Lax-Friedrich Scheme for the Numerical Simulation of a Traffic Flow Model Based on a Nonlinear Velocity Density Relation

Abstract

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.

Inflow Outflow Effect and Shock Wave Analysis in a Traffic Flow Simulation

“…Hence general finite difference methods are not well suited in this task. For this reason, Godunov method [12] [14] is applied in single conservation laws in order to find out the numerical solution. We consider a discrete spatial grid with mesh size x ∆ and a discretized dependent variable j ρ whose average over a grid cell is given by ( )…”

“…At the point where the solution switches from a characteristic branch to another, the solution ρ will be discontinuous. An example of this construction is shown in Figure 9 with a dotted line; the locus of discontinuity is called a shock wave [12] [14].…”