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December  2019, 14(4): 709-732. doi: 10.3934/nhm.2019028

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

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

* Corresponding author: Wen Shen

Received  September 2018 Revised  March 2019 Published  October 2019

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.

Citation: Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks & Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028
##### References:

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##### References:
Flux functions $f^-(\rho), f^+(\rho)$, and location of $\rho_1, \rho_2, \rho_3, \rho_4$ and $\hat\rho$
Sample traveling waves for Case A1, with $\kappa^- = 2, \kappa^+ = 1, h = 0.2$
Numerical simulation for model (M1) with Riemann initial data for Case A1
Typical traveling wave profile for Case A2
Numerical simulation for the PDE model with Riemann initial data for Case A2
Numerical simulation for the PDE model with Riemann initial data for Case A3
Numerical simulation for the PDE model with Riemann initial data for Case A4
Sample traveling waves for Case B1
Numerical simulation for the PDE model with Riemann initial data for Case B1
Sample traveling wave for Case B2
Numerical simulation for the PDE model with Riemann initial data for Case B2
Numerical simulation for the PDE model with Riemann initial data for Case B3
Numerical simulation for the PDE model with Riemann initial data for Case B4
Sample traveling wave for Case C1
Numerical simulation for the PDE model with Riemann initial data for Case C1
Sample traveling wave for Case C2
Numerical simulation for the PDE model with Riemann initial data for Case C2
Numerical simulation for the PDE model with Riemann initial data for Case C3
Numerical simulation for the PDE model with Riemann initial data for Case C4
Sample traveling wave for Case D1
Numerical simulation for the PDE model with Riemann initial data for Case D1
Sample traveling wave for Case D2
Solution of Riemann problem for Case D2
Solution of Riemann problem for Case D3
Solution of Riemann problem for Case D4
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