A note on the weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill–Whitham–Richards traffic flow model

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2009; 25:1120–1126Published online 5 June 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.1277

A note on the weighted essentially non-oscillatory numericalscheme for a multi-class Lighthill–Whitham–Richards

traffic flow model

Peng Zhang1, S. C. Wong2,∗,† and Shi-Qiang Dai1

1Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai,People’s Republic of China

2Department of Civil Engineering, The University of Hong Kong, Hong Kong SAR, People’s Republic of China

SUMMARY

In a recent paper, the weighted essentially non-oscillatory (WENO) numerical scheme was applied tosolve a multi-class Lighthill–Whitham–Richards (MCLWR) traffic flow model (J. Comput. Phys. 2003;191:639–659). We discuss and present an enhanced WENO scheme with Lax–Friedrichs flux splitting byimproving the estimation of the minimal characteristic speed of the MCLWR model, which is based ona set of inequalities of eigenvalues. Copyright q 2009 John Wiley & Sons, Ltd.

Received 7 April 2009; Revised 21 April 2009; Accepted 29 April 2009

KEY WORDS: hyperbolic conservation laws; characteristic speed; numerical stability

In recent years, the modeling of multi-class traffic flow problems has received much attention [1–9],as it more realistically reflects the characteristics of traffic movements on highways, and numericalalgorithms were developed to solve these problems [10–18]. A notable numerical method isthe weighted essentially non-oscillatory (WENO) scheme, which has proven to be robust andefficient in solving hyperbolic conservation laws [19–21]. This is particularly evident when itis coupled with the Lax–Friedrichs flux-splitting method. A typical example is that of Zhanget al. [10], who applied the WENO scheme to an extended multi-class Lighthill–Whitham–Richards(MCLWR) traffic flow model that was proposed by Wong and Wong [1].

∗Correspondence to: S. C. Wong, Department of Civil Engineering, The University of Hong Kong, Hong Kong SAR,People’s Republic of China.

†E-mail: [email protected]

Contract/grant sponsor: National Natural Science Foundation of China; contract/grant numbers: 70629001, 10771134Contract/grant sponsor: National Basic Research Program of China; contract/grant number: 2006CB705503Contract/grant sponsor: University of Hong Kong; contract/grant number: 10207394Contract/grant sponsor: Research Grants Council of the Hong Kong Special Administrative Region, China;contract/grant number: HKU7183/08E

Copyright q 2009 John Wiley & Sons, Ltd.

A NOTE ON THE WENO SCHEME FOR THE MCLWR TRAFFIC FLOW MODEL 1121

Generally, when the theoretical bound for the eigenvalues of the system is available, it is used toconstruct the numerical scheme directly. However, in [10], the theoretical bound was not available,and thus an estimate was used, in which reasonable results were obtained with the WENO scheme,especially when a small CFL number was used. Although the defect caused by this inaccurateestimate was small, this note proposes an enhanced scheme based on the theoretical findings froma mathematical study in [13], which helps to offer a better estimate of the minimal characteristicspeed of the MCLWR model, thus improving the numerical results.

In Wong and Wong [1], the vehicles in traffic are divided into M types (‘classes’). km(x, t),um(x, t), and qm(x, t) denote the density, velocity, and flow of the m-class in location x and attime t . Then, the mass conservation for each type of vehicles is

�km�t

+ �qm�x

=0, 1�m�M

Using the relation qm =kmum , and further assuming that um is the strictly decreasing function ofthe total density

k=M∑l=1

kl for m=1, . . . ,M

we have the MCLWR model:

�km�t

+ �kmum(k)

�x=0, 1�m�M (1)

In this note, the types of vehicles are numbered from the slowest to the fastest, i.e.

u1(k)� · · ·�uM (k)

Note that each um satisfies the basic property of a velocity–density relation, i.e. um(kjam)=0,um(0)=u f m , where kjam is the jam density of the whole flow and ufm is the free flow (the maximal)velocity of the mth type.

In [13], the eigen-polynomial of system (1) was explicitly obtained and given by

P(�)=M∏

m=1(um−�)Q(�), Q(�)=1+

M∑m=1

cmum−�

, cm =kmu′m(k)�0 (2)

The existence of M real eigenvalues was proven through the application of the intermediatevalue theorem, and they were estimated in the following sequence:

u1+M∑l=1

cl��1�u1� · · ·�um−1��m�um� · · ·��M�uM (3)

Moreover, it was verified that {�m}|Mm=1 are distinct if no two velocity are equal and km>0 for all m,and thus the system is strictly hyperbolic. Note that the system might be non-strictly hyperbolicfor some km =0. The wave structure was also investigated in [13].

