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A combined finite volume - finite element scheme for a dispersive shallow water system
Solutions of the Aw-Rascle-Zhang system with point constraints
1. | LMB Laboratoire de Mathématiques CNRS UMR6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France |
2. | Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex |
3. | LMB Laboratoire de Mathématiques CNRS UMR6623, Université de Franche-Comté, 16 route de Gray, 25030 Besancon Cedex, France |
4. | Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland |
References:
[1] |
B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks, ESAIM: M2AN (2016), appeared online.
doi: 10.1051/m2an/2015078. |
[2] |
B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Riemann problems with non-local point constraints and capacity drop, Mathematical Biosciences and Engineering, 12 (2015), 259-278, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10696. |
[3] |
B. Andreianov, C. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2685-2722.
doi: 10.1142/S0218202514500341. |
[4] |
B. Andreianov, C. Donadello and M. D. Rosini, A second order model for vehicular traffics with local point constraints on the flow, Mathematical Models and Methods in Applied Sciences (2016), appeared online.
doi: 10.1142/S0218202516500172. |
[5] |
B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numerische Mathematik, 115 (2010), 609-645.
doi: 10.1007/s00211-009-0286-7. |
[6] |
A. Aw, Existence of a global entropic weak solution for the Aw-Rascle model, Int. J. Evol. Equ., 9 (2014), 53-70. |
[7] |
A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938.
doi: 10.1137/S0036139997332099. |
[8] |
F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004.
doi: 10.1007/b93802. |
[9] |
A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000. |
[10] |
C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551, URL http://projecteuclid.org/euclid.cms/1188405667.
doi: 10.4310/CMS.2007.v5.n3.a2. |
[11] |
C. Chalons, P. Goatin and N. Seguin, General constrained conservation laws. application to pedestrian flow modeling, Networks and Heterogeneous Media, 8 (2013), 433-463, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=8648.
doi: 10.3934/nhm.2013.8.433. |
[12] |
G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118.
doi: 10.1007/s002050050146. |
[13] |
R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675.
doi: 10.1016/j.jde.2006.10.014. |
[14] |
R. M. Colombo, P. Goatin and M. D. Rosini, A macroscopic model for pedestrian flows in panic situations, Proceedings of the 4th Polish-Japanese Days. GAKUTO International Series. Mathematical Sciences and Applications, 32 (2010), 255-272. |
[15] |
R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567.
doi: 10.1002/mma.624. |
[16] |
R. M. Colombo and M. D. Rosini, Existence of nonclassical solutions in a Pedestrian flow model, Nonlinear Analysis: Real World Applications, 10 (2009), 2716-2728.
doi: 10.1016/j.nonrwa.2008.08.002. |
[17] |
R. M. Colombo, P. Goatin and M. D. Rosini, Conservation laws with unilateral constraints in traffic modeling, in Applied and Industrial Mathematics in Italy III, 82 (2010), 244-255.
doi: 10.1142/9789814280303_0022. |
[18] |
Colombo, M. Rinaldo, P. Goatin and M. D. Rosini, On the modelling and management of traffic, ESAIM: M2AN, 45 (2011), 853-872.
doi: 10.1051/m2an/2010105. |
[19] |
C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33-41.
doi: 10.1016/0022-247X(72)90114-X. |
[20] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2000.
doi: 10.1007/3-540-29089-3_14. |
[21] |
R. E. Ferreira and C. I. Kondo, Glimm method and wave-front tracking for the Aw-Rascle traffic flow model, Far East J. Math. Sci. (FJMS), 43 (2010), 203-223. |
[22] |
M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648.
doi: 10.1016/j.jmaa.2011.01.033. |
[23] |
M. Godvik and H. Hanche-Olsen, Existence of solutions for the Aw-Rascle traffic flow model with vacuum, Journal of Hyperbolic Differential Equations, 5 (2008), 45-63.
doi: 10.1142/S0219891608001428. |
[24] |
H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, Springer, New York, 2011.
doi: 10.1007/978-3-642-23911-3. |
[25] |
S. N. Kruzhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. |
[26] |
M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, in Royal Society of London. Series A, Mathematical and Physical Sciences, 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[27] |
E. Panov, Generalized solutions of the Cauchy problem for a transport equation with discontinuous coefficients, in Instability in models connected with fluid flows. II, vol. 7 of Int. Math. Ser. (N. Y.), Springer, New York, 2008, 23-84.
doi: 10.1007/978-0-387-75219-8_2. |
[28] |
P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[29] |
M. D. Rosini, Nonclassical interactions portrait in a macroscopic pedestrian flow model, Journal of Differential Equations, 246 (2009), 408-427, URL http://www.sciencedirect.com/science/article/pii/S002203960800140X.
doi: 10.1016/j.jde.2008.03.018. |
[30] |
M. Rosini, The initial-boundary value problem and the constraint, in Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer International Publishing, 2013, 63-91.
