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March  2016, 11(1): 107-121. doi: 10.3934/nhm.2016.11.107

Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity

Paola Goatin 1, and  Sheila Scialanga 2,

1. 

INRIA Sophia Antipolis - Méditerranée, EPI OPALE, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex
2. 

Inria Sophia Antipolis - Méditerranée, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France

Received  April 2015 Revised  October 2015 Published  January 2016

We consider an extension of the traffic flow model proposed by Lighthill, Whitham and Richards, in which the mean velocity depends on a weighted mean of the downstream traffic density. We prove well-posedness and a regularity result for entropy weak solutions of the corresponding Cauchy problem, and use a finite volume central scheme to compute approximate solutions. We perform numerical tests to illustrate the theoretical results and to investigate the limit as the convolution kernel tends to a Dirac delta function.
Keywords: finite volume schemes., Scalar conservation laws, macroscopic traffic flow models, non-local flux.
Mathematics Subject Classification: Primary: 35L65; Secondary: 65M12, 90B2.
Citation: Paola Goatin, Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Networks & Heterogeneous Media, 2016, 11 (1) : 107-121. doi: 10.3934/nhm.2016.11.107
References:
[1]

A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM Journal on Numerical Analysis, 53 (2015), 963-983. doi: 10.1137/140975255.  Google Scholar

[2]

D. Amadori and W. Shen, An integro-differential conservation law arising in a model of granular flow, J. Hyperbolic Differ. Equ., 9 (2012), 105-131. doi: 10.1142/S0219891612500038.  Google Scholar

[3]

P. Amorim, R. Colombo and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM M2AN, 49 (2015), 19-37. doi: 10.1051/m2an/2014023.  Google Scholar

[4]

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

[5]

S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numerische Mathematik, (2015), 1-25. doi: 10.1007/s00211-015-0717-6.  Google Scholar

[6]

C. Canudas De Wit, F. Morbidi, L. Leon Ojeda, A. Y. Kibangou, I. Bellicot and P. Bellemain, Grenoble Traffic Lab: An experimental platform for advanced traffic monitoring and forecasting, IEEE Control Systems, 35 (2015), 23-39, URL https://hal.archives-ouvertes.fr/hal-01059126. doi: 10.1109/MCS.2015.2406657.  Google Scholar

[7]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1150023, 34pp. doi: 10.1142/S0218202511500230.  Google Scholar

[8]

R. M. Colombo, M. Herty and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), 353-379. doi: 10.1051/cocv/2010007.  Google Scholar

[9]

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

[10]

G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 523-537. doi: 10.1007/s00030-012-0164-3.  Google Scholar

[11]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS, 2006.  Google Scholar

[12]

P. Goatin and S. Scialanga, The Lighthill-Whitham-Richards Traffic Flow Model with Non-Local Velocity: Analytical Study and Numerical Results, Research Report RR-8685, Inria Sophia Antipolis, 2015, URL https://hal.inria.fr/hal-01118734. Google Scholar

[13]

S. Göttlich, S. Hoher, P. Schindler, V. Schleper and A. Verl, Modeling, simulation and validation of material flow on conveyor belts, Applied Mathematical Modelling, 38 (2014), 3295-3313. doi: 10.1016/j.apm.2013.11.039.  Google Scholar

[14]

J. C. Herrera and A. M. Bayen, Incorporation of lagrangian measurements in freeway traffic state estimation, Transportation Research Part B: Methodological, 44 (2010), 460-481. doi: 10.1016/j.trb.2009.10.005.  Google Scholar

[15]

M. Herty and R. Illner, Coupling of non-local driving behaviour with fundamental diagrams, Kinetic and Related Models, 5 (2012), 843-855. doi: 10.3934/krm.2012.5.843.  Google Scholar

[16]

F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equations, SIAM Journal on Numerical Analysis, 53 (2015), 895-916. doi: 10.1137/140959997.  Google Scholar

[17]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar

[18]

A. Kurganov and A. Polizzi, Non-oscillatory central schemes for a traffic flow model with Arrehenius look-ahead dynamics, Netw. Heterog. Media, 4 (2009), 431-451. doi: 10.3934/nhm.2009.4.431.  Google Scholar

[19]

D. Li and T. Li, Shock formation in a traffic flow model with Arrhenius look-ahead dynamics, Networks and Heterogeneous Media, 6 (2011), 681-694. doi: 10.3934/nhm.2011.6.681.  Google Scholar

[20]

K.-A. Lie and S. Noelle, On the artificial compression method for second-order nonoscillatory central difference schemes for system of conservation laws, SIAM J. Sci. Comput., 24 (2003), 1157-1174. doi: 10.1137/S1064827501392880.  Google Scholar

