Fast integrators for dynamical systems with several temporal scales

TL;DR

This paper introduces a computationally efficient fast integrator for complex dynamical systems with multiple time scales, extending previous multiscale methods to handle many scales with linear complexity.

Contribution

The authors develop a new multiscale integrator that scales linearly with the number of time scales, improving efficiency over existing methods.

Findings

The method effectively handles systems with multiple scales.

Numerical tests show accurate results on dissipative and oscillatory problems.

The approach reduces computational complexity compared to traditional multiscale integrators.

Abstract

We propose a fast integrator to a class of dynamical systems with several temporal scales. The proposed method is developed as an extension of the variable step size Heterogeneous Multiscale Method (VSHMM), which is a two-scale integrator developed by the authors. While iterated applications of multiscale integrators for two different scales increase the computational complexity exponentially as the number of different scales increases, the proposed method, on the other hand, has computational complexity linearly proportional to the number of different scales. This efficiency is achieved by solving different scale components of the vector fields with variable time steps. It is shown that variable time stepping of different force components has an effect of fast integration for the effective force of the slow dynamics. The proposed fast integrator is numerically tested on problems with several different scales which are dissipative and highly oscillatory including multiscale partial differential equations with sparsity in the solution space.