Asymptotic method applied to the Greenberg traffic model

Abstract

In this work, the asymptotic method was applied to the Greenberg traffic model, which is represented by a conservation law, to obtain shock-type solutions. The text begins by presenting the polynomial function, which is an approximation of the flow function of the Greenberg model, from which the Hugoniot-Maslov chain is constructed, which is the asymptotic method. The mathematical basis for constructing it is the Colombeau subalgebra. Such a chain is a system of ordinary differential equations. This system was solved numerically using the Runge-Kutta method, thus obtaining the numerical solution using the asymptotic method. Finally, a graphical comparison was made with two finite difference methods, Lax-Wendroff e Lax-Friedrichs, to show the effectiveness of the asymptotic method.