Efficient numerical scheme for non-conservative form of a traffic flow model

Introduction

The number of vehicles is increasing rapidly, in recent years; traffic congestion has become especially an acute problem. Therefore, an efficient traffic control and management is essential in order to grid of such huge traffic congestion. The aims of this analysis are principally represented by the maximization of vehicles flow, and the minimization of traffic congestions, accidents and pollutions etc. In this paper, we consider a macroscopic non-conservative form of traffic flow model first developed by Lighthill and Whitham (1955) and Richard (1956) shortly called LWR model based on [1, 4, 5] . As presented, we study finite difference method for first order non-linear PDE [6, 8, 9, 10] and based on these, we develop a finite difference scheme for a non-conservative form of traffic flow model as an (IBVP) which has been presented in numerical simulation. We develop computer programming for the implementation of the numerical scheme and perform numerical experiments in order to verify some qualitative behaviour of non-conservative form of traffic flow for various traffic parameters.

Mathematical Formulation of Non-Conservative Form of Traffic Flow Model

The traffic flow model is used to study traffic flow by collective variables such as traffic flow rate (flux) ) , ( t x q , traffic speed . The well-known LWR model [4, 5, 7] based on the principle of mass conservation reads as

Inserting a linear velocity-density closure relationship

and equation ( 2) takes the form

 

The velocity decreases as the density increases, which is a statement about the slope of the v verse  curve.

Assume further that the traffic is bumper-to bumper, i.e., 0 v  , at some maximum density m a x  with m a x 1

, where L is the average length of a vehicle. We summarize these experience-born intuitions in mathematical requirements on the function,

The IVP ( 5) can be solved by the method of characteristics as follows. PDE in (5) can be written as

From equation ( 7), we have   m a x

is called the characteristics curve starting from 0 xx  . Equation (5) yields that along the characteristics curve the density  

which is the exact solution of the IVP (5) and is in implicit form. However, in reality it is very difficult to approximate the initial density   0 x  from given initial data, it may cause a huge error. Therefore, there is a demand of some efficient numerical methods for solving the IVP (5) as an initial boundary value problem (IBVP), which is also appended with boundary condition.

Finite Difference Scheme for Numerical Solution of Non-Conservative Form of Traffic Flow Model

In this section we present the derivation of finite difference scheme for our model with linear velocity-density relation appended with initial and boundary condition. We consider our specific non-linear PDE of non-conservative form of traffic flow model as an IBVP:

The explicit finite difference scheme (10) takes the form

The equation ( 12) implies that if 1,   the new solution is a convex combination of the two previous solutions. That is the solution at new time-step ( 1) n  at a spatial node is an average of the solutions at the previous time-step at the spatial nodes a n d

.

ii  This means that the extreme value of the new solution is the average of the extreme values of the previous two solutions at the two consecutive nodes. Therefore, the new solution continuously depends on the initial value

Error Estimation of Numerical Scheme

In order to perform error estimation for density ()  , we consider exact solution (8) We compute the relative error in 1 L -norm defined by

Conclusion

The computational result obtained by implementing the analogous version of non-conservative form of traffic flow model of the numerical solution by EUDS shows the accuracy up to four 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 diagram of traffic flow using this scheme, which is a very good qualitative agreement 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.

References

1. Bretti G, Natalini R, Piccoli B. A Fluid-Dynamic Traffic Model on Road Networks, Comput Methods Eng., CIMNE, Barcelona, Spain. 2007; 14:139-172.
2. Andallah LS, Ali S, Gani MO, Pandit MK, Akhter J, A Finite Difference Scheme for a Traffic Flow Model Based on a Linear Velocity-Density Function. Jahangirnagar University Journal of Science. 2009; 32:61-71.
3. Kuhne R, Michalopoulos P. Continuum Flow Models, 1997.
4. Haberman R. Mathematical Models. Prentice-Hall, Inc., Delhi, 1977.
5. Klar A, Kuhne RD, Wegener R. Mathematical Models for Vehicular Traffic. Technical University of Kaiserslautern, Dept. of Math., TU of Kaiserlautern, Germany, 1996.
6. Zhang HM. A finite difference approximation of non-equilibrium traffic flow model, Transportation Research Part-B: Methodological (Elsevier), 2001; 35(4):337-365.
7. Dym CL. Principles of Mathematical Modeling, Academic press, 2004.
8. Larsson S, Thomee V. Partial Differential Equations with Numerical Methods, Second Printing Springer-Verlag Berlin Heidelberg, 2005,
9. Leveque RJ. Numerical Methods for Conservation Laws. 2nd Edition, Springer, Berlin, 1992
10. Daganzo CF. A Finite Difference Approximation of the Kinematic Wave Model of Traffic Flow. Transportation Research Part B: Methodological. 1995; 29:261-276.