. . . . . .

.

...

## Hyperbolic Balance Laws for Multilane Traffic Flow Model

Hsin-Yi Lee^{∗}

Department of Mathematics National Central University

The 27th Annual Meeting on Differential Equations and Related Topics

January 27, 2019

*∗*Jointly with Shih-Wei Chou and John Hong

. . . . . .

**Content**

History: LWR mode *⇒ PW model ⇒ AR model*

Goals: Multilane model (applying AR model)

. . . . . .

**LWR model Lightihll, Whitham (1955) & Richards (1956)**
*ρ**t**+ (ρv**e**(ρ))**x* *= 0, where q**e**(ρ) = ρv**e**(ρ).*

(conservation law of vehicles)

We have two important results if traffic is in equilibrium.

*1. (Anisotropy) v*_{e}^{′}*(ρ)≤ 0 ⇒ q**e*^{′}*(ρ) = v*_{e}*(ρ) + ρv*_{e}^{′}*(ρ)≤ v**e**(ρ).*

*2. (Acceleration) a = dv /dt =−ρ**x**(v*_{e}^{′}*(ρ))*^{2}*ρ*

. . . . . .

**Drawbacks of LWR model: Traffic is not in equilibrium**
1. Instability at the vacuum：slow drivers in the light traffic
2. drivers’ behavior：a = v_{x}*(...), not a =−ρ**x**(...)*

cf.

. . . . . .

**PW moel Payne (1971) & Whitham (1974)**

*ρ*_{t}*+ (ρv )*_{x}*= 0,*

*v**t**+ vv**x*+^{A}^{′}^{e}_{ρ}^{(ρ)}*ρ**x* = ^{v}^{e}^{(ρ)}_{τ}^{−v}*.*

*NOTE. Different choices on A*_{e}*(ρ), "pressure" of traffic*
1. Payne *A*^{′}_{e}*(ρ) =* _{2τ}^{1} *|v**e*^{′}*(ρ)|*

*2. Whitham A*^{′}_{e}*(ρ) = D*

*3. Zhang (1998): A*^{′}_{e}*(ρ) = (ρv*_{e}^{′}*(ρ))*^{2}
4. Others ...

**But ...**

. . . . . .

PW model

*ρ*_{t}*+ (ρv )*_{x}*= 0,*

*v*_{t}*+ vv** _{x}* +

^{A}

^{′}

^{e}

_{ρ}

^{(ρ)}*ρ*

*=*

_{x}

^{v}

^{e}

^{(ρ)}

_{τ}

^{−v}*.*

**Drawbacks of PW model (Daganzo, 1995)**

*Violate anisotropy : v +*^{√}*A*^{′}_{e}*(ρ) (wave) > v (vehicle)*
Diffusion: negative speed

**Ref. Daganzo, Requiem (安魂曲) for second-order fluid**
approximation of traffic flow, Transportation Res., Part B,
1995.

. . . . . .

**Fortunately, AR model overcomes these drawbacks Daganzo**
mentioned.

{ *ρ**t**+ (ρv )**x* *= 0,*
*α*_{t}*+ v α*_{x}*= 0,*
*where α = v + cρ*^{γ}*( γ > 0).*

This model obeys anisotopy,

instability at the vacuum, non-diffusion effect

reasonable drivers’ behavior

**Ref. Aw and Rascle, Resurrection (復活，Risurrezione) of**

"second" order models of traffic flow, SIAM, J. Appl. Math, 2000.

. . . . . .

**A Multilane Model (Greenberg, Klar and Rascle, 2003)**
*For an unidirectional one-dimensional road with n lanes, the*
*macroscopic variables are the density ρ of vehicles and the*
*average speed v across all the lanes. Then*

*ρ = Σ*^{i =n}_{i =1}*ρ** _{i}* and

*ρv = Σ*

^{i =n}

_{i =1}*ρ*

_{i}*v*

*(total flux),*

_{i}*where ρ*

_{i}*and v*

*are respectively the density and the average*

_{i}*speed of vehicles in the i -th lane.*

When traffic is high, lane changing and passing is difficult.

