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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:
[1]

A. AggarwalR. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983.  doi: 10.1137/140975255.  Google Scholar

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F. BetancourtR. BürgerK. H. Karlsen and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity, 27 (2011), 855-885.  doi: 10.1088/0951-7715/24/3/008.  Google Scholar

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S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), 217-241.  doi: 10.1007/s00211-015-0717-6.  Google Scholar

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G.-Q. Chen and C. Christoforou, Solutions for a nonlocal conservation law with fading memory, Proc. Amer. Math. Soc., 135 (2007), 3905-3915.  doi: 10.1090/S0002-9939-07-08942-3.  Google Scholar

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J. Chien and W. Shen, Traveling Waves for nonlocal particle models of traffic flow on rough roads, Discrete Contin. Dyn. Syst., 39 (2019), 4001—4040, arXiv: 1902.08537. doi: 10.3934/dcds.2019161.  Google Scholar

[9]

M. Colombo, G. Crippa and L. V. Spinolo, On the singular local limit for conservation laws with nonlocal fluxes, Arch. Ration. Mech. Anal., 233 (2019), 1131–1167, arXiv: 1710.04547. doi: 10.1007/s00205-019-01375-8.  Google Scholar

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M. Colombo, G. Crippa and L. V. Spinolo, Blow-up of the total variation in the local limit of a nonlocal traffic model, Preprint, arXiv: 1808.03529. Google Scholar

[11]

R. M. Colombo and M. Lécureux-Mercier, Nonlocal crowd dynamics models for several populations, Acta Math. Sci., 32 (2012), 177-196.  doi: 10.1016/S0252-9602(12)60011-3.  Google Scholar

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Q. DuJ. R. KammR. B. Lehoucq and M. L. Parks, A new approach for a nonlocal, nonlinear conservation law, SIAM J. Appl. Math., 72 (2012), 464-487.  doi: 10.1137/110833233.  Google Scholar

[15]

J. Friedrich, O. Kolb and S. Göttlich, A Godunov type scheme for a class of LWR traffic flow models with non-local flux, Netw. Heterog. Media, 13 (2018), 531–547, arXiv: 1802.07484. doi: 10.3934/nhm.2018024.  Google Scholar

[16]

G. Guerra and W. Shen, Existence and stability of traveling waves for an integro-differential equation for slow erosion, J. Differential Equations, 256 (2014), 253-282.  doi: 10.1016/j.jde.2013.09.003.  Google Scholar

[17]

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

[18]

J. Ridder and W. Shen, Traveling waves for nonlocal models of traffic flow, Discrete Contin. Dyn. Syst., 39 (2019), 4001–4040, arXiv: 1808.03734. doi: 10.3934/dcds.2019161.  Google Scholar

[19]

W. Shen, Traveling wave profiles for a follow-the-leader model for traffic flow with rough road condition, Netw. Heterog. Media, 13 (2018), 449-478.  doi: 10.3934/nhm.2018020.  Google Scholar

[20]

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

[21]

W. Shen and T. Y. Zhang, Erosion profile by a global model for granular flow, Arch. Rational Mech. Anal., 204 (2012), 837-879.  doi: 10.1007/s00205-012-0499-2.  Google Scholar

[22]

K. Zumbrun, On a nonlocal dispersive equation modeling particle suspensions, Q. Appl. Math., 57 (1999), 573-600.  doi: 10.1090/qam/1704419.  Google Scholar

show all references

References:
[1]

A. AggarwalR. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983.  doi: 10.1137/140975255.  Google Scholar

[2]

D. AmadoriS.-Y. Ha and J. Park, On the global well-posedness of BV weak solutions to the Kuramoto-Sakaguchi equation, J. Differential Equations, 262 (2017), 978-1022.  doi: 10.1016/j.jde.2016.10.004.  Google Scholar

[3]

D. Amadori and W. Shen, Front tracking approximations for slow erosion, Dicrete Contin. Dyn. Syst., 32 (2012), 1481-1502.  doi: 10.3934/dcds.2012.32.1481.  Google Scholar

[4]

P. AmorimR. M. Colombo and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM Math. Model. Numer. Anal., 49 (2015), 19-37.  doi: 10.1051/m2an/2014023.  Google Scholar

[5]

F. BetancourtR. BürgerK. H. Karlsen and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity, 27 (2011), 855-885.  doi: 10.1088/0951-7715/24/3/008.  Google Scholar

[6]

S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), 217-241.  doi: 10.1007/s00211-015-0717-6.  Google Scholar

[7]

G.-Q. Chen and C. Christoforou, Solutions for a nonlocal conservation law with fading memory, Proc. Amer. Math. Soc., 135 (2007), 3905-3915.  doi: 10.1090/S0002-9939-07-08942-3.  Google Scholar

[8]

