9783540755593-c2

Chapter 2

Trac Flow Theoryfor 1-D

2.1 Introduction

Interest in modeling trac ow has been around since the appearanceof trac jams. Ideally, if you can correctly predict the behavior ofvehicle ow given an initial set of data, then, in theory, adjusting theow in crucial areas can maximize the overall throughput of tracalong a stretch of road. This is of particular interest in regions ofhigh trac density, which may be caused by high volume peak timetrac, accidents or closure of one or more lanes of the road.

The development of the pedestrian evacuation dynamic systemsfollows from the trac ow theory in 1-D space [2, 39, 51]. In manyways, the pedestrian evacuation system is similar to the vehicle tracow problem [73]. The main conservation equations used in modelingthe vehicle trac ow and the pedestrian evacuation ow are thesame, with the exception that vehicle trac is a 1-D space problemand the evacuation system is a 2-D space problem. Other similaritiesexist from having escape routes and escape times in both problems.In most of the situations a vehicle or a pedestrian has more than oneroute to a destination and each route has an associated cost, suchas time.

In this chapter we will give the necessary background on tracow theory and survey the existing macroscopic mathematical mod-els for single-lane, 1-D space trac ow. These models will be used

5

6 2 Trac Flow Theory for 1-D

for crowd ow in 1-D, and they will be modied in Chap. 3 for2-D ow. In Sect. 2.2, we start with the concept of macroscopic vsmicroscopic ways of modeling the trac ow problem, followed bySect. 2.3, where a microscopic model is introduced. The derivationof the trac ow theory based on conservation of mass law, and therelationships between velocity and density are given in Sect. 2.4. InSect. 2.5, four macroscopic trac ow models are presented, derived,and analyzed based on their mathematical characteristics. Finally,the exact and weak solutions to the scalar trac ow PDE, and theconcepts of shock wave, rarefaction wave, and the admissibility of asolution are considered.

2.2 Microscopic vs Macroscopic

In the trac ow problem, there are two classes of models: Macro-scopic, which is concerned with average behavior, such as trac den-sity, average speed and module area, and a second class of modelsbased on individual behavior referred to as microscopic models. Thelatter is classied into dierent types. The most famous one is theCar-Following models [6, 17, 57], where the driver adjusts his or heracceleration according to the conditions in front. In these models thevehicle position is treated as a continuous function and each vehicle isgoverned by an ordinary dierential equation (ODE) that depends onspeed and distance of the car in front. Another type of microscopicmodels are the Cellular Automata or vehicle hopping which diersfrom Car-Following in that it is a fully discrete model. It considersthe road as a string of cells which are either empty or occupied by onevehicle. One such model is the Stochastic Trac Cellular Automata,given in [75]. However, microscopic approaches are computationallyexpensive, as each car has an ODE to be solved at each time step,and as the number of cars increases, so does the size of the systemto be solved. On the other hand, the macroscopic models are com-putationally less expensive because they have fewer design details interms of interaction among vehicles and between vehicles and theirenvironment. Therefore, it is desirable to use macroscopic models ifa good model can be found satisfactorily to describe the trac ow.In addition, this idea provides exibility since detailed interactionsare overlooked, and the models characteristics are shifted toward

2.3 Car-Following Model 7

parameters such as ow rate f(, v), concentration (also known astrac density), and average speed v, all functions of 1-D or 2-D space(x, y), and time (t). This is also true for rst-order uid dynamicmodels of isothermal ow and gases through pipes.

Two main prototypes set the stage for macroscopic trac ow:the rst is called the LWR model which is a non-linear, rst-orderhyperbolic PDE based on law of conservation of mass. The secondone is a second-order model known as the PW model, which is basedon two coupled PDEs one given by the conservation of mass and asecond equation that mimics trac ow.

2.3 Car-Following Model

We present the well known car-following microscopic trac owmodel. In [93], a 2-D version of this model was used for pedestrianow in 2-D space. To derive the 1-D model, rst assume cars cannot pass each other. Then the idea is that a car in 1-D can move andaccelerate forward based on two parameters; the headway distancebetween the current car and the one in front, and their speed dier-ence. Hence, it is called following, where a car from behind followsthe one in front, and this is the anisotropic property. This propertyis also desirable in macroscopic models, since it reects the actualobserved behavior of trac ow [23].

