Explicit construction of entropy solutions for the Lighthill–Whitham–Richards traffic flow model with a piecewise quadratic flow–density relationship

Lu, Yadong; Wong, S. C.; Zhang, Mengping; Shu, Chi‐Wang; Chen, Wenqin

doi:10.1016/j.trb.2007.08.004

Cited by 28 publications

(36 citation statements)

References 24 publications

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“…The first traffic flow test example is taken from [18]. The flow-density function is given by a concave function qðqÞ ¼ À0:4q 2 þ 100q; 0 6 q 6 50; À0:1q 2 þ 15q þ 3500; 50 6 q 6 100; À0:024q 2 À 5:2q þ 4760; 100 6 q 6 350:…”

Section: Distance (Km)mentioning

confidence: 99%

On maximum-principle-satisfying high order schemes for scalar conservation laws

Zhang

Shu

2010

Journal of Computational Physics

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488

531

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Section: Distance (Km)mentioning

confidence: 99%

On maximum-principle-satisfying high order schemes for scalar conservation laws

Zhang

Shu

2010

Journal of Computational Physics

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488

531

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“…In this case k 0 = −11.6569 in (22)- (25). The exact solution of these problems can be worked out using the procedure in this paper, shown as solid lines in Figures 12 and 13, in comparison with the numerical solution obtained by the WENO scheme using N = 1000 uniform grid points, shown as circles.…”

Section: Example 2 (Rarefaction Waves)mentioning

confidence: 95%

“…As for each fixed ∆t, S(∆t) and S (∆t) can both be readily computed, we can easily set up a Newton iteration to solve for the root of S(∆t) with ∆t = 0 as the initial guess. We can follow the discussion in the appendix of Lu et al (2006) to discuss the uniqueness of the solution to equation (56) before the natural break time, thereby facilitating the convergence of the Newton iteration process.…”

Section: Solution Proceduresmentioning

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%

“…Therefore, we cannot use our results in (Lu et al, 2006) directly. It is not possible to connect the two discontinuous pieces of the flux in a continuous and globally concave fashion.…”

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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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Data mining using regularized adaptive B‐splines regression with penalization for multi‐regime traffic stream models

Sun

Pan

2013

J of Advced Transportation

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SUMMARYThis paper presents a new data mining method that integrates adaptive B-spline regression and traffic flow theory to develop multi-regime traffic stream models (TSMs). Parameter estimation is implemented adaptively and optimally through a constrained bi-level programming method. The slave programming determines positions of knots and coefficients of the B-spline by minimizing the error of B-spline regression. The master programming model determines the number of knots through a regularized function, which balances model accuracy and model complexity. This bi-level programming method produces the best fitting to speed-density observations under specific order of splines and possesses great flexibility to accommodate the exhibited nonlinearity in speed-density relationships. Jam density can be estimated naturally using spline TSM, which is sometimes hardly obtainable in many other TSM. Derivative continuity up to one order lower than the highest spline degree can be preserved, a desirable property in some application. A five-regime B-spline model is found to exist for generalized speed-density relationships to accommodate five traffic operating conditions: free flow, transition, synchronized flow, stop and go traffic, and jam condition. A typical two-regime B-spline form is also explicitly given, depending only on free-flow speed, optimal speed, optimal density, and jam density.

show abstract

Explicit construction of entropy solutions for the Lighthill–Whitham–Richards traffic flow model with a piecewise quadratic flow–density relationship

Cited by 28 publications

References 24 publications

On maximum-principle-satisfying high order schemes for scalar conservation laws

On maximum-principle-satisfying high order schemes for scalar conservation laws

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

Data mining using regularized adaptive B‐splines regression with penalization for multi‐regime traffic stream models

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