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A fluid dynamic traffic flow model based on a non-linear velocity-density function is considered. The model provides a quasi-linear first order hyperbolic partial differential equation which is appended with initial and boundary data and turns out an initial boundary value problem (IBVP). A first order explicit finite difference scheme of the IBVP known as Lax-Friedrich’s scheme for our model is presented and a well-posedness and stability condition of the scheme is established. The numerical scheme is implemented in order to perform the numerical features of error estimation and rate of convergence. Fundamental diagram, density, velocity and flux profiles are presented.

With the increasingly rapid economic globalization and urbanization, more problems are brought to our attention. One of them is traffic jams. Traffic jams are now a major problem in most of the cities. So at the core of traffic congestion, development of traffic management is the need of time. Therefore, an efficient traffic control and management is essential in order to get rid of such huge traffic congestion. Modeling and computer simulation play an increasing role in the flow management. Many scientists have been working to develop various mathematical models [

In this section, the general features of the model are shortly presented based on [

A non-linear velocity-density relationship [

In this paper, we will use the non-linear velocity-density relationship (for m = 2 in (2)' as

Now, substituting (3) in

Now, we put flow-density function (4) into the general non-linear model partial differential equation (PDE) (1), we obtain the specific non-linear partial differential equation in the form

There is a connection between traffic density and vehicle velocity. If there is more vehicles are on a road then their velocity will be slower. On the basis of observations of traffic flow, we make a basic simplifying assumption that the velocity of a car at any point along the highway depends only on the traffic density. Drivers speed up when traffic is sparse and they slow down when traffic is dense. Thus, there is a direct relationship between traffic density and traffic velocity as

Based on the intuition mentioned above, one may assume that a driver will drive fastest, with velocity, say

bumper-to-bumper, i.e. ν = 0 , at some maximum density

of a vehicle. We summarize these experience-born intuitions in mathematical requirements on the function,

The flow or flux given by Equation (4) is a cubic non-linear function. The maximum flow (flux) occurs when its

slope vanishes and

Therefore,

So,

the maximum flow (flux) is

The traffic flow model appended with the initial condition reads as initial value problem (IVP) is

The exact solution [

which is non-linear implicit form and therefore very complicated to evaluate at each

Moreover, in reality it is very complicated to approximate the initial density

For the numerical solution of the traffic flow model, we consider our specific non-linear traffic model problem as an initial and two points boundary value problem

For the model, the numerical solution based on [

where,

This difference equation is known as Lax-Friedrich’s scheme.

However for the flux function (12), the scheme is not straight forward to implement. One needs to work stability condition and sub-sequel physical constraint condition for the scheme. Now we will study the well-po- sedness and stability condition of this scheme for our model.

Rewrite the non-linear PDE in (8) as

Then the scheme (11) can be written as

For well-posedness,

Here,

Therefore, we have

This is the condition of well-posedness.

Now from Equation (13), we have

The Equation (18) implies that if

Then condition (17) can be guaranteed via (13) by

which is the stability condition involving the parameter

Thus whenever one employs the stability condition (20), the well-posedness condition (physical constraints) (14) can be guaranteed immediately by choosing

We implement the Lax-Friedrichs scheme by developing a computer programming code and perform numerical simulation as described below.

In order to perform error estimation, we consider the exact solution (9) with initial condition

We prescribe the corresponding two-sided boundary value by the equations

And

For the above initial and boundary conditions with v_{max} = 0.0167 km/sec = 60.12 km/hour; satisfying the physical constraint condition (21); ρ_{max} = 5 max_{i}ρ_{0}(x_{i}) = 550 vehicles/km in the spatial domain [5 km, 10 km] we perform the numerical experiment for 4 minutes in _{1}-norm defined by

for all time where

Now we consider the initial density using sine function and perform numerical computation in the spatial domain [0, 10] in km.

Finally,

of density which also verifies qualitative behavior, the well-known fundamental diagram as

The computational result obtained by implementing the analogous version of Lax-Friedrich’s scheme shows the accuracy up to five decimal places and a good rate of convergence. Performing numerical simulation, we have verified some qualitative traffic flow behavior for various traffic parameters. Finally, we have presented fundamental diagram of traffic flow using this scheme, which is a very good qualitative agreement of the Lax-Frie- drich’s scheme for traffic flow model. In our model, we have considered only single lane highway. The model can be extended for multi-lane traffic flow model which we leave as our future work.