The entropy solutions for the Lighthill-Whitham-Richards traffic flow model with a discontinuous flow-density relationship Yadong Lu 1 , S.C. Wong 2 , Mengping Zhang 3 , Chi-Wang Shu 4 Abstract 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 construct- ing 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 per- spectives. Key Words: LWR model; traffic flow; discontinuous flow-density relationship; entropy solution; WENO scheme 1 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China. E-mail: [email protected]2 Department of Civil Engineering, The University of Hong Kong, Hong Kong, China. E-mail: [email protected]3 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China. E-mail: [email protected]4 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail: [email protected]1
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The entropy solutions for the Lighthill-Whitham-Richards tra!c flow model
et al., 1984; Wong and Wong, 2002a; Zhang et al., 2003). The main di!culty in designing
e!cient and high order finite di"erence methods for the LWR model or in general for hy-
perbolic conservation laws is the inherent presence of discontinuities (shocks) in the solution
(Lebacque, 1996). Moreover, discontinuous weak solutions are not unique for hyperbolic con-
servation 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. The fundamental diagrams in their works are continuous.
However, when tra!c flow data are plotted on the fundamental diagram, the uncongested
and congested regimes may be separated by gaps or discontinuities as shown in Figure 1.
Edie (1961) was among the first to point out that tra!c behaved di"erently at di"erent
2
density regimes, and introduced the idea of a two-regime model leading to a discontinuous
fundamental diagram. The discontinuous fundamental diagrams have also been observed
from empirical works (e.g., Ceder, 1976; Ceder and May, 1976; Drake et al., 1967; May and
Keller, 1967). In particular, Koshi et al. (1983) suggested a ”reverse lambda” shape to
describe the characteristics of the data plotted on the discontinuous fundamental diagrams.
Further evidence of the discontinuous fundamental diagram was revealed by a series of papers
by Hall (1987), Hall and Gunter (1986) and Hall et al. (1986). In addition, Bank (1991a,b)
described this discontinuous fundamental diagram as a two-capacity phenomenon, with one
capacity corresponding to the tip of the left leg of the reverse lambda, and the other capacity
belonging to the tip of the right leg of the reverse lambda with a capacity drop from the
former tip. More recently, Cassidy (1998) and Cassidy and Bertini (1999) also confirmed
the capacity drop on highways. When such discontinuous fundamental diagram is embedded
into the LWR model, it is to our best knowledge that there is still no mathematical theory
on the existence and uniqueness of the entropy solutions for the resultant tra!c model.
In this paper we assume that the flow-density relationship q(!) is concave and is repre-
sented by a piecewise quadratic function, with any two adjacent pieces joining discontinu-
ously at a critical density !0. Such discontinuous fundamental diagrams were developed in
Drake et al. (1967) by fitting with observed data. Our procedure to construct the physically
relevant solutions for such conservation laws with discontinuous fluxes is as follows. We first
explicitly construct the entropy solutions to a sequence of approximate problems in which
the flow-density relationship q(!) is continuous but tends to the discontinuous flux when a
small parameter in this sequence tends to zero. We then explicitly construct the limit of
the entropy solutions for this sequence and consider this limit solution as the the entropy
solution associated with the discontinuous flux. In order to verify the physical relevancy
of such entropy solutions, we apply our results to a few typical tra!c flow examples and
comment on the implication of the solutions.
The organization of the paper is as follows. In Section 2 we obtain the explicit formulas
3
Density !
Flow
q(!)
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
3000
3500
4000
!0
q1(!) q2(!)
Figure 1: A typical flow with two di"erent concave quadratic functions joining discontinu-ously at a critical density !0 with decreasing derivative at !0.
for the entropy solutions to Riemann problems with discontinuous flow-density relationship,
in a limit process involving a sequence of approximate problems in which the flow-density
relationship q(!) is continuous but tends to the discontinuous flux when a small parameter in
this sequence tends to zero. In Section 3 we obtain the explicit formulas for the entropy solu-
tions with discontinuous flow-density relationship and with piecewise linear initial condition
and piecewise constant boundary conditions. In Section 4 we provide numerical examples
in tra!c flows to demonstrate the explicit solutions obtained in Sections 2 and 3. We also
compare these explicit solutions with numerical solutions obtained by using the high order
weighted essentially non-oscillatory (WENO) schemes (Jiang and Shu, 1996; Zhang et al.,
2003). Concluding remarks are given in Section 5.