In general, {�m}|Mm=1 cannot be explicitly solved for the case M>2 [10]. Therefore, an estima-tion of

�≡maxk

max1�m�M

|�m |

Copyright q 2009 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2009; 25:1120–1126DOI: 10.1002/cnm

1122 P. ZHANG, S. C. WONG AND S.-Q. DAI

should be properly made for the Lax–Friedrichs flux splitting, and for the CFL condition thatensures numerical stability. Here, the vector k=(k1, . . . ,km)T, and the maximum is taken over therelevant region of k. The flux-splitting parameter � should be chosen in a way that satisfies

�−��0 (4)

In addition, � should be as close to � as possible to minimize the numerical viscosity.In this note, the enhanced approximation of � is based on (3) and given by

�=maxk

max

(∣∣∣∣u1+M∑l=1

cl

∣∣∣∣ , |uM |)

(5)

which obviously satisfies (4). In contrast, based on numerical experience, the approximation of �in [10] (see Equation (31) in the article) was made by setting

�=maxk

max1�m�M

|um | i.e. �=maxk

uM (6)

In the following, we show that (5) could hardly be improved, and that (6) fails to satisfy (4) insome cases. In other words, (6) is deficient and (5) is optimal in general. To substantiate ourarguments, let us consider a trivial case:

�1=u1+ku′1=u1+

M∑l=1

cl , �2=·· ·=�M =uM for u1=·· ·=uM (7)

which are obtained according to (2), and which indicates that �=� with � given by (5).On the other hand, the trivial case of (7) can be used to show the deficiency of (6). Suppose

that the total density k approaches kjam. Then, we have, k=kjam, �1=kjamu′1(kjam)<0, and �M =

uM (kjam)=0. In such a case, � is obviously close to zero (given by (6)), and hence �<�=|u1+ku′1|.

To be more specific, let u1=·· ·=uM =ue, and we apply the following velocity–density relation:

ue(k)=u f [1−(k/kjam)�], �>0 (8)

we have �=|�1|=|kjamu′1(kjam)|=�u f , and �=ue(k), for k∈[k,kjam]. If we choose k>(1−

�)1/�kjam for ��1, and k=0 for �>1, then condition (4) is violated (i.e. �<� in these cases).For example, if �=2 in the latter case, then the violation is remarkable, in which �−�=u f .In general, the difference of (�−�) increases with �.

The same arguments can be applied to another trivial case:

�1=u1+k1u′1=u1+

M∑l=1

cl , �m =um, m�2 for k2=·· ·=kM =0 (9)

We remark that (4) could hardly be satisfied by adopting (6) when the total density k is close tokjam. This is also true for non-trivial cases other than (7) and (9).

The model equations of (1) are written in the following vector form:

�k�t

+ �q(k)

�x=0 (10)

Accordingly, its Lax–Friedrichs flux splitting reads

q(k)=q+(k)+q−(k), q+(k)= 12 (q(k)+�k), q−(k)= 1

2 (q(k)−�k)



Here, we note that � is required to satisfy q+k =qk+�I�0 and q−

k =qk−�I�0, and I is unitmatrix. These requirements are actually for ‘upwinding’. According to the previous discussion,the � that is given by (5) will surely satisfy, but the � that is given by (6) may not. Correspondingto the flux splitting, the spatial discretization of (10) reads

�ki�t

+ 1

�x(q+

i+1/2− q+i−1/2)+

1

�x(q−

i+1/2− q−i−1/2)=0

In [10], the numerical fluxes q±i+1/2 are approximated by the fifth-order WENO reconstruction.

Moreover, the third-order TVDRunge–Kutta method is adopted for time discretization. For stability,considerable numerical experience suggests (if � is correct) the following CFL condition:

��t

�x�0.6 (11)

Also see [19, 20] for more details about the approximation. These are exactly our numericalimplementations that for further comparison between the �’s are given by (5) and (6), respectively.