Suppose the nth car location is xn(t), then the nonlinear modelis given by

xn(t) = cxn(t) xn1(t)xn(t) xn1(t) , (2.1)

The acceleration of the current car xn(t) depends on the front carspeed and location, c is the sensitivity parameter. Integrating theabove yields

xn(t) = c ln (xn(t) xn1(t)) + dn. (2.2)

Since by the denition of the density (number of cars per unit area)

1(x, t)

= xn(t) xn1(t), (2.3)


and the integration constant dn is chosen such that at jam densitym, the velocity is zero. Then for steady-state we get

v = c ln m

. (2.4)

We see that for 0 we get in trouble, but from observations inlow trac densities, car speed is the maximum allowed speed, hencewe can assume v = vmax, which is the maximum allowed speed.

2.4 Trac Flow Theory

In this section we will cover the vehicle trac ow fundamentals forthe macroscopic modeling approach. The relation between density,velocity and ow is presented for trac ow. Then we derive theconservation of vehicles, which is the main governing equation forscalar macroscopic trac models. Finally, the velocitydensity func-tions that makes the conservation equation a function of only onevariable (density) are given.

2.4.1 Flow

In this section, we will illustrate the close relationship between thethree variables: density, velocity and trac ow. Suppose there isa road with cars moving with constant velocity v0, and constantdensity 0 such that the distance between the cars is also constantas shown in the Fig. 2.1a. Now let an observer measure the numberof cars per unit time that pass him (i.e. trac ow f). In time,each car has moved v0 distance, and hence the number of cars thatpass the observer in time is the number of cars in v0 distance, seeFig. 2.1b.

Since the density 0 is the number of cars per unit area and thereis v0 distance, then the trac ow is given by

f = 0v0 (2.5)

This is the same equation as in the time varying case, i.e.,

f(, v) = (x, t)v(x, t). (2.6)

2.4 Trac Flow Theory 9

Watcher

(a)

Watcher

(b)

0v W

Fig. 2.1. (a) Constant ow of cars; (b) Distance traveled in hoursfor a single car

To show this, consider the number of cars that pass point x = x0 in avery small time t. In this period of time the cars have not moved farand hence v(x, t), and (x, t) can be approximated by their constantvalues at x = x0 and t = t0. Then, the number of cars passing theobserver occupy a short distance, and they are approximately equalto (x, t)v(x, t)t, where the trac ow is given by (2.6).

2.4.2 Conservation Law

The models for trac, whether they are one-equation or system ofequations, are based on the physical principle of conservation. Whenphysical quantities remain the same during some process, these quan-tities are said to be conserved. Putting this principle into a mathe-matical representation will make it possible to predict the densitiesand velocities patterns at future time. In our case, the number ofcars in a segment of a highway [x1, x2] are our physical quantities,and the process is to keep them xed (i.e., the number of cars comingin equals the number of cars going out of the segment). The deriva-tion of the conservation law is given in [26, 37], and it is presentedhere for completion. Consider a stretch of highway on which cars are


U

1x 2x

Fig. 2.2. One-dimension ow

moving from left to right as show in Fig. 2.2. It is assumed herethat there are no exit or entrance ramps. The number of cars within[x1, x2] at a given time t is the integral of the trac density given by

N = x2x1

(x, t) dx. (2.7)

In the above equation, it is implied that the number of peoplewithin [x1, x2] is at maximum when trac density is equal to jamdensity m which is associated with the maximum number of carsthat could possibly t in a unit area.

The number of cars can still change (increase or decrease) in timedue to cars crossing both ends of the segment. Assuming no cars arecrated or destroyed, then the change of the number of cars is due tothe change at the boundaries only. Therefore, the rate of change ofthe number of cars is given by

dNdt

= fin(, v) fout(, v), (2.8)since the number of cars per unit time is the ow f(, v). Combining(2.7), and (2.8), yields the integral conservation law

ddt

x2x1

(x, t) dx = fin(, v) fout(, v). (2.9)

This equation represents the fact that change in number of cars is dueto the ows at the boundaries. Now let the end points be independentvariables (not xed with time), then the full derivative is replaced

2.4 Trac Flow Theory 11

by partial derivative to get

t

x2x1

(x, t) dx = fin(, v) fout(, v). (2.10)

The change in the number of cars with respect to distance is givenby

fin(, v) fout(, v) = x2

x1

f

x(, v) dx, (2.11)

and by setting the last two equations equal to each other, we get x2x1

[

t(x, t) +

f

x(, v)

]dx = 0. (2.12)

This equation states that the denite integral of some quantity isalways zero for all values of the independent varying limits of theintegral. The only function with this feature is the zero function.Therefore, assuming (x, t), and q(x, t) are both smooth, the 1-Dconservation law is found to be

t + fx(, v) = 0. (2.13)

We need to mention that this equation is valid for trac and manymore physical quantities. The idea here is conservation, and forvehicle trac ow, the ow is given by (2.6).