*⇒ the equilibrium speed for vehicles is low.*

When traffic is lower, these actions become easier.

*⇒ the equilibrium speed for vehicles is higher.*

. . . . . .

With the aid of Kerner’s three-phase traffic theory, Greenberg, Klar, and Rascle give the following figure.

*For simplicity, we apply Sopasakis’ argument (2002): ρ*_{1}*= ρ*_{2}.

. . . . . .

Multilane model (Greenberg, Klar, and Rascle, 2003)

*ρ**t**+ (m− cρ*^{2})*x* = 0
*m**t*+ (^{m}_{ρ}*(m− cρ*^{2}))*x* =

{ _{ρw}

1*(ρ)**−(m−cρ*^{2})

*τ* *, ρ < ρ*_{∗}*,*

*ρw*2*(ρ)**−(m−cρ*^{2})

*τ* *, ρ≥ ρ*_{∗}*,*
*where m≡ ρv + cρ*^{2} *and ρ̸= 0.*

*Multilane model (balance laws) U*_{t}*+ F (U)*_{x}*= G (U)*

*= AR model (conservation laws) U*_{t}*+ F (U)** _{x}* = 0

*+ Kerner’s theory, 1998 (source term) G (U)*

**Our goals are:**

(1) to solve Riemann problem for the AR model

(2) to get an approximate solution for the multilane model.

. . . . . .

.Definition (Riemann’s problem) ..

...

The Riemann problem is the initial-value problem for the conservation laws

*U*_{t}*+ F (U)** _{x}* = 0 in

*R × (0, ∞)*with the piecewise-constant initial data

*U(x , 0) =*

{ *U*_{L}*, x < 0,*
*U**R**, x > 0.*

*We call u*_{L}*and u** _{R}* the left and right initial data, respectively.

. . . . . .

*Since the flux F is a smooth function, the system we consider*
*is of the form U**t**+ DF (U)U**x* = 0. More precisely,

[
*ρ*
*m*

]

*t*

+

[ *−2cρ*^{2} 1

*−(*^{m}* _{ρ}*)

^{2}

*− mc*

^{2m}

_{ρ}*− cρ*] [

*ρ*
*m*

]

*x*

= [

0 0

]
*.*

*this gives the distinct real eigenvalues of DF (U)*
*λ*1*(U) =* ^{m}_{ρ}*− 2cρ < λ*2*(U) =* ^{m}_{ρ}*− cρ (= v)*
*so that the system is strictly hyperbolic since ρ̸= 0.*

Thus this model is anisotropic.

. . . . . .

Moreover, the corresponding eigenvectors can be taken as

*r*1*(U) =* ^{−1}* _{2c}*
[ 1

*m*
*ρ*

]

*and r*2*(U) =* ^{−1}* _{2c}*
[ 1

*m*
*ρ* *+ cρ*

]
*.*
.Definition

..

...

*The kth characteristic field is said to be genuinely nonlinear if*

*∇λ**k**(z)· r**k**(z)̸= 0 for all z.*

*The kth characteristic field is said to be linearly degenerate if*

*∇λ**k**(z)· r**k**(z) = 0 for all z.*

It follows from our system that

The first characteristic field is genuinely nonlinear since

*∇λ*1*(U)· r*1*(U) = 1̸= 0.*

The second characteristic field is linearly degenerate since

*∇λ*2*(U)· r*2*(U) = 0.*

. . . . . .

After some computations we have the three essential curves
*R*_{1}^{+}*(U** _{L}*),

*S*

_{1}

^{−}*(U*

*) and*

_{L}*C (U*

*).*

_{L}to help us to solve the Riemann problem of conservation laws.

