# American Institute of Mathematical Sciences

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

 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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, 20 Springer-Verlag, New York-Heidelberg, 1977. [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. [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. [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. [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. [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). [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. [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. [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. [21] W. Shen, Scilab codes for simulations and plots used in this paper www.personal.psu.edu/wxs27/SIM/Traffic-DDDE, 2017. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, 20 Springer-Verlag, New York-Heidelberg, 1977. [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. [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. [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. [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. [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). [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. [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. [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. [21] W. Shen, Scilab codes for simulations and plots used in this paper www.personal.psu.edu/wxs27/SIM/Traffic-DDDE, 2017. [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.
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
Flux functions $f_-, f_+$, and the locations of $\rho^-_1$, $\rho^+_1$, $\rho^-_2$, $\rho^+_2$, and $\rho^*$
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 $t = 2$
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 $t = 2$
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 $t = 2$
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 $t = 2$
Flux functions $f_-, f_+$, and the locations of $\rho^-_1, \rho^+_1, \rho^-_2, \rho^+_2$
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 $t = 2$
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 $t = 2$
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 $t = 2$
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 $t = 2$
(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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