Numerical Simulation Analysis

In this section, we would analysis the three macroscopic equations with numerical simulation. We would use “Upwind method to analysis”. It is a simple first-order partial differential equation and it belongs to an explicit finite-difference method. According to Helbing (1999) [25], we set flow Q=cv and traffic pressureP=cθ. Then the three macroscopic equations (4.12), (4.14) and (4.16) could be rewrite to be related to density c function as the following.

=0

∂ +∂

∂

∂ x Q t

c (5.14)

0 2

2 2

/ 1 2 / 3

2 / 1 2 / 1 2

/ 3 2 / 1 2

2 }]

3 2

max _ 2

1 2

3

4 11 max

_ 4 { 5 ) 1 ( [

Q cv

aPQ Q

c cQ c

a P cP

c QP c

c P a c p

P Q x t Q

λ

τ τ π

π τ λ π

−

− −

=

−

−

− +

−

+

− +

∂ + + ∂

∂

∂

(5.15)

If we rewrite the equations (5.1), (5.2), and (5.3) as vector form as the following.

f(u) s(u)

We use Dirichlet boundary condition, a period boundary condition, and it means that the simulation result would happen periodically. So we would simulate one cycle as present as the following.

) small quantity of space. We assume the equation (5.9).

) ( ) (x u L

u = (5.22)

We separate time to 4000 equal grids and every time grid is 0.0001 hour. We separate space to 250 equal grids and every space grid is 0.004 km. Assume average car length is 0.005 km, desired velocity is 100km per hour (v₀ =100 km/hr), relaxation time is 30 seconds (T =30s), reaction time is 0.75 seconds (τ =0.75s),

heavy density is 180 cars per km (c_max=180 cars/km), equilibrium density is 30 cars per hour (c_equi=30 cars/km), λ =0.0001 hour is the same as the time grid, covariance C=144 square km/ square hour, uniform deceleration a=60000 km/square hour, passing probability p=0.99. According to Helbing (1995) [26], he assumes the relation between equilibrium velocity and desired velocity as the equation (5.10) ,and the relation between covariance and variance as the equation (5.11).

) 10

* 3.72 -),-1.0) 0.25)/0.06

-i/c_max exp((c_equ

+ pow(1 (

_equi v₀ ^-6

v =

(5.23)

) 10

* 3.72 -),-1.0) 0.25)/0.06

-i/c_max exp((c_equ

+ pow(1 ( var _

variance equi= iance ^-6

(5.24)

<1>. Perturbation Simulation:

We add sine wave to the initial value of density, and see the result that perturbation causes. The following is the result of our research.

Fig. 5-3-2 Density Changing 2

Fig. 5-3-3 Density Changing 3

Fig. 5-3-4 Average Velocity Changing 1

Fig. 5-3-5 Average Velocity Changing 2

Fig. 5-3-6 Average Velocity Changing 3

Fig. 5-3-7 Traffic Pressure Changing 1

Fig. 5-3-8 Traffic Pressure Changing 2

Fig. 5-3-9 Traffic Pressure Changing 3

Fig. 5-3-10 Variance Changing 1

Fig. 5-3-11 Variance Changing 2

Fig. 5-3-12 Variance Changing 3

From the result of simulation, we could observe some result that match the real traffic, and we describe as the following.

(1). We could know the characteristic velocity is positive and it is the same result as the section 5.1. In dilute traffic, density wave could propagate downstream with time.

(2). When the traffic pressure becomes larger, the average velocity becomes slower.

It is the same as general traffic theory. When drivers anticipate the density of downstream becomes heavier, traffic pressure becomes larger, drivers would decelerate and this causes the average velocity slower.

(3). When the density becomes heavier and average velocity becomes slower, variance would become smaller. Because cars could not pass, density would become heavier. The latter car decelerates to the velocity of the former car, so the

<2>. Simulation of Different Passing Probability Effect:

In the same condition as the above, we could adjust different passing probability p to observe density, average velocity, traffic pressure and variance changing.

Fig. 5-3-13 Passing Probability =0.7

Fig. 5-3-14 Passing Probability =0.8

Fig. 5-3-15 Passing Probability =0.9

Fig. 5-3-16 Passing Probability =0.99

From figure 5-3-14 to figure 5-3-16, we could observe two characteristics.

