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This paper investigates the effect of inflow, outflow and shock waves in a single lane highway traffic flow problem. A constant source term has been introduced to demonstrate the inflow and outflow. The classical Lighthill Whitham and Richards (LWR) model combined with the Greenshields model is used to obtain analytical and numerical solutions. The model is treated as an IBVP and numerical solutions are presented using Lax Friedrichs scheme. Godunov method is also used to present shock wave analysis. The numerical procedures adopted in this investigation yield results which are very much consistent with real life scenario in terms of traffic density and velocity.

The problem of traffic congestion is becoming endemic due to increased levels of population. Traffic conditions in many major metropolitan areas are becoming increasingly congested, affecting the operational efficiency of whole networks as well as the travel cost of each trip. Therefore, traffic flow models are becoming more important in traffic engineering and the transportation policy making process. In an effort to minimize congestion, an accurate method for modeling the flow of traffic is imperative. Mainly two approaches are widely used in describing traffic flow phenomena mathematically. The first one is the microscopic model which describes flow by tracking individual vehicles using car-following logic. On the other hand, macroscopic models are concerned with describing the flow-density relationship for a traffic stream (a group of vehicles). Macroscopic models are more suitable for modeling traffic flow since less supporting data and computation are needed. In this paper, we have studied macroscopic traffic flow models.

Computer simulation of traffic is a widely used method in research of traffic modeling, planning and development of traffic networks and systems. Many research groups are involved in dealing with the problem with different kinds of traffic models for several decades [

If a vehicle in a single lane highway can be assumed to be a molecule, then the traffic can be defined to be an incompressible fluid which cannot be compressed after a certain density. In 1955 and 1956, Lighthill, Whitham and Richards proposed a macroscopic traffic flow model which is known as the LWR model [

where

A linear relationship between velocity and density proposed by Greenshields [

where

A source term in the Equation (4) for the vehicular density may represent entries or exits. We are concerned with the role of source terms in traffic flow models based on hyperbolic systems of conservation laws in order to account for entries and exits or local changes of the traffic in the considered road [

If we consider the source to be a constant, the Equation (5) becomes

We will examine the effects of inflow and outflow using Equation (6).

The flow of traffic along a stream can be considered similar to a fluid flow. Consider a stream of traffic flowing with steady state conditions, i.e., all the vehicles in the stream are moving with a constant velocity, density and flow. Suddenly due to some obstructions in the stream (like an accident or traffic block) the steady state characteristics changes and they acquire another state of flow. In the context of traffic flow theory, the boundary which distinguishes one flow state from another is called a shock wave. Now to analyze shocks in a single lane highway, we consider the traffic flow Equation (1). We propose that

Since the mean velocity

The non-linear PDE (6) can be solved if we know the traffic density at a given initial time, i.e. if we know the traffic density at a given initial time

We solve the IVP (9) using method of characteristics and the analytical solution is given by

Again for a first-order PDE, the method of characteristics discovers curves along which the PDE becomes an ordinary differential equation (ODE). We use the crossings of the characteristics to find shock waves. Intuitively, we can think of each characteristic line implying a solution to

In this section, we present the discretization of the traffic flow model by finite difference formula. According to [

The Lax-Friedrichs method is a numerical method for the solution of hyperbolic partial differential equations based on finite differences. The method can be described as the FTCS (forward in time, centered in space) scheme. For Lax-Friedrichs scheme, we consider our specific non-linear traffic model problem as an IBVP characterized with two sided boundary conditions using [

In order to develop the scheme, we discretize the space and time. We discretize the time derivative

obtained by first order central difference in space. The discrete version of the PDE (12) is given by:

But unfortunately, despite the quite natural derivation of the method (13) it suffers from severe stability prob-

lems and is useless in practice, i.e. the scheme becomes unstable. But if we replace

where,

This difference Equation (14) is known as Lax-Friedrichs scheme.

It is verified that the well-posedness and stability of the Lax-Friedrichs scheme is guaranteed by the simultaneous conditions

The main challenge in simulating the traffic flow Equation (7) numerically is that the solutions are typically not smooth, due to the presence of shockwaves. Hence general finite difference methods are not well suited in this task. For this reason, Godunov method [

Every time step

where

Since information travels along characteristics at a finite speed, non?neighboring cells will not interact with each other, provided that we pick

Based on the numerical methods discussed above, we now present outcomes from various simulations in this section.

Using the procedure for Lax-Friedrichs scheme presented in Section 4.1, we now demonstrate the effect of constant rate inflow in traffic flow simulation. We present the numerical solution for two specific cases. For this, we

use the initial condition

where x_{a} and x_{b} are the positions of left and right boundary respectively. We take

In ^{th} km position of our 10 km considered highway.

In this section we present numerical experiments using Lax-Friedrichs scheme for some specific cases of flow parameters like

Here, the maximum density is

At first we demonstrate the density profile figures with comparative initial and 6 minute position of cars with respect to the certain points of 10 km highway. For ^{th} km position and decreases at 8^{th} km position of our considered 10 km highway. This particular situation arises due to the effect of constant rate inflow and outflow at 5^{th} km and 8^{th} km position respectively. Since the maximum velocity in 4(a) is

In ^{th} km position where some vehicles are able to leave from our 10 km single lane highway. Consequently, the density of vehicles slowly decreases at outflow position. As a result, at outflow position, drivers are able to attain their own comfortable driving speed.

^{th} km position, as time goes on, the density of cars increase. This particular situation occurs because the rate of flow in the single lane highway is continuously increasing but the number of cars that leave through the sink is constant.

In ^{th} km, and vice-versa for outflow position.

According to the flux profile, it is clear that flux of traffic increases at 5^{th} km position due to the constant rate inflow and decreases at 8^{th} km position for constant rate outflow through a sink situated at that position.

We now compare two special cases to visualize the effect of inflow and outflow in a single lane highway. In each case we perform numerical simulation for 6 minutes using Lax-Friedrichs scheme.

From the construction of characteristics summarized in the implicit solution (11), it follows that the information

yielding a solution

Note that at approximately

Therefore, the blow up time is

In this paper, a modification of classical LWR model has been presented to investigate the significant effects of constant rate inflow and outflow in a single lane highway. Although the model has been developed for a single lane highway, due to the presence of off ramps and on ramps, this model can be extended for multilane highway

easily. This investigation also shows that Godunov method outperforms finite difference methods in presence of stationary shock waves. Various numerical simulations carried out through this investigation will obviously make important contributions to the existing models so that we can minimize traffic congestion problems in an efficient way.

Ahsan Ali,Laek Sazzad Andallah, (2016) Inflow Outflow Effect and Shock Wave Analysis in a Traffic Flow Simulation. American Journal of Computational Mathematics,06,55-65. doi: 10.4236/ajcm.2016.62007