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The Entropy Solutions for the Lighthill-Whitham-Richards Traffic Flow Model with a Discontinuous Flow-Density Relationship

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
Department of Civil Engineering, The University of Hong Kong, Hong Kong, P.R. China
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
luyd{at}mail.ustc.edu.cn
hhecwsc{at}hkucc.hku.hk
mpzhang{at}ustc.edu.cn
shu{at}dam.brown.edu

In this paper we explicitly construct the entropy solutions for the Lighthill-Whitham-Richards (LWR) traffic flow model with a flow-density relationship which is piecewise quadratic, concave, but not continuous at the junction points where two quadratic polynomials meet, and with piecewise linear initial condition and piecewise constant boundary conditions. The existence and uniqueness of entropy solutions for such conservation laws with discontinuous fluxes are not known mathematically. We have used the approach of explicitly constructing the entropy solutions to a sequence of approximate problems in which the flow-density relationship is continuous but tends to the discontinuous flux when a small parameter in this sequence tends to zero. The limit of the entropy solutions for this sequence is explicitly constructed and is considered to be the entropy solution associated with the discontinuous flux. We apply this entropy solution construction procedure to solve four representative traffic flow cases, compare them with numerical solutions obtained by a high order weighted essentially nonoscillatory (WENO) scheme, and discuss the results from traffic flow perspectives.

Key Words: LWR model; traffic flow; discontinuous flow-density relationship; entropy solution; WENO scheme
History: Received: April 2007; revised: January 2008; accepted: March 2009.