J. Chien and W. Shen, Traveling Waves for nonlocal particle models of traffic flow on rough roads, Discrete Contin. Dyn. Syst., 39 (2019), 4001—4040, arXiv: 1902.08537. doi: 10.3934/dcds.2019161.  Google Scholar

[9]

M. Colombo, G. Crippa and L. V. Spinolo, On the singular local limit for conservation laws with nonlocal fluxes, Arch. Ration. Mech. Anal., 233 (2019), 1131–1167, arXiv: 1710.04547. doi: 10.1007/s00205-019-01375-8.  Google Scholar

[10]

M. Colombo, G. Crippa and L. V. Spinolo, Blow-up of the total variation in the local limit of a nonlocal traffic model, Preprint, arXiv: 1808.03529. Google Scholar

[11]

R. M. Colombo and M. Lécureux-Mercier, Nonlocal crowd dynamics models for several populations, Acta Math. Sci., 32 (2012), 177-196.  doi: 10.1016/S0252-9602(12)60011-3.  Google Scholar

[12]

R. D. Driver, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal., 10 (1962), 401-426.  doi: 10.1007/BF00281203.  Google Scholar

[13]

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

[14]

Q. DuJ. R. KammR. B. Lehoucq and M. L. Parks, A new approach for a nonlocal, nonlinear conservation law, SIAM J. Appl. Math., 72 (2012), 464-487.  doi: 10.1137/110833233.  Google Scholar

[15]

J. Friedrich, O. Kolb and S. Göttlich, A Godunov type scheme for a class of LWR traffic flow models with non-local flux, Netw. Heterog. Media, 13 (2018), 531–547, arXiv: 1802.07484. doi: 10.3934/nhm.2018024.  Google Scholar

[16]

G. Guerra and W. Shen, Existence and stability of traveling waves for an integro-differential equation for slow erosion, J. Differential Equations, 256 (2014), 253-282.  doi: 10.1016/j.jde.2013.09.003.  Google Scholar

[17]

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

[18]

J. Ridder and W. Shen, Traveling waves for nonlocal models of traffic flow, Discrete Contin. Dyn. Syst., 39 (2019), 4001–4040, arXiv: 1808.03734. doi: 10.3934/dcds.2019161.  Google Scholar

[19]

W. Shen, Traveling wave profiles for a follow-the-leader model for traffic flow with rough road condition, Netw. Heterog. Media, 13 (2018), 449-478.  doi: 10.3934/nhm.2018020.  Google Scholar

[20]

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

[21]

W. Shen and T. Y. Zhang, Erosion profile by a global model for granular flow, Arch. Rational Mech. Anal., 204 (2012), 837-879.  doi: 10.1007/s00205-012-0499-2.  Google Scholar

[22]

K. Zumbrun, On a nonlocal dispersive equation modeling particle suspensions, Q. Appl. Math., 57 (1999), 573-600.  doi: 10.1090/qam/1704419.  Google Scholar

Figure 1.  Flux functions $ f^-(\rho), f^+(\rho) $, and location of $ \rho_1, \rho_2, \rho_3, \rho_4 $ and $ \hat\rho $
Figure 2.  Sample traveling waves for Case A1, with $ \kappa^- = 2, \kappa^+ = 1, h = 0.2 $
Figure 3.  Numerical simulation for model (M1) with Riemann initial data for Case A1
Figure 4.  Typical traveling wave profile for Case A2
Figure 5.  Numerical simulation for the PDE model with Riemann initial data for Case A2
Figure 6.  Numerical simulation for the PDE model with Riemann initial data for Case A3
Figure 7.  Numerical simulation for the PDE model with Riemann initial data for Case A4
Figure 8.  Sample traveling waves for Case B1
Figure 9.  Numerical simulation for the PDE model with Riemann initial data for Case B1
Figure 10.  Sample traveling wave for Case B2
Figure 11.  Numerical simulation for the PDE model with Riemann initial data for Case B2
Figure 12.  Numerical simulation for the PDE model with Riemann initial data for Case B3
Figure 13.  Numerical simulation for the PDE model with Riemann initial data for Case B4
Figure 14.  Sample traveling wave for Case C1
Figure 15.  Numerical simulation for the PDE model with Riemann initial data for Case C1
Figure 16.  Sample traveling wave for Case C2
Figure 17.  Numerical simulation for the PDE model with Riemann initial data for Case C2
Figure 18.  Numerical simulation for the PDE model with Riemann initial data for Case C3
Figure 19.  Numerical simulation for the PDE model with Riemann initial data for Case C4
Figure 20.  Sample traveling wave for Case D1
Figure 21.  Numerical simulation for the PDE model with Riemann initial data for Case D1
Figure 22.  Sample traveling wave for Case D2
Figure 23.  Solution of Riemann problem for Case D2
Figure 24.  Solution of Riemann problem for Case D3
Figure 25.  Solution of Riemann problem for Case D4
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