A Two-Level Galerkin Reduced Order Model for the Steady Navier-Stokes Equations

TL;DR

This paper introduces a novel two-level Galerkin reduced order model for steady Navier-Stokes equations that improves computational efficiency while maintaining accuracy, by solving a low-dimensional nonlinear system followed by a linearized higher-dimensional system.

Contribution

The paper develops and analyzes a new two-level ROM approach that reduces computational cost compared to standard methods for steady Navier-Stokes equations.

Findings

2L-ROM reduces computational cost by a factor of 2 to 3.

The error bound for 2L-ROM is established and validated.

Numerical tests on the steady Burgers equation demonstrate effectiveness.

Abstract

We propose, analyze, and investigate numerically a novel two-level Galerkin reduced order model (2L-ROM) for the efficient and accurate numerical simulation of the steady Navier-Stokes equations. In the first step of the 2L-ROM, a relatively low-dimensional nonlinear system is solved. In the second step, the Navier-Stokes equations are linearized around the solution found in the first step, and a higher-dimensional system for the linearized problem is solved. We prove an error bound for the new 2L-ROM and compare it to the standard one level ROM (1L-ROM) in the numerical simulation of the steady Burgers equation. The 2L-ROM significantly decreases (by a factor of $2$ and even $3$) the 1L-ROM computational cost, without compromising its numerical accuracy.