Numerical Computation of 1-D Traffic Flow on the Payne-Weber Compressible Fluid Model

The problem of traffic congestion is becoming endemic due to increased levels of populations owning automobiles. In an effort to minimize congestion, an accurate method for modeling the flow of traffic is imperative. While randomized microscopic models can be very accurate, they are computationally much slower than macroscopic models. Furthermore, macroscopic models can be very accurate in modeling observed driver behavior. Unfortunately, a greater problem in practical modeling is with the numerical method used to solve the equations, rather than the model used. It has been shown that for most cases the fluid-flow model of traffic is a very accurate one. The most simple, one-dimensional compressible fluid flow model for traffic is a surprisingly complex one. Despite being only one-dimensional in x and t, the general-purpose PDE solver pdepe in MATLAB cannot accurately calculate it in many conditions. Four numerical integration methods will be studied in this paper: Lax-Friedrichs, Upwind method, MacCormack method, and the Lax-Wendroff method. Additionally, small modifications of parameters, or additional terms in the model can necessitate changing the method of integration.