*R*_{1}^{+}*(U** _{L}*)=

*{(ρ, m)*

*:*

^{T}

^{m}

_{ρ}

^{L}*L* = ^{m}_{ρ}*, ρ < ρ*_{L}*, m < m*_{L}*},*
*The slope = m**L**/ρ**L* *= v + cρ = v*0*+ c· 0, where v*0 is the
vehicle speed at the vacuum.

. . . . . .

*The three curves can be expressed in the v− ρ plane. That is,*
*R*_{1}^{+}*(v , ρ)*=*{(v, ρ)*^{T}*: v = v*_{L}*+ c(ρ*_{L}*− ρ), 0 < ρ < ρ**L**},*
*S*_{1}^{−}*(v , ρ)*=*{(v, ρ)*^{T}*: v = v**L**+ c(ρ**L**− ρ), ρ**L* *≤ ρ ≤ ρ*max*},*
*C (v , ρ)* =*{(v, ρ)*^{T}*: v = v*_{L}*, 0 < ρ≤ ρ**max**}.*

. . . . . .

Let*D = {(v, ρ) : 0 ≤ v ≤ w*1*(ρ)), 0 < ρ≤ ρ*max*, 0≤ v ≤ v*max*}.*

Then it is the triangle region.

.Theorem (Aw and Rascle 2000) ..

...

*Given the left initial state U**L**. If the right initial state U**R* *is in the*
*regionD, then there exists a unique integral solution U of*

*Riemann’s problem, which is constant on lines through the origin.*

NOTE. ThisD, the triangle region, is called the invariant region.

. . . . . .

But the Riemann’s problem obviously has no solution for such initial data in the following figure.

. . . . . .

**Fortunately, extending their work, we can define the trapezoid**

*D(v**L**, ρ** _{L}*) =

*D ∩ {v : 0 ≤ v ≤ v*

*L*

*+ cρ*

_{L}*}*

as a new invariant region such that the Riemann’s problem has a
*unique solution for any (v**R**, ρ**R*)*∈ D(v**L**, ρ**L*).

.

. . . . . .

*In this case, u*_{L}^{1}*−→ u*^{S}^{1}_{B}*−→ u*^{C}^{1}_{R}*denoted by (1, 1, 1)** _{S}*.
.Example

..

...

*For (1, 1, 1)** _{S}* we have

*u(x , t) =*

*u*_{L}*, x < st,* *s =* ^{λ}^{1}^{(u}^{L}^{)+λ}_{2} ^{1}^{(u}^{B}^{)}*,*
*u*_{B}*, st < x < σt,*

*u*_{R}*, σt < x ,* *σ =* ^{λ}^{2}^{(u}^{B}^{)+λ}_{2} ^{2}^{(u}^{R}^{)}*.*

*u** _{B}* =

^{(}

^{1}

_{c}*((cρ*

_{L}*+ v*

_{L}*)(cρ*

_{L}*+ v*

_{L}*− v*

*R*

*)),*

^{1}

_{c}*(cρ*

_{L}*+ v*

_{L}*− v*

*R*)

^{)}

. . . . . .

*In general, the horizontal line ρ = ρ** _{∗}* divides the invariant region

*D(u*

*L*) into two subregions, called Ω1

*if ρ < ρ*

*, and Ω2*

_{∗}*if ρ*

_{∗}*≤ ρ.*

Thus there are four cases that need to be considered in terms of
*(u*_{L}^{i}*, u*^{j}* _{R}*)

*∈ (Ω*

*i*

*, Ω*

_{j}*), where i , j = 1, 2.*

(1) *For the case (u*_{L}^{1}*, u*_{R}^{1})*∈ (Ω*1*, Ω*1), there are three possibilities:

*u*^{1}_{L}*−→ u*^{R}*B*^{1}

*−→ u**C* *R*^{1}*, u*^{1}_{L}*−→ u*^{S}*B*^{1}

*−→ u**C* *R*^{1}*, and u*_{L}^{1} *−→ u*^{S}*B*^{2}

*−→ u**C* ^{1}*R**,*
*which are denoted by (1, 1, 1)**R**, (1, 1, 1)**S**, and (1, 2, 1)**S*, resp..

