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Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS, 2006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[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. |
[25] |
M. Treiber and A. Kesting, Traffic Flow Dynamics, Springer-Verlag, Berlin Heidelberg, 2013.
doi: 10.1007/978-3-642-32460-4. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS, 2006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[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. |
[25] |
M. Treiber and A. Kesting, Traffic Flow Dynamics, Springer-Verlag, Berlin Heidelberg, 2013.
doi: 10.1007/978-3-642-32460-4. |
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