Continuous Riemann solvers for traffic flow at a junction

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September 2015, 35(9): 4149-4171. doi: 10.3934/dcds.2015.35.4149

Continuous Riemann solvers for traffic flow at a junction

Alberto Bressan ^1, and Fang Yu ^2,

1.	Department of Mathematics, Penn State University, University Park, Pa.16802
2.	Department of Mathematics, Penn State University, University Park, PA. 16802, United States

Received May 2014 Revised August 2014 Published April 2015

Abstract
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The paper studies a class of conservation law models for traffic flow on a family of roads, near a junction. A Riemann Solver is constructed, where the incoming and outgoing fluxes depend Hölder continuously on the traffic density and on the drivers' turning preferences. However, various examples show that, if junction conditions are assigned in terms of Riemann Solvers, then the Cauchy problem on a network of roads can be ill posed, even for initial data having small total variation.

Keywords: network of roads., Riemann solvers, traffic flow.

Mathematics Subject Classification: Primary: 35Q93, 35L65; Secondary: 90B2.

Citation: Alberto Bressan, Fang Yu. Continuous Riemann solvers for traffic flow at a junction. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4149-4171. doi: 10.3934/dcds.2015.35.4149

References:

[1]	A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem, Oxford University Press, 2000.
[2]	A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flow on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2.
[3]	A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks, Networks & Heter. Media, 8 (2013), 627-648. doi: 10.3934/nhm.2013.8.627.
[4]	A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads, Networks & Heter. Media, to appear.
[5]	A. Bressan and K. Nguyen, Optima and equilibria for traffic flow on networks with backward propagating queues, Networks & Heter. Media, submitted.
[6]	A. Bressan and W. Shen, Uniqueness for discontinuous O.D.E. and conservation laws, Nonlinear Analysis, T.M.A., 34 (1998), 637-652. doi: 10.1016/S0362-546X(97)00590-7.
[7]	G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.
[8]	C. 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.
[9]	C. Daganzo, Fundamentals of Transportation and Traffic Operations, Pergamon-Elsevier, Oxford, U.K., 1997.
[10]	L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, American Mathematical Society, Providence, RI, 2010.
[11]	M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst., 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915.
[12]	M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, in AIMS Series on Applied Mathematics, Springfield, Mo., 2006.
[13]	M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincaré, 26 (2009), 1925-1951. doi: 10.1016/j.anihpc.2009.04.001.
[14]	M. Herty, S. Moutari and M. Rascle, Optimization criteria for modeling intersections of vehicular traffic flow, Netw. Heter. Media., 1 (2006), 275-294. doi: 10.3934/nhm.2006.1.275.
[15]	M. Herty, J. P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media., 4 (2009), 813-826. doi: 10.3934/nhm.2009.4.813.
[16]	C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002.
[17]	M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London: Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.
[18]	P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.
[19]	J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

show all references

References:

[1]	A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem, Oxford University Press, 2000.
[2]	A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flow on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2.
[3]	A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks, Networks & Heter. Media, 8 (2013), 627-648. doi: 10.3934/nhm.2013.8.627.
[4]	A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads, Networks & Heter. Media, to appear.
[5]	A. Bressan and K. Nguyen, Optima and equilibria for traffic flow on networks with backward propagating queues, Networks & Heter. Media, submitted.
[6]	A. Bressan and W. Shen, Uniqueness for discontinuous O.D.E. and conservation laws, Nonlinear Analysis, T.M.A., 34 (1998), 637-652. doi: 10.1016/S0362-546X(97)00590-7.
[7]	G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.
[8]	C. 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.
[9]	C. Daganzo, Fundamentals of Transportation and Traffic Operations, Pergamon-Elsevier, Oxford, U.K., 1997.
[10]	L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, American Mathematical Society, Providence, RI, 2010.
[11]	M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst., 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915.
[12]	M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, in AIMS Series on Applied Mathematics, Springfield, Mo., 2006.
[13]	M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincaré, 26 (2009), 1925-1951. doi: 10.1016/j.anihpc.2009.04.001.
[14]	M. Herty, S. Moutari and M. Rascle, Optimization criteria for modeling intersections of vehicular traffic flow, Netw. Heter. Media., 1 (2006), 275-294. doi: 10.3934/nhm.2006.1.275.
[15]	M. Herty, J. P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media., 4 (2009), 813-826. doi: 10.3934/nhm.2009.4.813.
[16]	C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002.
[17]	M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London: Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.
[18]	P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.
[19]	J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

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