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Second order reduced model via incremental projection for Navier Stokes

Mejdi Azaïez, Yayu Guo, Carlos Núñez Fernández, Samuele Rubino, Chuanju Xu

TL;DR

To address efficient simulation of incompressible flows, the authors integrate incremental projection methods with POD-based reduced-order modeling for the Stokes equations. They develop semi-discrete and fully discrete schemes using BDF2 time stepping and FE spatial discretization, derive a POD-ROM with explicit reduced velocity and pressure, and prove stability with error estimates showing second-order temporal convergence. Numerical experiments validate the theory, showing rapid decay of POD eigenvalues, high accuracy with only a few modes, and robust performance under time extrapolation and parameter variation (e.g., Reynolds number in lid-driven cavity). The study suggests that this POD-ROM framework provides a computationally efficient, stable approach for reduced-order modeling of incompressible flows and can serve as a stepping stone toward full Navier–Stokes applications and data-driven extensions.

Abstract

The numerical simulation of incompressible flows is challenging due to the tight coupling of velocity and pressure. Projection methods offer an effective solution by decoupling these variables, making them suitable for large-scale computations. This work focuses on reduced-order modeling using incremental projection schemes for the Stokes equations. We present both semi-discrete and fully discrete formulations, employing BDF2 in time and finite elements in space. A proper orthogonal decomposition (POD) approach is adopted to construct a reduced-order model for the Stokes problem. The method enables explicit computation of reduced velocity and pressure while preserving accuracy. We provide a detailed stability analysis and derive error estimates, showing second-order convergence in time. Numerical experiments are conducted to validate the theoretical results and demonstrate computational efficiency.

Second order reduced model via incremental projection for Navier Stokes

TL;DR

To address efficient simulation of incompressible flows, the authors integrate incremental projection methods with POD-based reduced-order modeling for the Stokes equations. They develop semi-discrete and fully discrete schemes using BDF2 time stepping and FE spatial discretization, derive a POD-ROM with explicit reduced velocity and pressure, and prove stability with error estimates showing second-order temporal convergence. Numerical experiments validate the theory, showing rapid decay of POD eigenvalues, high accuracy with only a few modes, and robust performance under time extrapolation and parameter variation (e.g., Reynolds number in lid-driven cavity). The study suggests that this POD-ROM framework provides a computationally efficient, stable approach for reduced-order modeling of incompressible flows and can serve as a stepping stone toward full Navier–Stokes applications and data-driven extensions.

Abstract

The numerical simulation of incompressible flows is challenging due to the tight coupling of velocity and pressure. Projection methods offer an effective solution by decoupling these variables, making them suitable for large-scale computations. This work focuses on reduced-order modeling using incremental projection schemes for the Stokes equations. We present both semi-discrete and fully discrete formulations, employing BDF2 in time and finite elements in space. A proper orthogonal decomposition (POD) approach is adopted to construct a reduced-order model for the Stokes problem. The method enables explicit computation of reduced velocity and pressure while preserving accuracy. We provide a detailed stability analysis and derive error estimates, showing second-order convergence in time. Numerical experiments are conducted to validate the theoretical results and demonstrate computational efficiency.

Paper Structure

This paper contains 15 sections, 83 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Second order incremental scheme
  3. Semi-discrete scheme
  4. Fully discrete scheme
  5. The POD reduced order model
  6. The proper orthogonal decomposition
  7. Formulating projection ROM
  8. Stability and error estimate of pressure projection ROM
  9. The Numerical Results
  10. Accuracy Test
  11. POD Reduced Order Model
  12. Lid-driven cavity flow
  13. Snapshots Generation for POD construction
  14. Numerical Results
  15. Conclusion

Figures (7)

  • Figure 1: Decay of the normalized POD eigenvalues (left) and captured energy (right).
  • Figure 2: l2l2 errors of velocity and pressure.
  • Figure 3: Comparative between the theoretical error estimator \ref{['rom_ea_th']} and the velocity and pressure error depending on the number of modes $(r).$
  • Figure 4: Decay of the normalized POD eigenvalues (left) and captured energy (right).
  • Figure 5: l2l2 errors of velocity and pressure.
  • ...and 2 more figures

Theorems & Definitions (2)

  • proof
  • proof