Implicit Methods for Equation-Free Analysis: Convergence Results and Analysis of Emergent Waves in Microscopic Traffic Models

TL;DR

This paper develops an implicit equation-free method with proven convergence for slow-fast systems and applies it to analyze emergent traffic waves, revealing bifurcation phenomena in microscopic traffic models.

Contribution

It introduces a rigorous convergence analysis for an implicit equation-free approach and demonstrates its application to traffic jam modeling and bifurcation analysis.

Findings

Convergence of the coarse-level time stepper to true dynamics with exponentially small error.

Identification of saddle-node bifurcation in traffic jam dynamics.

Successful macroscopic analysis of microscopic traffic models.

Abstract

We introduce a general formulation for an implicit equation-free method in the setting of slow-fast systems. First, we give a rigorous convergence result for equation-free analysis showing that the implicitly defined coarse-level time stepper converges to the true dynamics on the slow manifold within an error that is exponentially small with respect to the small parameter measuring time scale separation. Second, we apply this result to the idealized traffic modeling problem of phantom jams generated by cars with uniform behavior on a circular road. The traffic jams are waves that travel slowly against the direction of traffic. Equation-free analysis enables us to investigate the behavior of the microscopic traffic model on a macroscopic level. The standard deviation of cars' headways is chosen as the macroscopic measure of the underlying dynamics such that traveling wave solutions correspond to equilibria on the macroscopic level in the equation-free setup. The collapse of the traffic jam to the free flow then corresponds to a saddle-node bifurcation of this macroscopic equilibrium. We continue this bifurcation in two parameters using equation-free analysis.