Parametric Reduced Order Models for the Generalized Kuramoto--Sivashinsky Equations

TL;DR

This paper develops parametric reduced order models for the generalized Kuramoto--Sivashinsky equations, aiming to efficiently capture diverse dynamical regimes and transient behaviors across parameter ranges.

Contribution

It introduces strategies for constructing ROMs that effectively represent multiple regimes and transient dynamics of the gKS equations using POD and POD-DEIM methods.

Findings

ROMs can accurately capture weakly chaotic, transitional, and quasi-periodic regimes.

Snapshot collection strategies are crucial for ROM accuracy across parameters.

ROMs are effective in modeling short-time transient behaviors.

Abstract

The paper studies parametric Reduced Order Models (ROMs) for the Kuramoto--Sivashinsky (KS) and generalized Kuramoto--Sivashinsky (gKS) equations. We consider several POD and POD-DEIM projection ROMs with various strategies for parameter sampling and snapshot collection. The aim is to identify an approach for constructing a ROM that is efficient across a range of parameters, encompassing several regimes exhibited by the KS and gKS solutions: weakly chaotic, transitional, and quasi-periodic dynamics. We describe such an approach and demonstrate that it is essential to develop ROMs that adequately represent the short-time transient behavior of the gKS model.