2.4.3 VelocityDensity Relationship(s)

Trac density and vehicle velocity are related by one equation, con-servation of vehicles,

t(x, t) + ((x, t)v(x, t))x = 0, (2.14)

where the notation () = () will be used from here on. If the initialdensity and the velocity eld are known, the above equation can beused to predict future trac density. This leads us to choose thevelocity function for the trac ow model to be dependent on densityand call it V (). The choice of such function depends on the behaviorthe model is trying to mimic. The following is a brief description ofmodels that have been recognized and used by researchers [57], withemphasis on Greenshield model that will be used in several trac(crowd) models throughout this book.


Greenshields Model [34]

This model is simple and widely used. It is assumed here that thevelocity is a linearly decreasing function of the trac ow density,and it is given by

V () = vf (1 m

) (2.15)

where vf is the free ow speed and m is the maximum density.Figure 2.3 shows the speed V ((x, t)) as a monotonically decreasingfunction. For zero density the model allows free ow speed vf , whilefor maximum density m no car can move in or out.

The uxdensity relationship for Greenshields model (2.15) isgiven in Fig. 2.4, where it shows the ux increases to a maximumwhich occurs at some density and then it goes back to zero. Thiskind of behavior is due to the fact that f () < 0 (note that f() =V () is the ux ow).

Greenberg Model [33]

In this model the speeddensity function is given by

V () = vf ln(

m) (2.16)

Underwood Model [33]

In the Underwood model the velocitydensity function is representedby

V () = vf exp(m

) (2.17)

fv

( ( , ))V x tU

( , )x tUmU

Fig. 2.3. Greenshields model for trac ow speed

2.5 Trac Flow Model 1-D 13

( ( , ))f x tU

mU

mf

( , )x tUU

Fig. 2.4. Trac ow ux as a function of density

Diusion Model [14, 74]

Diusion is a good extension to the model given by (2.15), where theeect of gradual rather than instantaneous reduction of speed by thedriver takes place in response to shock waves. This kind of reactioncan be accomplished by adding an extra term such that the modiedGreenshield model will become

V () = vf (1 m

) D(

x) (2.18)

where D is a diusion coecient given by

D = v2r

and vr is a random velocity, is a relaxation parameter.

2.5 Trac Flow Model 1-D

In this section, we will present four dierent models for trac owin 1-D space. The rst model is one equation model, and the restare systems of two-equation models. All the models are described bypartial dierential equations and based on conservation of mass and asecond equation that is intended to capture the complex interactionsobserved in trac ow motion. In addition, this second equationprovides another way to couple velocity and density. Hence the owis not in equilibrium like the one equation model. This is one of themain dierences between the scalar and the system models.


2.5.1 LWR Model

The rst model used in describing the trac ow problem knownas the LWR model, named after the authors in [68] and [87]. TheLWR model is a scalar, time-varying, non-linear, hyperbolic partialdierential equation. The model governing equation is (2.14). In thisequation, trac density is the conserved quantity, and we rewrite themodel as

t(x, t) + ((x, t)V ((x, t)))x = 0 (2.19)

with the ux being replaced by the velocitydensity relationship

f(x, t) = (x, t)V ((x, t)) (2.20)

and V ((x, t)) is the velocity function given by (2.15).One of the basic assumptions in the LWR model regarding the

velocity is its dependence on density alone. Any changes to densitywill be reected in the velocity. The drawback for this assumption,as pointed out in [23], is that trac is in equilibrium when suchvelocitydensity functions are used, i.e., given a particular density,especially for light trac, the velocity will be xed and the modeldoes not recognize that there is a distribution of desired velocitiesacross vehicles. Therefore the model is not able to describe observedbehavior in light trac, although one can argue that vf is an averagespeed which might take care of this issue. On the plus side, the modelis anisotropic as the nature of the observed trac ow, i.e. vehiclebehavior is aected by mostly the car in front. This can be foundfrom the model eigenvalue given by

f ((x, t)) =f

((x, t)) = V ((x, t)) + V ((x, t)), (2.21)

which means that the model allows information to travel as fast asthe ow of trac, and not more, since it satises 0 < f () < V (),because

V () = vfm

. (2.22)

The LWR model given by (2.19) and (2.15) is a simple model andit is unable to capture all of the complex interactions for a realistictrac ow model. For this reason, modications to the LWR modelhave been suggested. One way is by the various velocitydensity


functions we gave in Sect. 2.4.3. The second way is by couplingthe conservation of mass with a second equation that tries to mimictrac motion instead of the velocitydensity models as given next.