4
2 A sequence of approximate problems with continu-ous fluxes: Riemann problems
The governing equation for the LWR model is the following scalar hyperbolic conservation
law
!t + q(!)x = 0 (1)
with suitable initial and boundary conditions. Here ! ! (0, !max) is the density, !max is the
maximum (jam) density, and q(!) is the tra!c flow on a homogeneous highway, which is
assumed to be a function of the density ! only in the LWR model. More specifically, the
flow q, the density ! and the equilibrium speed u are related by
q(!) = u(!) !. (2)
In this paper, the flow q(!) is considered to be piecewise quadratic and locally concave in
each piece. Without loss of generality, we will concentrate our discussion on the situation
where the flow q is defined by two di"erent quadratic functions in di"erent regimes
Figure 10: The exact entropy solution obtained by the procedure in Section 3 (solid line)and the numerical solution obtained by using WENO scheme with N = 1000 uniform gridpoints (circles) at the time t=30 min for Example 1 (a-1) and (a-2), respectively.
Figure 11: The exact entropy solution obtained by the procedure in Section 3 (solid line)and the numerical solution obtained by using WENO scheme with N = 1000 uniform gridpoints (circles) at the time t=12 min for Example 1 (b-1) and (b-2), respectively.
The exact solution of these problems can be worked out using the procedure in this
paper, shown as solid lines in Figures 10 and 11, in comparison with the numerical solution
obtained by the WENO scheme using N = 1000 uniform grid points, shown as circles. In
both cases, we can see that the two results agree very well.
4.2 Example 2 (Rarefaction waves)
In this case k0 = $11.6569 in (22)-(25).
Example 2 (a-1): $1 = 300, %1 = 0,$2 = 20, %2 = 0
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q!2(300) < k0 and the solution is shown in Figure 12.
q!2(310) < k0 and the solution is also shown in Figure 12.
x
!
0 5 10 15 200
40
80
120
160
200
240
280
320
x
!
0 5 10 15 200
40
80
120
160
200
240
280
320
Figure 12: The exact entropy solution obtained by the procedure in Section 3 (solid line)and the numerical solution obtained by using WENO scheme with N = 1000 uniform gridpoints (circles) at the time t=6 min for Example 2 (a-1) and (a-2), respectively.
Example 2 (b-1): $1 = 80, %1 = 0,$2 = 20, %2 = 0
q!2(80) > k0 and the solution is shown in Figure 13.
q!2(70) > k0 and the solution is also shown in Figure 13.
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. Again, in both
cases, we can see that the two results agree very well.
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x
!
0 5 10 15 200
10
20
30
40
50
60
70
80
x
!
0 5 10 15 200
10
20
30
40
50
60
70
80
Figure 13: The exact entropy solution obtained by the procedure in Section 3 (solid line)and the numerical solution obtained by using WENO scheme with N = 1000 uniform gridpoints (circles) at the time t=6 min for Example 2 (b-1) and (b-2), respectively.
4.3 Example 3 (Wide moving jam)
Consider a long homogeneous freeway of length 30 km. We now assume the following initial
density
!(x, 0) =
!250, 15 < x < 17!d, otherwise
which represents the tra!c condition after an incident (recurrent or non-recurrent) at x = 17.
The left-hand entrance of the highway is always kept at a density of !d all time. We consider
three cases: (a) !d = 50, (b) !d = 45, and (c) !d = 40 veh/km, with entrance (demand) flows
of 4000, 3690, and 3360 veh/h, respectively. The demand of the first two cases exceed the
capacity of Flux II (3600 veh/h), whereas the demand of the third case is below this capacity.
Figures 14-16 plot the exact entropy solution (solid line) and the numerical solution obtained
by the fifth order WENO finite di"erence scheme with N = 1000 uniform grid points (circles)
for all three cases. We can see that the two results agree very well.
From the figures, we can clearly see the formation of a wide moving jam with two very
sharp shock fronts on both ends of the moving jam, which is a tra!c phenomenon that is
commonly observed on highways (Kerner and Rehborn, 1996), and was well studied using
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the three-phase tra!c theory (Kerner, 2004), cell transmission model (Lin and Lo, 2003),
and traveling wave analysis (Jin and Zhang, 2003; Kerner and Konhauser, 1994; Zhang and
Wong, 2006; Zhang et al., 2006). In Examples 3 (a) and 3 (b), when the demand from the
left-hand highway entrance is higher than the capacity of the congested regime (Flux II in
this example), the wide moving jam lasts for more than 30 minutes, whereas in Example 3
(c) when the demand is lower than this reduced capacity, the jam dissolves very quickly. It
is interesting that the wide moving jam can also be formed using the first order LWR model,
and does not require a linear flow-density relationship in the congested regime as discussed
in Lin and Lo (2003).
We also show in Figure 17 the time-space diagrams and three-dimensional speed plots of
these three cases to illustrate the results.
5 Conclusions
In this paper we consider the explicit construction of physically relevant entropy solutions of
a class of conservation laws with discontinuous flux functions, which are piecewise quadratic
and locally concave in each piece. We treat this problem as the limit of a sequence of approx-
imate problems in which the fluxes are continuous functions but with progressively sharper
gradients. We have presented explicit formulas for such entropy solutions for both the simple
Riemann initial conditions and for piecewise linear initial conditions and piecewise constant
boundary conditions. We demonstrate these explicitly constructed entropy solutions to rep-
resentative tra!c flow problems and compare them with numerical solutions obtained with
high order WENO schemes.
