Networks & Heterogeneous Media, 13 (2018). Google Scholar

[18]

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

[19]

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

[20]

W. Shen, On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 37, 25pp. doi: 10.1007/s00030-017-0461-y.  Google Scholar

[21]

W. Shen, Scilab codes for simulations and plots used in this paper www.personal.psu.edu/wxs27/SIM/Traffic-DDDE, 2017. Google Scholar

[22]

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

Figure 1.  Graph of $Q(x)$ on the interval $[z_k, z_{k+2}]$. Illustration of the locations for $y_k, y^\sharp_k$ and $y'_{k+1}$, used in the proof of Lemma 2.8
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Figure 2.  Flux functions $f_-, f_+$, and the locations of $\rho^-_1$, $\rho^+_1$, $\rho^-_2$, $\rho^+_2$, and $\rho^*$
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Figure 3.  Case 1A: (1) Plot of the flux functions $f_-, f_+$ and the locations of $\rho_-, \rho_+$; (2) Plots of various profiles of $Q(x)$, with different values of $Q(0)$; (3) Plots of various viscous traveling waves $\rho^\varepsilon(x)$, with different values of $\rho^\varepsilon(0)$; (4) Plots of various solutions of the FtL model $\{z_i(t), \rho_i(t)\}$, with 3 different initial Riemann data. Here the thick dots denote the locations of cars at $t = 2$
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Figure 4.  Case 1B. (1) Plots of the flux functions and the locations of $\rho_-, \rho_+$; (2) Plot of the unique stationary profile $Q(x)$ with $Q(0) = \rho_+$; (3) Plot of the unique viscous profile $\rho^\varepsilon(x)$ with $\rho^\varepsilon(0) = \rho_+$; (4) Plot of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots are the locations of cars at $t = 2$
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Figure 5.  Case 1C. (1) Plots of the flux functions and the locations of $\rho_-, \rho_+$; (2) Plot of the unique viscous profile $\rho^\varepsilon(x)$ with $\rho^\varepsilon(0) = \rho_-$; (3) Plot of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots are the locations of cars at $t = 2$
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Figure 6.  Case 1D. (1): Plots of the flux functions and the locations of $\rho_-, \rho_+$; (2): Plot of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots are the locations of cars at $t = 2$
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Figure 7.  Flux functions $f_-, f_+$, and the locations of $\rho^-_1, \rho^+_1, \rho^-_2, \rho^+_2$
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Figure 8.  Case 2A. (1) Flux functions and the locations of $\rho_-, \rho_+$; (2) Plots of various profiles of $Q(x)$, with different values of $Q(0)$; (3) Plots of various viscous traveling waves $\rho^\varepsilon(x)$, with different values of $\rho^\varepsilon(0)$; (4) Plots of various solutions of the FtL model $\{z_i(t), \rho_i(t)\}$, with 3 different initial Riemann data. Here the thick dots denote the locations of cars at $t = 2$
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Figure 9.  Case 2B. (1) Flux functions and the locations of $\rho_-, \rho_+$; (2) Plots of the unique profile of $Q(x)$, with $Q(0) = \rho_+$; (3) Plots of various viscous traveling waves $\rho^\varepsilon(x)$, with $\rho^\varepsilon(0) = \rho_+$; (4) Plots of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots denote the locations of cars at $t = 2$
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Figure 10.  Case 2C. (1) Plot of the flux functions $f_-, f_+$ and the locations of $\rho_-, \rho_+$; (2) Plot of the unique viscous profile $\rho^\varepsilon$ with $\rho^\varepsilon(0) = \rho_-$; (3) Plot of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots are the locations of cars at $t = 2$
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Figure 11.  Case 2D. (1): Plots of the flux functions and the locations of $\rho_-, \rho_+$; (2): Plot of the solution of the FtL model $\{z_i(t), \rho_i(t)\}$ with a Riemann initial data. Here the thick dots are the locations of cars at $t = 2$
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Figure 12.  (1). Plots of the flux functions and the location of the left (L), right (R) and middle (M) states in the solution of the Riemann problem; (2). Numerical simulation results $\{z_i(t), \rho_i(t)\}$ with FtL model with Riemann initial data, at $t = 1$; (3) Numerical simulation results $\rho^\varepsilon(t)$ for the viscous conservation law at $t = 1$, with the same Riemann initial data
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