Implicit methods for equation-free analysis: convergence results and analysis of emergent waves in microscopic traffic models

Christian Marschler, Jan Sieber, Rainer Berkemer, Atsushi Kawamoto, Jens Starke

Research output: Contribution to journalJournal articleResearchpeer-review

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.
Original languageEnglish
JournalS I A M Journal on Applied Dynamical Systems
Volume13
Issue number3
Pages (from-to)1202-1238
ISSN1536-0040
DOIs
Publication statusPublished - 2014

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