6 Acknowledgements
The work of the second author was partially supported by a grant from the Research Grants
Council of the Hong Kong Special Administrative Region, China (HKU7187/05E). The re-
search of the third author was partially supported by the NSFC grant 10671190. The re-
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search of the fourth author was partially supported by the NSFC grant 10671190 while this
author was visiting the Department of Mathematics, University of Science and Technology
of China, Hefei, Anhui 230026, P.R. China. Additional support was provided by ARO grant
W911NF-04-1-0291 and NSF grant DMS-0510345.
References
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portation Research Part B, 24 (2), 133-143.
[2] Bank, J.H., 1991a. Two-capacity phenomenon at freeway bottlenecks: a basis for ramp
metering? Transportation Research Record, 1320, 83-90.
[3] Bank, J.H., 1991b. The two-capacity phenomenon: some theoretical issues. Transporta-
by finite-di"erence methods. Transportation Research Part B, 18 (4-5), 409-421.
[28] Richards, P.I., 1956. Shock waves on the highway. Operations Research, 4 (1), 42-51.
[29] Velan, S., Florian, M., 2002. A note on the entropy solutions of the hydrodynamic model
of tra!c flow. Transportation Science, 36 (4), 435-446.
37
[30] Whitham, G.B., 1974. Linear and Nonlinear Waves, John Wiley & Sons, New York,
USA.
[31] Wong, G.C.K., Wong, S.C., 2002a. A multi-class tra!c flow model - an extension of
LWR model with heterogeneous drivers. Transportation Research Part A, 36 (9), 827-
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[32] Wong, S.C., Wong, G.C.K., 2002b. An analytical shock-fitting algorithm for LWR kine-
matic wave model embedded with linear speed-density relationship. Transportation Re-
search Part B, 36 (8), 683-706.
[33] Zhang, M., Shu, C.W., Wong, G.C.K., Wong, S.C., 2003. A weighted essentially non-
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[34] Zhang, P. and Wong, S.C., 2006. Essence of conservation forms in the traveling wave
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Distance(km)
Density(veh/km)
0 10 20 300
25
50
75
100
125
150
175
200
225
250
Distance(km)
Density(veh/km)
0 10 20 300
25
50
75
100
125
150
175
200
225
250
Distance(km)
Density(veh/km)
0 10 20 300
25
50
75
100
125
150
175
200
225
250
Distance(km)
Density(veh/km)
0 10 20 300
50
100
150
200
250
Figure 14: The exact entropy solution (solid line) and the numerical solution obtained byWENO scheme (circles) for Example 3 (a) !d = 50. Top left: t = 5 min; top right: t = 15min; bottom left: t = 30 min; bottom right: t = 60 min.
39
Distance(km)
Density(veh/km)
0 10 20 300
25
50
75
100
125
150
175
200
225
250
Distance(km)
Density(veh/km)
0 10 20 300
25
50
75
100
125
150
175
200
225
250
Distance(km)
Density(veh/km)
0 10 20 300
25
50
75
100
125
150
175
200
225
250
Distance(km)
Density(veh/km)
0 10 20 300
50
100
150
200
250
Figure 15: The exact entropy solution (solid line) and the numerical solution obtained byWENO scheme (circles) for Example 3 (b) !d = 45. Top left: t = 5 min; top right: t = 15min; bottom left: t = 30 min; bottom right: t = 60 min.
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Distance(km)
Density(veh/km)
0 10 20 300
50
100
150
200
250
Distance(km)
Density(veh/km)
0 10 20 300
50
100
150
200
250
Distance(km)
Density(veh/km)
0 10 20 300
50
100
150
200
250
Distance(km)
Density(veh/km)
0 10 20 300
50
100
150
200
250
Figure 16: The exact entropy solution (solid line) and the numerical solution obtained byWENO scheme (circles) for Example 3 (c) !d = 40. Top left: t = 5 min; top right: t = 15min; bottom left: t = 30 min; bottom right: t = 60 min.
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Distance(km)
Time(min)
0 10 20 300
10
20
30
40
50
60
S1
S2
t0
t1
t2
t3
Distance(km)
Time(min)
0 10 20 300
10
20
30
40
50
60
S1
S2
S3
t0
t1t2
t3
t4
t5
Distance(km)
Time(min)
0 10 20 300
10
20
30
40
50
60
S1
S2
S3
t0
t1t2
t3
t4
t5
t6
Figure 17: The exact entropy solutions for Example 3. Top: case (a); middle: case (b);bottom: case (c). Left: time-space diagram; right: 3D speed plot.