Accuracy Analysis of Time Integration Schemes for Stiff Multiscale Problems

In the context of multiscale computations, techniques have recently been developed that enable microscopic simulators to perform macroscopic level tasks (equation-free multiscale computation). The main tool is the so-called coarse-grained time-stepper, which implements an approximation of the unavailable macroscopic time-stepper using only the microscopic simulator. Several schemes were developed to accelerate the coarse-grained time-stepper, exploiting the smoothness in time of the macroscopic dynamics. To date, mainly the stability of these methods was analysed. In this paper, we focus on their accuracy properties in the context of parabolic problems. We study the global error of the different methods, compare with explicit stiff ODE solvers, and use the theoretical results to develop more accurate variants. Our theoretical results are confirmed by various numerical experiments.