A Stochastic Approach for Parameterizing Unresolved Scales in a System with Memory

TL;DR

This paper introduces a stochastic parameterization method for unresolved scales in complex systems with memory, using large eddy simulations of a nonlinear PDE with a time-integral term, validated through numerical experiments.

Contribution

It develops a novel stochastic scheme to incorporate unresolved scales effects in systems with memory, enhancing large eddy simulation accuracy.

Findings

Stochastic parameterization effectively captures unresolved scales.

The stochastic LES model closely matches the original system.

Numerical experiments validate the approach.

Abstract

Complex systems display variability over a broad range of spatial and temporal scales. Some scales are unresolved due to computational limitations. The impact of these unresolved scales on the resolved scales needs to be parameterized or taken into account. One stochastic parameterization scheme is devised to take the effects of unresolved scales into account, in the context of solving a nonlinear partial differential equation with memory (a time-integral term), via large eddy simulations. The obtained large eddy simulation model is a stochastic partial differential equation. Numerical experiments are performed to compare the solutions of the original system and of the stochastic large eddy simulation model.