A linear filter regularization for POD-based reduced order models of the quasi-geostrophic equations

TL;DR

This paper introduces a BV-alpha regularization method for POD-based reduced order models of the quasi-geostrophic equations, improving accuracy especially with limited POD modes, and compares its performance to unregularized models.

Contribution

The paper presents a novel BV-alpha regularization technique for POD-Galerkin ROMs of QGE, enhancing model accuracy and stability with limited modes.

Findings

Regularized ROM yields more accurate solutions than unregularized ROM.

The regularized ROM remains computationally feasible compared to full simulations.

Regularization improves model performance even with few POD modes.

Abstract

We propose a regularization for Reduced Order Models (ROMs) of the quasi-geostrophic equations (QGE) to increase accuracy when the Proper Orthogonal Decomposition (POD) modes retained to construct the reduced basis are insufficient to describe the system dynamics. Our regularization is based on the so-called BV-alpha model, which modifies the nonlinear term in the QGE and adds a linear differential filter for the vorticity. To show the effectiveness of the BV-alpha model for ROM closure, we compare the results computed by a POD-Galerkin ROM with and without regularization for the classical double-gyre wind forcing benchmark. Our numerical results show that the solution computed by the regularized ROM is more accurate, even when the retained POD modes account for a small percentage of the eigenvalue energy. Additionally, we show that, although computationally more expensive that the ROM with no regularization, the regularized ROM is still a competitive alternative to full order simulations of the QGE.