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

Abstract

This paper considers a macroscopic non-conservative form of traffic flow model known as Lighthill, Whitham and Richards (LWR) model appended with a linear velocity-density function. The model reads as a quasi-linear first order hyperbolic partial differential equation (PDE) and in order to incorporate initial and boundary data treated as an initial boundary value problem (IBVP). We presents the exact solution of the PDE as a Cauchy problem and the derivation of a finite difference scheme of non-conservative form of traffic flow model which is a first order explicit upwind difference scheme and establish well-posed-ness and stability condition for the scheme. Computer programs for the implementation of the numerical scheme and perform numerical experiments in order to verify some qualitative behaviour of traffic flow for various traffic parameters and relative errors and verify convergence of errors.