doi: 10.1007/978-3-319-00155-5_6. |
[31] |
M. Rosini, Numerical applications, in Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer International Publishing, 2013, 167-173. |
[32] |
H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.
doi: 10.1016/S0191-2615(00)00050-3. |
show all references
References:
[1] |
B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks, ESAIM: M2AN (2016), appeared online.
doi: 10.1051/m2an/2015078. |
[2] |
B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Riemann problems with non-local point constraints and capacity drop, Mathematical Biosciences and Engineering, 12 (2015), 259-278, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10696. |
[3] |
B. Andreianov, C. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2685-2722.
doi: 10.1142/S0218202514500341. |
[4] |
B. Andreianov, C. Donadello and M. D. Rosini, A second order model for vehicular traffics with local point constraints on the flow, Mathematical Models and Methods in Applied Sciences (2016), appeared online.
doi: 10.1142/S0218202516500172. |
[5] |
B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numerische Mathematik, 115 (2010), 609-645.
doi: 10.1007/s00211-009-0286-7. |
[6] |
A. Aw, Existence of a global entropic weak solution for the Aw-Rascle model, Int. J. Evol. Equ., 9 (2014), 53-70. |
[7] |
A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938.
doi: 10.1137/S0036139997332099. |
[8] |
F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004.
doi: 10.1007/b93802. |
[9] |
A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000. |
[10] |
C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551, URL http://projecteuclid.org/euclid.cms/1188405667.
doi: 10.4310/CMS.2007.v5.n3.a2. |
[11] |
C. Chalons, P. Goatin and N. Seguin, General constrained conservation laws. application to pedestrian flow modeling, Networks and Heterogeneous Media, 8 (2013), 433-463, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=8648.
doi: 10.3934/nhm.2013.8.433. |
[12] |
G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118.
doi: 10.1007/s002050050146. |
[13] |
R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675.
doi: 10.1016/j.jde.2006.10.014. |
[14] |
R. M. Colombo, P. Goatin and M. D. Rosini, A macroscopic model for pedestrian flows in panic situations, Proceedings of the 4th Polish-Japanese Days. GAKUTO International Series. Mathematical Sciences and Applications, 32 (2010), 255-272. |
[15] |
R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567.
doi: 10.1002/mma.624. |
[16] |
R. M. Colombo and M. D. Rosini, Existence of nonclassical solutions in a Pedestrian flow model, Nonlinear Analysis: Real World Applications, 10 (2009), 2716-2728.
doi: 10.1016/j.nonrwa.2008.08.002. |
[17] |
R. M. Colombo, P. Goatin and M. D. Rosini, Conservation laws with unilateral constraints in traffic modeling, in Applied and Industrial Mathematics in Italy III, 82 (2010), 244-255.
doi: 10.1142/9789814280303_0022. |
[18] |
Colombo, M. Rinaldo, P. Goatin and M. D. Rosini, On the modelling and management of traffic, ESAIM: M2AN, 45 (2011), 853-872.
doi: 10.1051/m2an/2010105. |
[19] |
C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33-41.
doi: 10.1016/0022-247X(72)90114-X. |
[20] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2000.
doi: 10.1007/3-540-29089-3_14. |
[21] |
R. E. Ferreira and C. I. Kondo, Glimm method and wave-front tracking for the Aw-Rascle traffic flow model, Far East J. Math. Sci. (FJMS), 43 (2010), 203-223. |
[22] |
M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648.
doi: 10.1016/j.jmaa.2011.01.033. |
[23] |
M. Godvik and H. Hanche-Olsen, Existence of solutions for the Aw-Rascle traffic flow model with vacuum, Journal of Hyperbolic Differential Equations, 5 (2008), 45-63.
doi: 10.1142/S0219891608001428. |
[24] |
H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, Springer, New York, 2011.
doi: 10.1007/978-3-642-23911-3. |
[25] |
S. N. Kruzhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. |
[26] |
M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, in Royal Society of London. Series A, Mathematical and Physical Sciences, 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[27] |
E. Panov, Generalized solutions of the Cauchy problem for a transport equation with discontinuous coefficients, in Instability in models connected with fluid flows. II, vol. 7 of Int. Math. Ser. (N. Y.), Springer, New York, 2008, 23-84.
doi: 10.1007/978-0-387-75219-8_2. |
[28] |
P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[29] |
M. D. Rosini, Nonclassical interactions portrait in a macroscopic pedestrian flow model, Journal of Differential Equations, 246 (2009), 408-427, URL http://www.sciencedirect.com/science/article/pii/S002203960800140X.
doi: 10.1016/j.jde.2008.03.018. |
[30] |
M. Rosini, The initial-boundary value problem and the constraint, in Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer International Publishing, 2013, 63-91.
doi: 10.1007/978-3-319-00155-5_6. |
[31] |
M. Rosini, Numerical applications, in Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer International Publishing, 2013, 167-173. |
[32] |
H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.
doi: 10.1016/S0191-2615(00)00050-3. |
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