# Congestion on Multilane Highways

## Introduction

All computations in this section were run with the following equilibrium relations:

 (1)

These transform to

 (2)

where . The specific parameter used were

 (3)

 (4)

and

 (5)

The latter number corresponds to a maximum car density of

 (6)

We used the constant switch curve introduced by Sopasakis [1]:

 (7)

with feet. For initial data we chose 3 sets of data:

 (8)

for and , and 3. The observation that

 (9)

and

 (10)

implies we may interpret the data as initial data for a ring-road with 400 cars which is of length 1.5151... miles. We chose constant initial velocities

 (11)

or

 (12)

These data guarantee points on both sides of the switch curve. Simulations were run with relaxation times

 (13)

A word is inorder about the simulations which follow. The first two frames in each figure are self-explanatory. In the third frame of each figure we plot the curve . This curve is shown in black. The blue curves are the equilibrium curves and and the red curve is the image of . The red dot - o - is the image of . The discontinuities in the profiles propagate at the speed

 (3.14)

## Simulation

 e = 1 e = 2 e = 4 e = 8 k = 1 view view view view k = 2 view view view view k = 3 view view view view

 e = 1 e = 2 e = 4 e = 8 k = 1 view view view view k = 2 view view view view k = 3 view view view view