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

Wen Shen

doi:10.3934/nhm.2018020

Article Contents

2018, Volume 13, Issue 3: 449-478. Doi: 10.3934/nhm.2018020

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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
* Corresponding author: Wen Shen

Received: October 31, 2017

Revised: December 31, 2017

Published: August 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 65M20, 35L02, 35L65; Secondary: 34B99, 35Q99.

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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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