A simple closure approximation for slow dynamics of a multiscale system: nonlinear and multiplicative coupling

TL;DR

This paper introduces a new closure approximation method for multiscale systems with nonlinear and multiplicative coupling, enabling efficient reduced modeling of slow dynamics while maintaining statistical accuracy.

Contribution

The authors develop a novel reduced model based on the fluctuation-dissipation theorem that effectively captures slow variable statistics in complex multiscale systems with nonlinear coupling.

Findings

The new method produces statistics comparable to the original multiscale model.

Constant coupling parameterization yields less accurate results.

Applicable to systems with quadratically nonlinear and multiplicative coupling.

Abstract

Multiscale dynamics are ubiquitous in applications of modern science. Because of time scale separation between relatively small set of slowly evolving variables and (typically) much larger set of rapidly changing variables, direct numerical simulations of such systems often require relatively small time discretization step to resolve fast dynamics, which, in turn, increases computational expense. As a result, it became a popular approach in applications to develop a closed approximate model for slow variables alone, which both effectively reduces the dimension of the phase space of dynamics, as well as allows for a longer time discretization step. In this work we develop a new method for approximate reduced model, based on the linear fluctuation-dissipation theorem applied to statistical states of the fast variables. The method is suitable for situations with quadratically nonlinear and multiplicative coupling. We show that, with complex quadratically nonlinear and multiplicative coupling in both slow and fast variables, this method produces comparable statistics to what is exhibited by an original multiscale model. In contrast, it is observed that the results from the simplified closed model with a constant coupling term parameterization are consistently less precise.