.

. . . . . .

(1) *(u*_{L}^{1}*, u*^{1}* _{R}*)

*∈ (Ω*1

*, Ω*1

*): (1, 1, 1)*

*R*

*, (1, 1, 1)*

*S*

*, (1, 2, 1)*

*S*

Analogous arguments can be applied to the others. That is,

(2) *(u*_{L}^{1}*, u*^{2}* _{R}*)

*∈ (Ω*1

*, Ω*

_{2}

*): (1, 1, 2)*

_{R}*, (1, 1, 2)*

_{S}*, (1, 2, 2)*

_{S}(3) *(u*_{L}^{2}*, u*^{1}* _{R}*)

*∈ (Ω*2

*, Ω*

_{1}

*): (2, 2, 1)*

_{R}*, (2, 1, 1)*

_{R}*, (2, 2, 1)*

_{S}(4) *(u*_{L}^{2}*, u*^{2}* _{R}*)

*∈ (Ω*2

*, Ω*2

*): (2, 2, 2)*

*R*

*, (2, 1, 2)*

*R*

*, (2, 2, 2)*

*S*

. . . . . .

*In fact, there are twelve cases as follows. (R←→ S and 1 ←→ 2)*

NOTE. We can get the geometrical information from the algebraic
*notations (i , j, k)** _{R,S}*, and the converse is also true.

. . . . . .

Back to the balance laws (multiland model),

*ρ*_{t}*+ (m− cρ*^{2})* _{x}* = 0

*m*

*t*+ (

^{m}

_{ρ}*(m− cρ*

^{2}))

*x*=

*ρw*1*(ρ)−(m−cρ*^{2})

*τ* *, ρ < R(*^{m}^{−cρ}_{ρ}^{2}*),*

*ρw*2*(ρ)**−(m−cρ*^{2})

*τ* *, ρ≥ R(*^{m}^{−cρ}_{ρ}^{2}*),*
we consider

*U*e*t**+ F (U)*^{e} *x* *= 0,*
*U(x , 0) =*e

{

*U**L**, x < 0,*
*U*_{R}*, x > 0,*

&

*U**t**+ F (U)**x* *= G (U),*
*U(x , 0) =*

{

*U*_{L}*, x < 0,*
*U*_{R}*, x > 0.*

*Let U = U−U. Then U(x , 0) = 0 for any nonzero x .*^{e}

*Our goal is to obtain an approximate solution of U and use it to*
*get an approximate solution of U.*

. . . . . .

*It follows from U**t**+ F (U)**x* *= G (U) and U =U + U that*^{e}
(*U + U)*^{e} _{t}*+ F (U + U)*^{e} _{x}*= G (U + U).*^{e}

Doing linearization on F and G, and applying the operator-splitting
*method, the perturbation U is the solution of the initial-value*
problem of ODE.

{

*U*_{t}*= G (U) + DG (*^{e} *U)U,*^{e}
*U(x , 0) = 0.*

*To solve it, we need to introduce a new parameter µ(x ) which*
*depends only on the location x .*

. . . . . .

.Definition ..

...

*Fixed a location x , the parameter is defined to be the ratio*
*µ(x ) = T*_{x}*/T ,*

*where T** _{x}* is the observed time when the solution

*U exists in Ω*

^{e}

_{1},

*and T is the observed time when the solutionU exists in Ω*

^{e}

_{1}

*∪ Ω*2.

NOTE. Obviously, the parameter is decreasing from 1 to 0, and increasing from 0 to 1.

. . . . . .

.Theorem (Approximate Solution) ..

...

