Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads

Wen Shen

doi:10.3934/nhm.2019028

Article Contents

2019, Volume 14, Issue 4: 709-732. Doi: 10.3934/nhm.2019028

This issue Previous Article A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method Next Article Asymptotic structure of the spectrum in a Dirichlet-strip with double periodic perforations

Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads

Wen Shen^,

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

^* Corresponding author: Wen Shen
^* Corresponding author: Wen Shen

Received: August 31, 2018

Revised: February 28, 2019

Published: November 2019

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Abstract

We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model one considers an averaged downstream velocity. The road condition is piecewise constant with a jump at $ x = 0 $. We study stationary traveling wave profiles cross $ x = 0 $, for all possible cases. We show that, depending on the case, there could exit infinitely many profiles, a unique profile, or no profiles at all. Furthermore, some of the profiles are time asymptotic solutions for the Cauchy problem of the conservation laws under mild assumption on the initial data, while other profiles are unstable.

Keywords:

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

Citation:

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Figure 1. Flux functions $ f^-(\rho), f^+(\rho) $, and location of $ \rho_1, \rho_2, \rho_3, \rho_4 $ and $ \hat\rho $

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Figure 2. Sample traveling waves for Case A1, with $ \kappa^- = 2, \kappa^+ = 1, h = 0.2 $

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Figure 3. Numerical simulation for model (M1) with Riemann initial data for Case A1

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Figure 4. Typical traveling wave profile for Case A2

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Figure 5. Numerical simulation for the PDE model with Riemann initial data for Case A2

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Figure 6. Numerical simulation for the PDE model with Riemann initial data for Case A3

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Figure 7. Numerical simulation for the PDE model with Riemann initial data for Case A4

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Figure 8. Sample traveling waves for Case B1

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Figure 9. Numerical simulation for the PDE model with Riemann initial data for Case B1

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Figure 10. Sample traveling wave for Case B2

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Figure 11. Numerical simulation for the PDE model with Riemann initial data for Case B2

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Figure 12. Numerical simulation for the PDE model with Riemann initial data for Case B3

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Figure 13. Numerical simulation for the PDE model with Riemann initial data for Case B4

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Figure 14. Sample traveling wave for Case C1

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Figure 15. Numerical simulation for the PDE model with Riemann initial data for Case C1

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Figure 16. Sample traveling wave for Case C2

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Figure 17. Numerical simulation for the PDE model with Riemann initial data for Case C2

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Figure 18. Numerical simulation for the PDE model with Riemann initial data for Case C3

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Figure 19. Numerical simulation for the PDE model with Riemann initial data for Case C4

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Figure 20. Sample traveling wave for Case D1

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Figure 21. Numerical simulation for the PDE model with Riemann initial data for Case D1

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Figure 22. Sample traveling wave for Case D2

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Figure 23. Solution of Riemann problem for Case D2

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Figure 24. Solution of Riemann problem for Case D3

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Figure 25. Solution of Riemann problem for Case D4

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