1. When passing probability p becomes smaller, variance becomes larger. Because smaller passing probability p means cars could not pass each other easily, they could not attain their desired velocity easily. Therefore, variance becomes larger.

2. The whole changing trend does not change with passing probabilityp . Because we construct the model by assuming in dilute traffic, macroscopic model could approach local equilibrium quickly. Hence, the whole changing trend does not change with passing probabilityp .

<3>. Simulation of Different Uniform Deceleration Effect:

In the same conditions as the above, we compare density, average velocity, traffic pressure and variance with different uniform deceleration a and the same passing probability p . We use two kinds of passing probability p , 99p=0. andp=0.7, and three kinds of uniform decelerationa, a=60000, a=600 and a=6. We show

60000

=

a with blue, a=600 with green and a=6 with red.

Fig. 5-3-17 Density in p=0.99 and Different a

Fig. 5-3-18 Density in p=0.7 and Different a

Fig. 5-3-19 Average Velocity in p=0.99 and Different a

Fig. 5-3-20 Average Velocity in p=0.7 and Different a

Fig. 5-3-21 Variance in p=0.99 and Different a

Fig. 5-3-22 Variance in p=0.7 and Different a

From figure 5-3-17 to figure 5-3-22, we could find that when passing probability becomes smaller, uniform deceleration a=60000 affects density and average velocity more. By the way, we also could observe that when passing probability becomes smaller, different uniform deceleration a causes density and average velocity more different. In the view of mathematic, the value of uniform deceleration

60000

=

a is larger, so it substitutes to affect interaction term more. Hence, the average velocity becomes slower and variance becomes smaller. In the view of traffic flow, the uniform deceleration becomes larger in dilute traffic, and it means that the letter car could approach to the velocity of the former car quickly. Compare to the traffic flow of smaller uniform deceleration, average velocity of the traffic flow of bigger uniform deceleration is slower and variance is smaller.

Besides, when passing probability is smaller, a=60000 causes less effect to density than other two smaller uniform decelerationsa. Because we assume approach

local equilibrium quickly in dilute traffic, there have no heavy situation. However, different uniform decelerations would result in different velocities and variances, and they also affect the propagation of density wave. When the effect of interaction term becomes larger, it would vibrate the propagation of density wave more seriously.

Therefore, we could compare the effect of different uniform decelerations with different passing probability from figure 5-3-17 to figure 5-3-22, and we could observe that there has more difference in smaller passing probability. Because the number of cars use uniform deceleration is fewer with free passing, the whole has little difference. We could see the difference that cause by introducing uniform deceleration. This is the special point that other researches do not contain. Besides, it also shows that introducing uniform deceleration could describe traffic phenomenon more in heavy traffic.

5.4 Summary

From the first and second sections, we could know that our model could describe traffic flow more reasonable. The equations (5.6), (5.9), (5.12) and figure 5-1-1 show this research could describe more phenomenon than the model of C. Wagner and our model could explain the extreme condition. For example, because our model considers deceleration, when uniform deceleration a=0that shows drivers have no deceleration when the faster (the latter) car could not pass according to the equation (5.6). If the uniform decelerationa=0, it means there have no deceleration to avoid collision without passing. It also means that the velocity of the former is the same as the latter one, so it has no deceleration. Because every driver has his desired velocity, they would accelerate until he attains his desired velocity. It means that drivers have attained their desired velocity, so they have no acceleration. So the average velocity is

definitions of covariance and variance, so covariance is equal to variance in equilibrium. These results are special results that the researches of other scholars do not have. Besides, we also discuss the extreme values of passing probability p and relaxation time T .

We analysis the three macroscopic equations that obtain from chapter 4 with numerical simulation “Upwind method” From figure 5-3-14 to figure 5-3-16, we could observe that when passing probability p becomes smaller, variance become larger. But the whole changing trend does not change with passing probability p . Besides, we could observe that there has more difference in smaller passing probability and see the difference that cause by introducing uniform deceleration from figure 5-3-17 to figure 5-3-22. This is the special point that other researches do not contain. Besides, it also shows that introducing uniform deceleration could describe traffic phenomenon more in heavy traffic.

Therefore, we could know that our model could describe more traffic phenomenon that other model could not explain. This means that it is meaningful to introduce uniform deceleration to relax instant velocity and consider finite space in gas-kinetic traffic flow model.