A Priori Error Bounds and Parameter Scalings for the Time Relaxation Reduced Order Model

TL;DR

This paper provides a rigorous a priori error analysis and parameter scaling insights for the time relaxation reduced order model (TR-ROM), enhancing understanding of its stability and convergence in fluid flow simulations.

Contribution

It introduces the first stability and error bounds for TR-ROM, along with parameter scaling laws validated through numerical experiments.

Findings

Theoretical convergence rates match numerical results.

Parameter scaling laws are consistent across different ROM dimensions.

Scaling with filter radius is observed in predictive regimes.

Abstract

The a priori error analysis of reduced order models (ROMs) for fluids is relatively scarce. In this paper, we take a step in this direction and conduct numerical analysis of the recently introduced time relaxation ROM (TR-ROM), which uses spatial filtering to stabilize ROMs for convection-dominated flows. Specifically, we prove stability, an a priori error bound, and parameter scalings for the TR-ROM. Our numerical investigation shows that the theoretical convergence rate and the parameter scalings with respect to ROM dimension and filter radius are recovered numerically. In addition, the parameter scaling can be used to extrapolate the time relaxation parameter to other ROM dimensions and filter radii. Moreover, the parameter scaling with respect to filter radius is also observed in the predictive regime.