Shocks and instabilities in traffic

Author: Shinn-Mendoza, Rachel

Year: 1990

Degree: Dissertation (Ph.D.)

Advisor: Unknown, Unknown

Committee Member: Unknown, Unknown

Option: Applied And Computational Mathematics

DOI: 10.7907/dw5a-sw09

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.

In this thesis, we study several models for traffic flow. Our interest is in finding periodic solutions and to study the effect of including a time lag on the propagation of shocks through a line of cars. The periodic solution was stimulated by a problem from water waves in which a periodic solution is created in the unstable region of the parameters by connecting segments of the growing solution with shocks. This results in a finite amplitude solution in the region of instability. The analysis of this is presented and then applied to a continuum model for traffic flow. We look for a smooth version of this periodic shock solution by considering a car following model for traffic. Car following models define the [...] car's velocity in a line of cars as a function of the distance between the [...] and [...] cars and are thus a system of differential-difference equations which define the motion of the cars.

The model we study is attributed to G.F. Newell who found a transformation which makes the nonlinear equation linear. We discuss this exact solution and in particular, look at the shock solutions. These solutions, however, do not include the effect of a time lag. When this is included, we have the possibility of instabilities. We look at the shock solutions with the time lag included numerically and find that after some critical value, the smooth shock profile breaks up into oscillations about the final velocity state. We modify the equation by modeling the time lag continuously and look at these same shock solutions. We then find periodic solutions to this in the form of steady profile waves and compare the results with a continuum theory which also has smooth periodic solutions.

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