A finite difference scheme for a fluid dynamic traffic flow model appended with two-point boundary condition

Abstract

A fluid dynamic traffic flow model with a linear velocity-density closure relation is considered. The model reads as a quasi-linear first order hyperbolic partial differential equation (PDE) and in order to incorporate initial and boundary data the PDE is treated as an initial boundary value problem (IBVP). The derivation of a first order explicit finite difference scheme of the IBVP for two-point boundary condition is presented which is analogous to the well known Lax-Friedrichs scheme. The Lax-Friedrichs scheme for our model is not straight-forward to implement and one needs to employ a simultaneous physical constraint and stability condition. Therefore, a mathematical analysis is presented in order to establish the physical constraint and stability condition of the scheme. The finite difference scheme is implemented and the graphical presentation of numerical features of error estimation and rate of convergence is produced. Numerical simulation results verify some well understood qualitative behavior of traffic flow.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10307GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 43-52

Numerical Simulation of a Second Order Traffic Flow Model

“…For the numerical solution of the IBVP, we investigate explicit central difference scheme and implement the numerical scheme by developing computer programming code. Error estimation is produced which shows the numerical solution is accurate up to eight decimal places and this result is much better than the work presented in [6]. The numerical feature of rate of convergence observed by graphical presentation is also found better than the previous work.…”

“…In order to investigate efficient numerical scheme for the Lighthill-Whitham-Richards (LWR) traffic flow model, Gani et al [6] studied a finite difference scheme for the LWR model appended with a linear velocity density relation. This paper performs stability analysis and establishes a physical constraint condition for the implementation of the first order LWR model.…”

Modelling of distribution of the "Matatu" traffic flow using Poisson distribution in a highway in Kenya

“…He solved the p.d.e of higher order with a stiff source term. Gani [29] solved the traffic flow p.d.e. as compressible fluid.Johana et al [25]solved p.d.e governing the traffic situations using finite volume method to determine the wave propagation for traffic flow at a point when roads merges or diverges.…”