Variable Step Size Multiscale Methods for Stiff and Highly Oscillatory Dynamical Systems

TL;DR

This paper introduces a variable step size multiscale integrator for stiff and oscillatory systems, improving accuracy over existing methods while maintaining similar computational costs.

Contribution

It develops a new multiscale integrator with variable mesoscopic steps that enhances accuracy for stiff systems compared to prior methods.

Findings

Higher accuracy in approximating slowly changing quantities.

Maintains computational complexity similar to existing methods.

Provides analytical and numerical comparisons.

Abstract

We present a new numerical multiscale integrator for stiff and highly oscillatory dynamical systems. The new algorithm can be seen as an improved version of the seamless Heterogeneous Multiscale Method by E, Ren, and Vanden-Eijnden and the method FLAVORS by Tao, Owhadi, and Marsden. It approximates slowly changing quantities in the solution with higher accuracy than these other methods while maintaining the same computational complexity. To achieve higher accuracy, it uses variable mesoscopic time steps which are determined by a special function satisfying moment and regularity conditions. Detailed analytical and numerical comparison between the different methods are given.