*The approximate solution of the balance laws (multilane model) is*
{

*ρ =ρ,*e

*v = θv + (1*e *− θ)w(µ,ρ),*e

*where θ = e*^{−t/τ}*and w (µ,ρ)*e *≡ µw*1(*ρ) + (1*e *− µ)w*2(*ρ).*e

*If τ* *→ 0, then θ = e*^{−t/τ}*→ 0, and thus the above theorem shows*
*that the approximate speed v approaches w (µ,ρ), i.e.,*e

*τ*lim*→0**v = lim*

*θ**→0*(1*− θ)w(µ,ρ) + θ*^{e} e*v = w (µ,ρ).*e

*Clearly, this gives the asymptotic behavior as the relaxation time τ*
tends to zero.

. . . . . .

In particular, if we consider the AR model with a single source term

*w*1*−v*

*τ* , then our result on the asymptotic behavior is similar to
the result of Liu.

*w*1(*ρ)*e *≥*e*v⇒ v ≥v*e

lim*τ**→0**v = w*1(*ρ) , similar to the result of Liu. (1991)*e

.

. . . . . .

For Solving the initial-value problem of the balance laws

*U*_{t}*+ F (U)*_{x}*= G (U),*
*U(x , 0) = U*_{0}*(x ).*

,

we has three steps.

Step 1. Solving the Riemann problem for the conservation
laws*U*^{e}*t**+ F (U)*^{e} *x* *= 0. (U =U + U)*^{e}

Step 2. Using the solution*U of Riemann problem to obtain an*^{e}
*approximate solution for the solution U of balance*
laws.

Step 3. Apply the approximate solutions to construct the building blocks for the generalized Glimm scheme to solve the initial-value problem for the balance laws.

. . . . . .

**Thank** **you**

. . . . . .

**Thank** **you**

. . . . . .

**Thank** **you**

. . . . . .

.Example (Shock Wave) ..

...

*Given U**L**, if U**R* *lies in S*_{1}* ^{−}*, then

*U(x , t) =*

{

*U**L**, x < st,*
*U*_{R}*, x > st,*
*where s =* ^{m}_{ρ}^{L}

*L* *− c(ρ**R* *+ ρ**L**).*

*NOTE. Shock speed s =* ^{λ}^{1}^{(U}^{L}^{)+λ}_{2} ^{1}^{(U}^{R}^{)}*.*

. . . . . .

.Example (Rarefaction Wave) ..

...

*Given U**L**, if U**R* *lies in R*_{1}^{+}, then

*U(x , t) =*

*U*_{L}*,* ^{x}_{t}*≤ λ*1*(U*_{L}*),*

*V (*^{x}_{t}*), λ*1*(U**L*)*≤* ^{x}_{t}*≤ λ*2*(U**R**),*
*U*_{R}*,* ^{x}_{t}*≥ λ*2*(U*_{R}*).*

*NOTE. V (*^{x}* _{t}*) =
[ 1

*2c*(^{m}_{ρ}^{L}

*L* *−* ^{x}* _{t}*)

_{2c}^{1}

^{m}

_{ρ}

^{L}*L*(^{m}_{ρ}^{L}

*L* *−*^{x}* _{t}*)

^{]}

^{T}. . . . . .

.Example (2-contact discontinuity) ..

...

*Given U**L**, if U**R* *lies in S*2*(= R*2), then
*U(x , t) =*

{

*U*_{L}*, x < σt,*
*U*_{R}*, x > σt,*
*where σ = λ*2*(U**L**) = λ*2*(U**R**).*

*NOTE. λ*2*(U) =* ^{m}_{ρ}*− cρ = v, the average speed of vehicles.*

. . . . . .

*Define ϵ*1 =*−2((*^{m}_{ρ}_{L}^{L}*− cρ**L*)*− (*^{m}_{ρ}_{R}^{R}*− ρ**R**)) and ϵ*2 = ^{m}_{ρ}^{L}

*L* *−* ^{m}_{ρ}_{R}* ^{R}*.