2.5.2 PW Model

The rst system model to be presented is a two-equation model pro-posed in the 1970s independently in [82] and [102]. Their model wasthe rst model to couple velocity dynamics as a second equation,and it is referred to as the PW model. The rst equation is theconservation of mass as discussed in the previous sections

t + (v)x = 0, (2.23)

where the ux function f(, v) = v. In the LWR scalar equationmodel, a particular form of v was assumed where velocity is a functionof density, but in high order models, v and are assumed to beindependent and a second equation is formed to link them, as in uidand gas models. The second equation is derived from the Navier-Stokes equation of motion for a 1-D compressible ow, but with thepressure term replaced by P = C20, where C0 is the anticipation termthat describes the response of macroscopic driver to trac density,i.e. space concentration, and the pressure now is not pressure assuch. The model also includes a trac relaxation term that keepsspeed concentration in equilibrium

V () v

,

where is a relaxation time, and the velocity V () is the maximumout-of-danger velocity meant to mimic drivers behavior given ear-lier by (2.15) to (2.18). The second equation of the PW model innonconservative form is then given by

vt + v vx =V () v

C

20

x. (2.24)

To study this model, we have to nd its eigenvalues by rst rewritingthe model in conservation form. The rst step is to use the productrule

(v)t = vt + vt, (2.25)


and by multiplying (2.23) by v, we get

vt + v(v)x = 0. (2.26)

Then by substituting in for vt from the product rule (2.25), we get

v(v)x + (v)t vt = 0, (2.27)

and by substituting (2.27) into (2.24), and multiply the result by we get

vt + v vx = (

V () v

) C20x.

Then by substituting for vt from (2.27) we get

v(v)x + (v)t + v vx = (

V () v

) C20x. (2.28)

Again using the product rule on (vv)x, i.e.,

(vv)x = (v)xv + (v)vx (2.29)

and substituting in (2.28) we obtain

(v2)(v)t + (v2)x = (

V () v

) C20x. (2.30)

Hence we obtain (2.24) with the lefthand side in conservationform

(v)t + (v2 + C20)x = (

V () v

)(2.31)

where now and v are the conserved variables. Equations (2.23)and (2.31) can be written in vector form as

Qt + F (Q)x = S (2.32)

where,

Q =[

v

], F (Q) =

[v

v2 + C20

], S =

0

(V () v

) .

(2.33)


Setting the source term S = 0, we can rewrite the system inquasi-linear form as

Qt + A(Q)Qx = 0, (2.34)

where

A(Q) =F

Q=

[0 1

C20 v2 2 v]. (2.35)

Finally by solving for the eigenvalues from

|A(Q) I| = 0 (2.36)

we get two distinct and real eigenvalues

1,2 = v C0, (2.37)

therefore, the system is strictly hyperbolic.The model has a major drawback that researchers (see for exam-

ple [23]) are concerned about, mainly, that the model strongly followsthe uid ow theory. In uids, the behavior of a particle is aectedby its surrounding particles. Thus the anisotropic nature of tracis not preserved since the vehicles are allowed to move with nega-tive velocity, i.e. against the ow. This is clear from the eigenvalue,where one of 1,2 = vC0 is always greater than the vehicle speed v.So, information from behind aects the behavior of the driver, andthis is not true for observed trac ow. This is called the isotropicproperty.

2.5.3 AR Model

A new model in [4] and improved in [84] is argued to be an im-provement on the PW model. The authors of this model say thatother researchers have stuck too closely to uid ow models andhave not allowed for a signicant dierence between trac and u-ids, e.g., trac is more concerned with the ow in front, rather thanbehind. Therefore in order to move away from uids and towardthe anisotropic property of trac, they argue that replacing thepressure term with an anticipation term describing how the av-erage driver behaves is not a sucient x for the dierences betweenthe two types of ow. They claim that the drawback in the PWmodel (letting information travel faster than the ow) is due to an


incorrect anticipation factor involving the derivative of the pressurew.r.t. x. Therefore they suggest the correct dependence must involvethe convective derivative (full derivative) of the pressure term. Theconvective derivative in its general form is given by

D

Dt=

t+ (v ), (2.38)

for (x, t), and x n. They support their claim by the followingexample: Assuming that in front of a driver traveling with speedv the density is increasing with respect to x, but decreasing withrespect to (xvt). Then the PW type models predict that this driverwould slow down, since the density ahead is increasing with respectto x! On the contrary, any reasonable driver would accelerate, sincethis denser trac travels faster than him.

