An analytical shock-fitting algorithm for LWR kinematic wave model embedded with linear speed–density relationship

Wong, S. C.; Wong, G. C. K.

doi:10.1016/s0191-2615(01)00023-6

Cited by 51 publications

(23 citation statements)

References 15 publications

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“…Using Greenshields' Relation we can determine the velocity at each spacetime point, from which we can obtain vehicle trajectory curves and trip costs. Although this solution method works in principle, it is computationally difficult to calculate shock paths; however, some recent work has been done in designing a robust algorithm for use with Greenshields' Relation, Wong and Wong (2002).…”

Section: Methods Of Characteristics For the Single-entry Corridor Problemmentioning

confidence: 99%

Morning commute in a single-entry traffic corridor with no late arrivals

DePalma

Arnott

2012

Transportation Research Part B: Methodological

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This paper analyzes a model of early morning traffic congestion, that is a special case of the model considered in Newell (1988). A fixed number of identical vehicles travel along a single-lane road of constant width from a common origin to a common destination, with LWR flow congestion and Greenshields' Relation. Vehicles have a common work start time, late arrivals are not permitted, and trip cost is linear in travel time and time early. The paper explores traffic dynamics for the Social Optimum, in which total trip cost is minimized, and for the User Optimum, in which no vehicle's trip cost can be reduced by altering its departure time. Analytical and, when possible, closed-form solutions are presented, along with numerical examples.

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Section: Methods Of Characteristics For the Single-entry Corridor Problemmentioning

confidence: 99%

Morning commute in a single-entry traffic corridor with no late arrivals

DePalma

Arnott

2012

Transportation Research Part B: Methodological

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“…Otherwise, this is identical to the situation studied in the two previous subsections for an internal generalized Riemann problem. Again, we would need to consider the situation where ρ r and the linear function in the first element with end values ρ l and ρ r belong to different regimes in (4), for otherwise the solution is the one obtained in (Wong and Wong, 2002b). …”

Section: Sub-case B (B)mentioning

confidence: 99%

“…Moreover, discontinuous weak solutions are not unique for hyperbolic conservation laws and entropy conditions must be satisfied to obtain physically valid solution that is consistent with human behavior (such as the driver's ride impulse) (Ansorge, 1990;Velan and Florian, 2002). Recently, the analytical solution for specific classes of LWR model was derived, which assumed that the flow-density relationship is governed by a quadratic function throughout the density regime (Wong and Wong, 2002b), and then extended to the case of a piecewise quadratic function (Lu et al, 2006). Their constructed entropy solutions are exact if the initial condition is piecewise linear and the boundary condition is piecewise constant.…”

Section: Introductionmentioning

confidence: 99%

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

Lu¹,

Wong²,

Zhang³

et al. 2007

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In this paper we explicitly construct the entropy solutions for the Lighthill-WhithamRichards (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 three representative traffic flow cases, compare them with numerical solutions obtained by a high order weighted essentially non-oscillatory (WENO) scheme, and discuss the results from traffic flow perspectives.Key Words: LWR model; traffic flow; discontinuous flow-density relationship; entropy solution; WENO scheme Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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“…where F 1 (t), F 2 (t), F 3 (t) are the three different time-dependent functions that were explicitly derived in Wong and Wong [18] and expressed in terms of initial coordinates and the density of the element. Readers are referred to Wong and Wong [18] for the detailed expressions.…”

Section: Propagation Of a Shockmentioning

confidence: 99%

“…Readers are referred to Wong and Wong [18] for the detailed expressions. The trajectory of the shock path can then be traced by Figure 5.…”

Section: Propagation Of a Shockmentioning

confidence: 99%

Efficient implementation of the shock‐fitting algorithm for the Lighthill–Whitham–Richards traffic flow model

Chen

Wong

Shu

2007

Numerical Meth Engineering

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SUMMARYThis paper firstly presents the existence and uniqueness properties of the intersection time between two neighboring shocks or between a shock and a characteristic for the analytical shock-fitting algorithm that was proposed to solve the Lighthill-Whitham-Richards (LWR) traffic flow model with a linear speeddensity relationship in accordance with the monotonicity properties of density variations along a shock, which have greatly improved the robustness of the analytical shock-fitting algorithm. Then we discuss the efficient evaluation of the measure of effectiveness (MOE) of the analytical shock-fitting algorithm. We develop explicit expressions to calculate the MOE-which is the total travel time that is incurred by travelers, within the space-time region that is encompassed by the shocks and/or characteristic lines. A numerical example is used to illustrate the effectiveness and efficiency of the proposed method compared with the numerical solutions that are obtained by a fifth-order weighted essentially non-oscillatory scheme.

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An analytical shock-fitting algorithm for LWR kinematic wave model embedded with linear speed–density relationship

Cited by 51 publications

References 15 publications

Morning commute in a single-entry traffic corridor with no late arrivals

Morning commute in a single-entry traffic corridor with no late arrivals

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

Efficient implementation of the shock‐fitting algorithm for the Lighthill–Whitham–Richards traffic flow model

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