*Then the next result (U*

_{L}*→ U → U*

*R*) is an application of the previous theorem.

NOTE.

*U = U**L**+ ϵ*1*r*1*(U**L**),*

*U*_{R}*= U + ϵ*_{2}*r*_{1}*(U) +*^{ϵ}_{2}^{2}^{2}*Dr*_{k}*(U)· r**k**(U).*

. . . . . .

For Riemann’s problem, we have in general .Theorem (local solution of Riemann problem) ..

...

*Assume each kth characteristic field is either genuinely nonlinear or*
*linearly degenerate. Suppose further that the left initial state U*_{L}*is*
*given. Then for each right initial state U**R* *sufficiently close to U**L*

*there exists an integral solution U of Riemann’s problem, which is*
*constant on lines through the origin.*

To our system,
[ *ρ*

*m*
]

*t*

+

[ *−2cρ*^{2} 1

*−(*^{m}* _{ρ}*)

^{2}

*− mc*

^{2m}

_{ρ}*− cρ*] [

*ρ*

*m*
]

*x*

= [ 0

0 ]

*,*

we can give a complete solution of Riemann’s problem for any
*two "reasonable" constant states (U**L**= (ρ**L**, m**L*)* ^{T}* and

*U*_{R}*= (ρ*_{R}*, m** _{R}*)

*).*

^{T}. . . . . .

To our system, we have the following theorem.

.Theorem ..

...

*Assume that each kth characteristic field is normalized, that is,*

*∇λ**k**(U)· r**k**(U) = 1 and l*_{k}*(U)· r**k**(U) = 1 for each k, and the left*
*initial state U**L* *is given. Then*

**(a) there exists a parametrization of R**_{1}^{+}*(U** _{L}*)：

*ϵ7→ U**L**+ ϵr*_{1}*(U*_{L}*), where ϵ = 2c(ρ*_{L}*− ρ) > 0,*
**(b) there exists a parametrization of S**_{1}^{−}*(U**L*)：

*ϵ7→ U**L**+ ϵr*1*(U*_{L}*), where ϵ = 2c(ρ*_{L}*− ρ) < 0,*

**(c) there exists a parametrization of R**_{2}*(U*_{L}*) = S*_{2}*(U** _{L}*)：

*ϵ7→ U**L**+ ϵr*_{2}*(U** _{L}*) +

^{ϵ}_{2}

^{2}

*Dr*

_{2}

*(U*

*)*

_{L}*· r*2

*(U*

_{L}*), where ϵ = c(ρ*

_{L}*− ρ).*

. . . . . .

As a consequence of the previous result, we give a rigorous proof on the following theorem about instability.

.Theorem (Aw-Rascle, 2000) ..

...

*When one of Riemann data is near the vacuum, that is, the light*
*traffic, if and only if the solution presents instabilities.*

NOTE.

Near the vacuum (i.e., the density is very low), the solution of Riemann’s problem is very sensitive to the data.

For example, any driver in everyday life can observe that there is ahead very light traffic of very slow drivers.

. . . . . .

Ref. B.-C Huang, S.-W. Chou, J. M. Hong, and C.-C. Yen, Global transonic solutions of planetary atmospheres in a hydrodynamics region–hydrodynamic escape problem due to gravity and heat, SIAM, J. Math, Anal., Vol. 48. No. 6 (2016), pp. 4268–4310.

. . . . . .

.Theorem (Approximate Solution) ..

...

*The approximate solution of the balance laws (multilane model) is*
{

*ρ =ρ,*e

*v = (1− θ)w(µ,ρ) + θ*e e*v ,*

*where θ = e*^{−t/τ}*and w (µ,ρ)*e *≡ µw*1(*ρ) + (1*e *− µ)w*2(*ρ).*e
*By the theorem, it is clear that v−v = (1*^{e} *− θ)(w(µ,ρ)*^{e} *−v ).*^{e}
A necessary and sufficient condition for the inequality

*v* *≥v*e
to be true is that the following holds

*ω(µ,ρ)*e *≥*^{e}*v .*
*In addition, v =v if and only if ω(µ,*e *ρ) =*e *v .*e

. . . . . .