We call the model AR for short, and the rst PDE equation is thesame conservation of cars given by (2.23). However, in the AR modelthe next lagrangian equation replaces the second PDE equation givenin the PW model. This second equation is found by applying thefull derivative (2.38) to describe trac motion dynamics, and it isgiven by

(v + P ())t + v (v + P ())x = 0, (2.39)

where P () is an increasing function of density. This choice is to en-sure that this model caries the anisotropic property, and it is given by

P () = C20 , (2.40)

where > 0, and C0 = 1. Next we will put the second equation inconservation form, and nd the system eigenvalues. First multiply(2.39) by , then by using the product rule

( (v + P ()))t = t(v + P ()) + (v + P ())t, (2.41)(v (v + P ()))x = (v)x(v + P ()) + (v)(v + P ())x, (2.42)

we obtain

( (v+P ())tt(v+P ())+(v (v+P ()))x (v)x(v+P ()) = 0.(2.43)

Now, using the conservation law (2.23) for t, we can simplify theabove equation to

( (v + P ()))t + (v (v + P ()))x = 0, (2.44)


which is the conservation form of (2.39). Then our conserved vari-ables are , and (v + P ()). We proceed now to nd the systemeigenvalues, let X = (v + P ()) for simplication, then the ARmodel given by (2.23), and (2.44) can be rewritten as

t + (X P ())x = 0Xt +

(X2

XP ()

)x

= 0(2.45)

and in vector form (2.32), the stats and the ux are given by

Q =[

X

], F (Q) =

X P ()X2

XP ()

. (2.46)

For the quasi-linear form (2.34), the Jacobian is given by

A(Q) =F

Q=

( + 1) 1

(X2

2+

XP ()

) (2X P ()

) . (2.47)

Finally, solving for the eigenvalues from |A(Q)I| = 0, we nd twodistinct and real eigenvalues

1 = v P () & 2 = v. (2.48)Therefore, the system is strictly hyperbolic and since the pressureis an increasing function, then it is guaranteed that 1 < 2 due tothe fact that the maximum wave speed is equal to the velocity of theow v. Hence, the anisotropic property of trac is preserved.

2.5.4 Zhang Model

We present here a model that was proposed in [104, 105], and itclaims not to be of uid, or gas-like behavior. The model caries theanisotropic property, because the second equation is derived fromthe microscopic car-following model. Hence, a micro-to-macro linkis established for this model. Again, as in the PW and AR models,the Zhang model is also a system consisting of the conservation ofcars (2.23), and coupled with a second PDE that describe car motiongiven by

vt + vvx + V ()vx = 0 (2.49)


We start the micro-to-macro derivation from the homogeneous mi-croscopic car-following model (the relaxation term can be added for2-D crowd ow given in the next chapter) given by

(sn(t)) xn(t) = xn1(t) xn(t), (2.50)

wheresn(t) = xn1(t) xn(t), (2.51)

and sn(t) is a function of the local spacing between cars, xn(t) is theposition of the nth car, xn(t) is the acceleration, xn(t) is the velocity,and (sn(t)) is the average response time to the headway distance.Using the above notations, we rewrite (2.50), and dene the velocityas v(x, t) = x(t) to obtain the following

(s(x(t), t))dv(x, t)

dt=

d(s(x(t), t))dt

, (2.52)

and by using convective derivative t + vx on the velocity compo-nent, we get

(s) (vt + vvx) = (st + vsx). (2.53)

From the conservation law (2.23), let = 1/s, and by using thefollowing derivative form

Dx(a

b) =

bDxa aDxbb2

, (2.54)

for any a and b = 0, we get

st + vsx + usy = svx + suy, (2.55)

and by direct substituting in the right hand side of (2.53), we obtainour desired equation in the following form

(vt + vvx) =s

(s)vx, (2.56)

wheres

(s)= C() = V () 0 (2.57)

is the sound wave speed. This completes the derivation of the macro-scopic model (2.49) from its microscopic counterpart.