Combining with (8), we write all functions of velocities as follows:

u1(k)=b1ue(k), . . . ,uM (k)=bMue(k), b1� · · ·�bM−1�bM =1 (12)

where all b′ms are positive constants, and we specify u f =20m/s. For comparison, it suffices to

show the total density in each example; and for simplicity the jam density kjam and the length Lof the computational interval are all scaled to unity.

Figure 1 shows the first example, with initial data given by k(x,0)=(0.25,0.05,0.1) for x<0.5,and k(x,0)=(0.2,0.5,0.3) for x>0.5. The functions um(k) are given by (12) and (8), with �=2,M=3, and (b1,b2,b3)=(0.8,0.9,1).Figure 2 shows the second example, in which we simulate the traffic flow at a signal-controlled

junction. The stop line is located at x=0.5, and the signal switches to green at t=0. The functionsum(k) are given by (12) and (8), with �=4, M=5, and (b1,b2,b3,b4,b5)=(0.8,0.85,0.9,0.95,1).Initially, the traffic states before and after the stop line are set as follows: k(x,0)=0.5 for x<0.3

x

Tota

l den

sity

(k)

0 0.25 0.5 0.75 1

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0st=30s

L=1000m

Δx=5m

α=1.82u

Δt=0.6Δx/α

(a)

f

x

Tota

l den

sity

(k)

0 0.25 0.5 0.75 1

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0st=20s

L=1000m

Δx=5m

α=0.84u

Δt=0.6Δx/α

f

(b)

Figure 1. (a) With �=1.82u f given by (5) and �t≈0.0824s and (b) with�=0.84u f given by (6) and �t≈0.1786s.



x

Tota

l den

sity

(k)

0 0.25 0.5 0.75 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1L=4000m

Δx=20m

α=3.6u

Δt=0.6Δx/α

f

t=0s

t=10s

t=20s

(a) x

Tota

l den

sity

(k)

0 0.25 0.5 0.75 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

L=4000m

Δx=20m

α=u

Δt=0.6Δx/α

f

t=0s

t=10s

t=20s

(b)

Figure 2. (a) With �=3.6u f given by (5) and �t=0.1617s and (b) with �=u f given by (6) and �t=0.6s.

x

Tota

l den

sity

(k)

0 0.5 1-40

-30

-20

-10

0

10

20

30

40

50

60

70

80

90

t=10s

t=0s

L=400m

Δx=4m

α=0.25u

Δt=0.6Δx/α

f

(a) x

Tota

l den

sity

(k)

0 0.5 1

0.75

0.8

0.85

0.9

0.95

1

t=10s

t=0s

L=400m

Δx=4m

α=0.25u

Δt=0.1Δx/α

f

(b)

Figure 3. For �=0.25u f that is given by (6), the stability is improved for much smaller �t with �x fixed,but the results are still unacceptable due to the improper flux splitting: (a) �t≈0.48s and (b) �t≈0.08s.

and x>0.7, k(x,0)=1 for 0.3<x<0.5, and k(x,0)=0 for 0.5<x<0.7. Furthermore, the densitiesof all five types are initially equal.

In the above two examples, non-physical oscillations are observed when the values of � aregiven by (6) (see Figures 1(b) and 2(b)). In contrast, if the values of � are given by (5), then thenumerical tests show that non-physical oscillation does not occur in both examples (see Figures1(a) and 2(a)). Moreover, the quality of results deteriorates for smaller �. This is also true in allother numerical tests, which indicates that the estimation of � in (5) is sufficiently good. Note thatthe equality of (11) is used for the determination of �t , i.e. �t=0.6�x/�.

With � given by (6), some improvement is expected if a smaller �t than 0.6�x/� is applied.Figure 3 shows such comparison, where Figure 3(a) indicates drastic oscillations (instability), butthe improved result in Figure 3(b) is still unacceptable; even�t is chosen to be sufficiently small andsatisfying the correct CFL condition that is determined by (5) and (11). Obviously, this means that



the deterioration of Figure 3(b) is completely due to the non-upwind Lax–Friedrichs flux splitting.In contrast, oscillation does not occur (not shown in the figure), when the value of � is chosenaccording to (5). In this third example, we set �=1 in (8), M=5 in (12) with (b1,b2,b3,b4,b5)=(0.8,0.85,0.9,0.95,1). Initially, k(x,0)=0.75 for x<0.5, k(x,0)=1 for x>0.5, and the densitiesof all five types are equal.

