Full and Reduced Order Model Consistency of the Nonlinearity Discretization in Incompressible Flows

TL;DR

This paper explores the impact of nonlinear discretization consistency between full order and reduced order models in incompressible flows, demonstrating that consistency leads to better accuracy and error bounds.

Contribution

It provides a theoretical analysis of FOM-ROM consistency effects and numerically validates that consistent discretization improves model accuracy in flow simulations.

Findings

Consistent discretization yields optimal error bounds.

Inconsistent discretization introduces additional error terms.

Numerical results show higher accuracy with FOM-ROM consistency.

Abstract

We investigate both theoretically and numerically the consistency between the nonlinear discretization in full order models (FOMs) and reduced order models (ROMs) for incompressible flows. To this end, we consider two cases: (i) FOM-ROM consistency, i.e., when we use the same nonlinearity discretization in the FOM and ROM; and (ii) FOM-ROM inconsistency, i.e., when we use different nonlinearity discretizations in the FOM and ROM. Analytically, we prove that while the FOM-ROM consistency yields optimal error bounds, FOM-ROM inconsistency yields additional terms dependent on the FOM divergence error, which prevent the ROM from recovering the FOM as the number of modes increases. Computationally, we consider channel flow around a cylinder and Kelvin-Helmholtz instability, and show that FOM-ROM consistency yields significantly more accurate results than FOM-ROM inconsistency.