The conservative form of this model is derived next. First collectterms and rewrite (2.49) to get

vt + (v + V ()) vx = 0, (2.58)

then expand the conservation of mass equation

t + vx + vx = 0. (2.59)

We substitute for vx from (2.59) into (2.58) to obtain

vt + vvx + V () (t vx) = 0, (2.60)which can be rewritten as

vt + vvx (V ())t v (V ())x = 0, (2.61)or in lagrangian form as

(v V ())t + v (v V ())x = 0. (2.62)We now proceed to nd the conservation form by multiplying (2.62)by and using the product rules

((v V ()))t = t(v V ()) + (v V ())t, (2.63)( v (v V ()))x = (v)x(v V ()) + v (v V ())x,(2.64)

to get

((vV ()))tt(vV ())+(v (vV ()))x(v)x(vV ()) = 0.(2.65)

From (2.59), we substitute for t in the above equation to obtain ournal conservation form given by

( (v V ()))t + (v (v V ()))x = 0, (2.66)where our states are given by and (vV ()). For the quasi-linearform (2.34), let X = (v V ()), and write the system in vectorform (2.32), such that

Q =[

X

], F (Q) =

X + V ()X2

XV ()

. (2.67)


The Jacobian is then can be found to be

A(Q) =F

Q=

V () + V

() 1

(

X2

2XV ()

) (2X

+ V ())

. (2.68)

Finally by solving for the eigenvalues from |A(Q) I| = 0, we ndtwo distinct and real eigenvalues

1 = v + V () & 2 = v (2.69)

therefore the system is strictly hyperbolic. Since V () is negativeand given by (2.22), then the maximum the information can travelis equal to the vehicle speed v.

2.5.5 Models Summary

The one-equation LWR model consists of a single wave whose velocityis given by the derivative of the ux function, and information travelsforward at a maximum not faster than the speed of trac. There-fore the model behavior is anisotropic, i.e. only reacts to conditionsahead. For the two-equation models, the PW has two waves travelingat speeds given by vC0, one of them will always be traveling fasterthan the current speed v. This is a major cause of criticism of thismodel. The AR model has wave speeds given by v and v P ().This seems reasonable since, as in the LWR model, the faster wavewill move at the same speed as the trac v. This is also true for theZhang model, whose wave speeds are given by v and v+ V () withV () 0. This demonstrates the desirable anisotropic nature of theLWR, AR and Zhang models, and the isotropic nature of the PWmodel, for which it is severely criticized. In addition, Zhang modelhas a microscopic counterpart, which is not true for the PW and ARmodels. Although, the AR model has an indirect micro-to-macro re-lationship [4], where a numerical discretization of the AR model anda microscopic car-following model gave the same numerical formula.This suggest that the macroscopic AR model can be considered asan upper limit to the microscopic model.

2.6 Method of Characteristics 23

2.6 Method of Characteristics

A typical problem in partial dierential equation consists of ndingthe solution of a PDE subject to boundary conditions (BVP), initialconditions (IVP), or both (IBVP). In most cases it is dicult to ndthe exact (classical) solution of a hyperbolic PDE, but due to thesimplicity of the LWR model and the fact it is a scalar 1-D spacemodel we are able to nd the exact solution by method of character-istics. The method of characteristics is a widely used technique tosolve hyperbolic PDEs [24, 58, 77, 86].

2.6.1 LWR Model Classication

The partial derivative scalar conservation law in (2.19) is classiedas rst-order quasi-linear partial dierential equation. This is due tothe fact that the derivative of the highest partial occurs linearly. Wecan rewrite (2.19) as

t(x, t) + f ((x, t)) x = 0 (2.70)

where f () is the vehicle speed, and it is called the characteristicslop or the eigenvalue of the PDE. By using Greenshields model(2.15), we get

f ((x, t)) =df((x, t))

dx= vf 2 vf (x, t)

m, (2.71)

where we see that this eigenvalue is real. Therefore, the LWR modelis classied as strictly hyperbolic PDE. Figure 2.5 below shows thechanges in speed f () with respect to the changes in density .This relationship is important in nding the solution to the tracow model (2.70) by using method of characteristics discussed in thefollowing section.