ACKNOWLEDGEMENTS

The work that is described in this paper was jointly supported by grants from the National NaturalScience Foundation of China (70629001,10771134), the National Basic Research Program of China(2006CB705503), the University of Hong Kong (10207394), and the Research Grants Council of theHong Kong Special Administrative Region, China (Project No.: HKU7183/08E).

REFERENCES

1. Wong GCK, Wong SC. A multi-class traffic flow model—an extension of LWR model with heterogeneous drivers.Transportation Research Part A 2002; 36:827–841.

2. Bagnerini P, Rascle M. An homogenized hyperbolic model of multiclass traffic flow: a few examples. NinthInternational Conference on Hyperbolic Problems, California Institute of Technology, Pasadena, CA, 25 March2002–29 March 2003. Source: Hyperbolic Problems: Theory, Numerics, Applications, 113–123.

3. Bagnerini P, Rascle M. A multiclass homogenized hyperbolic model of traffic flow. SIAM Journal on MathematicalAnalysis 2003; 35(4):949–973.

4. Benzoni-Gavage S, Colombo RM. An n-populations model for traffic flow. European Journal of AppliedMathematics 2003; 14:587–612.

5. Herty M, Kirchner C, Moutari S. Multi-class traffic models on road networks. Communications in MathematicalSciences 2006; 4(3):591–608.

6. Gupta AK, Katiyar VK. A new multi-class continuum model for traffic flow. Transportmetrica 2007; 3(1):73–85.7. Ngoduy D, Liu R. Multiclass first-order simulation model to explain non-linear traffic phenomena. Physica

A—Statistical Mechanics and its Applications 2007; 385(2):667–682.8. Logghe S, Immers LH. Multi-class kinematic wave theory of traffic flow. Transportation Research Part B 2008;

42(6):523–541.9. Ngoduy D. Multiclass first order modeling of traffic networks using discontinuous flow–density relationships.

Transportmetrica, DOI: 10.1080/18128600902857925.10. Zhang M, Shu CW, Wong GCK, Wong SC. A weighted essentially non-oscillatory numerical scheme for a

multi-class Lighthill–Whitham-Richards traffic flow model. Journal of Computational Physics 2003; 191:639–659.11. Zhang P, Dai SQ, Liu RX. Description and WENO numerical approximation to nonlinear waves of a multi-class

traffic flow LWR model. Applied Mathematics and Mechanics—English Edition 2005; 26(6):691–699.12. Xie ZM, Zhu ZJ, Hu JJ. Numerical study of mixed freeway traffic flows. Communications in Numerical Methods

in Engineering 2006; 22(1):33–39.13. Zhang P, Liu RX, Wong SC, Dai SQ. Hyperbolicity and kinematic waves of a class of multi-population partial

differential equations. European Journal of Applied Mathematics 2006; 17:171–200.14. Zhang P, Wong SC, Shu CW. A weighted essentially non-oscillatory numerical scheme for a multi-class traffic

flow model on an inhomogeneous highway. Journal of Computational Physics 2006; 212(2):739–756.15. Burger R, Kozakevicius A. Adaptive multiresolution WENO schemes for multi-species kinematic flow models.

Journal of Computational Physics 2007; 224:1190–1222.16. Burger R, Gracıa A, Karlsen KH, Towers JD. Difference schemes, entropy solutions, and speedup impulse for

an inhomogeneous kinematic traffic flow model. Networks and Heterogeneous Media 2008; 3(1):1–41.17. Burger R, Gracıa A, Karlsen KH, Towers JD. A family of numerical schemes for kinematic flows with

discontinuous flux. Journal of Engineering Mathematics 2008; 60(3–4):387–425.18. Zhang P, Wong SC, Xu ZL. A hybrid scheme for solving a multi-class traffic flow model with complex wave

breaking. Computer Methods in Applied Mechanics and Engineering 2008; 197(45–48):3816–3827.



19. Shu CW. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservationlaws. In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Cockburn B, Johnson C,Shu C-W, Tadmor E (eds). Lecture Notes in Mathematics, vol. 1697. Springer: Berlin, 1998; 325–432.

20. Gottlieb S, Shu CW, Tadmor E. Strong stability preserving high order time discretization method. SIAM Review2001; 43:89–112.

21. Shu CW. High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAMReview 2009; 51(1):82–126.


Documents

A note on the weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill–Whitham–Richards traffic flow model