*Since we introduce the parameter µ, the convex combination speed*
*w (µ,ρ) appears. So v is a convex combination of*e e*v and w (µ,ρ).*e
In particular,

*(1, 1, 1)**R,S* *⇒ µ = 1 ⇒*
{

lim*τ**→0**v = w*1(*ρ),*e
*w*1(*ρ)*e *≥v*^{e}*⇒ v ≥*^{e}*v .*

*(2, 2, 2)*_{R,S}*⇒ µ = 0 ⇒*
{

lim*τ**→0**v = w*2(*ρ),*e
see the following figure.

**Hsin-Yi Lee**. **Hyperbolic Balance Laws for Multilane Traffic Flow Model** **January 27, 2019**

. . . . . .

*It follows from U**t**+ F (U)**x* *= G (U) and U =U + U that*^{e}
(*U + U)*^{e} *t**+ F (U + U)*^{e} *x* *= G (U + U).*^{e}

*Performing the Taylor series expansion on F and G atU yields that*^{e}
*F (U + U) = F (*^{e} *U) + DF (*^{e} *U)U + higher order terms,*^{e}
and

*G (U + U) = G (*^{e} *U) + DG (*^{e} *U)U + higher order terms*^{e}
respectively. Removing higher order terms from two expansions
and putting the remaining terms into the balance laws yields

*U*e_{t}*+ U*_{t}*+ F (U)*^{e} _{x}*+ (DF (U)U) = G (*^{e} *U) + DG (*^{e} *U)U.*^{e}
This with the conservation laws*U*^{e}_{t}*+ +F (U)*^{e} * _{x}* = 0 gives

*U*_{t}*+ (DF (U)U)*^{e} _{x}*= G (U) + DG (*^{e} *U)U.*^{e}

. . . . . .

We now have

*U**t**+ (DF (U)U)*^{e} *x* *= G (U) + DG (*^{e} *U)U.*^{e}

*Apply the operator-splitting method, i.e., the term ∂*_{x}*(DF (U)U)*^{e}
can be erased from the above equations:

*U*_{t}*= G (U) + DG (*^{e} *U)U*^{e}

Hence we have the initial-value problem of ODE as follows.

{

*U**t* *= G (U) + DG (*^{e} *U)U,*^{e}
*U(x , 0) = 0.*

*We introduce a new parameter µ = µ(x ), a function depending*
*only on the space x so that the initial-value problem of ODE can*
be solved easily.

. . . . . .

**2. Diffusion**

.Example (Traffic light on the red) ..

...

Consider the stationary PW model with the initial condition
*ρ(x , 0) =*

{

*1, if− 1 ≤ x ≤ 0,*

*0, otherwise,* *and v (x , 0) = 0,*
and the boundary condition

*ρ(0, t) = 1 and v (0, t) = 0 for all t* *≥ 0.*

The natural traffic evolution should be that nothing happens if the red light remains. However, there are some cars in the interval (−∞, −1). This shows that these cars travel with negative speed, which is completely unrealistic.

. . . . . .

Before introducing the multilane model, we mention that both of the PW model and AR model are derived from the car-following model.

On the other hand, recall the acceleration equation from the LWR model:

*a = v** _{x}*(

*−ρv*

*e*

^{′}*(ρ)) =−ρ*

*x*

*(v*

_{e}

^{′}*(ρ))*

^{2}

*ρ.*

*If we choose a =−ρ**x**(v*_{e}^{′}*(ρ))*^{2}*ρ, this gives the PW model, and*
*if we choose a = v** _{x}*(

*−ρv*

*e*

^{′}*(ρ)), this gives the AR model.*