2.6.2 Exact Solution

Here we will use the method of characteristics to solve the initialvalue problem (IVP) and also called the Cauchy problem given by

{t(x, t) + f((x, t))x = 0

(x, 0) = 0(x)(2.72)


mU

fv

fv

/ 2mU ( , )x tU

'( ( , ))f x tU

Fig. 2.5. Characteristic slops vs density

where x and time t +. Flow rate f : is assumed tobe a smooth function, at least C2 (i.e., twice dierentiable) and theinitial condition 0 : is continuous. For the single conserva-tion law, the eigenvalue of the PDE in (2.72) is given by the slope ofthe characteristic curve found from the quasi-linear form (2.70) as

() = f () (2.73)

Theorem 2.6.1 Any C1 solution of the single conservation law in(2.72) is constant along its characteristics. Accordingly, character-istic curves for the partial derivative conservation law in (2.72) arestraight lines.

Proof See [86].The above theorem implies that any curve of the form

x(t) = k t + x(0) (2.74)

is a characteristic curve where x(t) is the solution, k = f ((x(t), t))is the constant slope of the characteristic rays and x(0) is the initialposition of the characteristics rays. To show that (2.74) is indeed asolution to our Cauchy problem, let us rst dene what is meant bya solution.

Denition 2.6.1 Let f : be smooth, and let 0 : becontinuous. We say that (x, t) : (+) is a classical solutionof the Cauchy problem if (x, t) C1 ( +) C0 ( +) and(2.70) is satised.


We can verify that it is indeed a solution by substituting forx(0) = x f t from (2.74) to get

(x, t) = 0(x f t), (2.75)

then, by taking partial derivatives with respect to t and x, respec-tively, we obtain

t = 0(x f t)(f ), and x = 0(x f t).

Substituting the above in (2.70), we get

0(x f t)(f ) + f 0(x f t) = 0,

which shows (2.72) is satised. So, the exact solution is basicallythe initial data shifted by the slope of the characteristic as shown inFig. 2.6.

From the initial data we are able to generate the slopes of thecharacteristic rays originating from the x-axis. For some certain pro-les of initial data, this gives us a method for solving the Cauchyproblem. To illustrate this method, let us consider the initial datain Fig. 2.7, where we have the density prole of heavy trac densityat one end and light at the other end. Substituting for the initialdensity values in (2.71) will give the slopes of each of the character-istics. Then the solution follows from (2.74) along the rays of thecharacteristics as shown on the same gure.

2.6.3 Blowup of Smooth Solutions

Unfortunately, other examples with dierent initial data show howeasily the procedure above fails. In Fig. 2.8, we see a density prole

x

0 ( )x

'0 ( )x f t

( , )x t

Fig. 2.6. Exact solution is achieved by shifting the initial densityprole


x

0 ( )x

/ 2m

t

x

kx

kx

1( , )x t

1t

0

Fig. 2.7. Density distribution in the upper part, and the correspond-ing characteristic rays in the lower part

that describes road condition when cars are approaching red traf-c light. Although initial density is continuous, the characteristicsoverlap at some later point in time. Since our solution cannot bemulti-valued, we must conclude (in light of Theorem 2.6.1) that thesolution shown cannot be smooth. For this type of initial data, atheory of discontinuous solutions, or shock wave solution is used.

Moreover, in Fig. 2.9, we face a dierent kind of problem. Thistime we have initial prole corresponding to heavy trac at thebeginning, then at some point xk it is lighter. This example can

x

0 ( )xU

t

x

/ 2mUmU

kx

Fig. 2.8. Overlapping characteristics from continuous initial data


0 ( )xU

xkx

/ 2mUmU

x

t?

Fig. 2.9. Characteristics do not specify solution in the wedge

be related to conditions of a red light turns to green. As we all ob-serve in real situation, cars start to accelerate from high density (lowspeed) to low density (higher speed). The exact solution for this datashows that there is a region untouched by any characteristics fromthe given initial data. Thus, the method of characteristics did notidentify a solution in this region. As we shall see next, for this casewe will be able to identify a continuous solution called a rarefactionor fan wave solution to ll the wedge.

2.6.4 Weak Solution

From the discussion above, smooth solutions of a single conservationlaws can blow up (develop discontinuities or singularities) in nitetime which fails to make any sense. Therefore, one cannot follow thepractice of accepting solutions to the hyperbolic partial dierentialequations as given directly by method of characteristics. In order tounderstand discontinuous solutions, one needs to extend the notionof solution itself. One of the main features of the quasi-linear theoryfor hyperbolic PDEs is the notion of weak solutions. For a giveninitial data,

Denition 2.6.2 Let 0 L. Then is a weak solution or asolution in the distributional sense of (2.72) if and only if L(+) and,

28 2 Trac Flow Theory for 1-D 0

[(x, t)t(x, t) + f((x, t))x(x, t)] dxdt

+

0(x)(x, 0) dx = 0 (2.76)

is satised for every C0 ( [0,[)and C0 ( [0,[) := {C0 ( [0,[) | r > 0 s.t. support of Br(0, 0) ( [0,[)}.