[21]

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

[22]

H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 408-463. doi: 10.1016/0021-9991(90)90260-8.  Google Scholar

[23]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar

[24]

A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 66 (2006), 921-944 (electronic). doi: 10.1137/040617790.  Google Scholar

[25]

M. Treiber and A. Kesting, Traffic Flow Dynamics, Springer-Verlag, Berlin Heidelberg, 2013. doi: 10.1007/978-3-642-32460-4.  Google Scholar

show all references

References:
[1]

A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM Journal on Numerical Analysis, 53 (2015), 963-983. doi: 10.1137/140975255.  Google Scholar

[2]

D. Amadori and W. Shen, An integro-differential conservation law arising in a model of granular flow, J. Hyperbolic Differ. Equ., 9 (2012), 105-131. doi: 10.1142/S0219891612500038.  Google Scholar

[3]

P. Amorim, R. Colombo and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM M2AN, 49 (2015), 19-37. doi: 10.1051/m2an/2014023.  Google Scholar

[4]

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

[5]

S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numerische Mathematik, (2015), 1-25. doi: 10.1007/s00211-015-0717-6.  Google Scholar

[6]

C. Canudas De Wit, F. Morbidi, L. Leon Ojeda, A. Y. Kibangou, I. Bellicot and P. Bellemain, Grenoble Traffic Lab: An experimental platform for advanced traffic monitoring and forecasting, IEEE Control Systems, 35 (2015), 23-39, URL https://hal.archives-ouvertes.fr/hal-01059126. doi: 10.1109/MCS.2015.2406657.  Google Scholar

[7]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1150023, 34pp. doi: 10.1142/S0218202511500230.  Google Scholar

[8]

R. M. Colombo, M. Herty and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), 353-379. doi: 10.1051/cocv/2010007.  Google Scholar

[9]

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

[10]

G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 523-537. doi: 10.1007/s00030-012-0164-3.  Google Scholar

[11]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS, 2006.  Google Scholar

[12]

P. Goatin and S. Scialanga, The Lighthill-Whitham-Richards Traffic Flow Model with Non-Local Velocity: Analytical Study and Numerical Results, Research Report RR-8685, Inria Sophia Antipolis, 2015, URL https://hal.inria.fr/hal-01118734. Google Scholar

[13]

S. Göttlich, S. Hoher, P. Schindler, V. Schleper and A. Verl, Modeling, simulation and validation of material flow on conveyor belts, Applied Mathematical Modelling, 38 (2014), 3295-3313. doi: 10.1016/j.apm.2013.11.039.  Google Scholar

[14]

J. C. Herrera and A. M. Bayen, Incorporation of lagrangian measurements in freeway traffic state estimation, Transportation Research Part B: Methodological, 44 (2010), 460-481. doi: 10.1016/j.trb.2009.10.005.  Google Scholar

[15]

M. Herty and R. Illner, Coupling of non-local driving behaviour with fundamental diagrams, Kinetic and Related Models, 5 (2012), 843-855. doi: 10.3934/krm.2012.5.843.  Google Scholar

[16]

F. James and N. Vauchelet, Numerical methods for one-dimensional aggregation equations, SIAM Journal on Numerical Analysis, 53 (2015), 895-916. doi: 10.1137/140959997.  Google Scholar

[17]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar

[18]

A. Kurganov and A. Polizzi, Non-oscillatory central schemes for a traffic flow model with Arrehenius look-ahead dynamics, Netw. Heterog. Media, 4 (2009), 431-451. doi: 10.3934/nhm.2009.4.431.  Google Scholar

[19]

D. Li and T. Li, Shock formation in a traffic flow model with Arrhenius look-ahead dynamics, Networks and Heterogeneous Media, 6 (2011), 681-694. doi: 10.3934/nhm.2011.6.681.  Google Scholar

[20]

K.-A. Lie and S. Noelle, On the artificial compression method for second-order nonoscillatory central difference schemes for system of conservation laws, SIAM J. Sci. Comput., 24 (2003), 1157-1174. doi: 10.1137/S1064827501392880.  Google Scholar

[21]

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

[22]

H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 408-463. doi: 10.1016/0021-9991(90)90260-8.  Google Scholar

[23]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar

[24]

A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 66 (2006), 921-944 (electronic). doi: 10.1137/040617790.  Google Scholar

[25]

M. Treiber and A. Kesting, Traffic Flow Dynamics, Springer-Verlag, Berlin Heidelberg, 2013. doi: 10.1007/978-3-642-32460-4.  Google Scholar

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