Momentum-conserving ROMs for the incompressible Navier-Stokes equations

TL;DR

This paper introduces a new reduced-order model for incompressible Navier-Stokes equations that exactly conserves mass and energy, improving stability and adherence to physical laws compared to existing models.

Contribution

The paper presents a novel ROM that exactly satisfies the projected ODE and conservation laws, enhancing stability and physical fidelity in fluid dynamics simulations.

Findings

The new ROM conserves mass exactly over subdomains.

It enables derivation of kinetic energy conservation.

It demonstrates improved nonlinear stability.

Abstract

Projection-based model order reduction of an ordinary differential equation (ODE) results in a projected ODE. Based on this ODE, an existing reduced-order model (ROM) for finite volume discretizations satisfies the underlying conservation law over arbitrarily chosen subdomains. However, this ROM does not satisfy the projected ODE exactly but introduces an additional perturbation term. In this work, we propose a novel ROM with the same subdomain conservation properties which indeed satisfies the projected ODE exactly. We apply this ROM to the incompressible Navier-Stokes equations and show with regard to the mass equation how the novel ROM can be constructed to satisfy algebraic constraints. Furthermore, we show that the resulting mass-conserving ROM allows us to derive kinetic energy conservation and consequently nonlinear stability, which was not possible for the existing ROM due to the presence of the perturbation term.