Here (x, t) is a test function with compact support on the bound-ary (i.e., (x, t) is zero outside the boundary). In the weak solution(2.76), the partial derivative is moved to the test function that isguaranteed to be smooth. In addition, the denition above is anextension of the classical solution according to

Theorem 2.6.2 Suppose C1( [0,[) is a classical solutionof (2.72). Then is also a weak solution.

Proof is given in [86].We have to keep in mind that a weak solution might not be a

classical solution. So we need necessary and sucient conditionsfor the weak solutions to be the correct solution. We start by thenecessary condition for a piecewise-smooth weak solution known asthe Rankine-Hugoniot condition given by

s =[f()][]

=f(R) f(L)

R,L (2.77)

where s is the shock speed. Lets look at the earlier example inFig. 2.8, where the initial density was given by

(x, 0) ={ m

2 x < 0,m x 0, (2.78)

and we seek a solution to our Cauchy problem (2.72), using themethod of characteristics. Since the characteristics do overlap atsome point in time, the shock speed is calculated from (2.77)

s =(vfR vf2R/m) (vfL vf2L/m)

R L= vf vf

m(R + L)

= vf vfm

(m +m2

)

= 0.5 vf


and the solution is given by

(x, t) ={ m

2 x < s t,m x s t. (2.79)

This solution is shown in Fig. 2.10.Lets look now at the example of Fig. 2.9, and try to solve the

trac ow problem there. We will give two methods to nd the solu-tion and discuss which one must be used to get the correct solution.Using the initial density values

(x, 0) ={

m x < 0,m/2 x 0, (2.80)

we get the rst solution as a shock wave given by

s = vf vfm

(R + L) = vf vfm

(m/2 + m) = 0.5 vf

As we can see 0.5 vf is the same shock speed as in the previousexample and it is plotted in the Fig. 2.11b. The second solution iscontinuous and it provides another way to ll the wedge. It is calledthe rarefaction wave solution. The general form of the solution fortrac ow is given by

(x, t) =

L x < f(L) t,

f (x

t)1 f (L) t x < f (R) t,

R x f (R) t,(2.81)

0 ( )xU

x

x

t0x

mU

0x

/2mU

Fig. 2.10. Shock solution


0 ( )xU

x

/ 2mUmU

0x

x

t

(a)

x

t

0x

(b)

0x

(c)

Fig. 2.11. Initial density prole followed by two weak solutions, shockand fan respectively

where the 1 is an inverse mapping. For the trac ow problem,f (

x

t)1 can be found by letting

f (x

t)1 = vf vf

m=

x

t, (2.82)

and solving for the density solution to get

(x, t) = f (x

t)1 =

m2 m x

2vf t. (2.83)

Then, the continuous solution for the PDE is given by

(x, t) =

L x < vf t,f (

x

t)1 vf t x < 0,

R x 0.(2.84)

For the initial data given in Fig. 2.11a, the rarefaction wave solutionis plotted in Fig. 2.11c.


R

S

L R

S

L

(a) (b)

Fig. 2.12. Lax shock condition for trac ow problem: (a) not ashock, (b) shock

Such multiplicity of solutions is unacceptable. Thus we need aselection criterion that picks out the physically reasonable solutionfrom among the possible weak solutions (2.81), and (2.84). Lax shockcondition or Lax entropy condition [66] is a sucient condition forscalar conservation laws. The condition for the trac ow PDEstates that the discontinuous solution for the trac ow problemis admissible (i.e., a shock solution is selected with the direction ofincreased entropy) if

L < s < R, (2.85)

otherwise, the rarefaction wave solution is the admissible one (seeFig. 2.12 for easy interpretation). In the example of Fig. 2.11a, andaccording to the Lax condition, the rarefaction solution is the ad-missible one and the shock solution is not admissible. Finally, wesummarize the solution for the LWR model by mentioning the fol-lowing two points:

The solution is piece-wise smooth as t with jumps indensity (shocks) separating the pieces.

This means trac is predicted to be stable with transition be-tween stable regions approximated by discontinuous shocks.

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