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We introduce a general formulation for an implicit equation-free method inthe setting of slow-fast systems. First, we give a rigorous convergence resultfor equation-free analysis showing that the implicitly defined coarse-leveltime stepper converges to the true dynamics on the slow manifold within anerror that is exponentially small with respect to the small parameter measuringtime scale separation. Second, we apply this result to the idealized trafficmodeling problem of phantom jams generated by cars with uniform behavior on acircular road. The traffic jams are waves that travel slowly against thedirection of traffic. Equation-free analysis enables us to investigate thebehavior of the microscopic traffic model on a macroscopic level. The standarddeviation of cars' headways is chosen as the macroscopic measure of theunderlying dynamics such that traveling wave solutions correspond to equilibriaon the macroscopic level in the equation-free setup. The collapse of thetraffic jam to the free flow then corresponds to a saddle-node bifurcation ofthis macroscopic equilibrium. We continue this bifurcation in two parametersusing equation-free analysis.

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