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Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations
Giovanni Stabile, Gianluigi Rozza
TL;DR
The paper addresses pressure-instability and computational-cost challenges in reduced Galerkin models for parametrized unsteady incompressible Navier-Stokes equations. It combines finite-volume full-order discretization with POD-Galerkin reduction and compares supremizer enrichment against a pressure Poisson equation. Both ROMs capture the main velocity and pressure flow features, while PPE-ROM is more reliable for long-time integrations and SUP-ROM uses a larger reduced system.
Problem
Standard discretizations can become infeasible for many configurations or limited computational budgets, while Galerkin ROMs can develop inf-sup pressure instabilities from spurious pressure modes.
Method
The study applies POD-Galerkin reduction to finite-volume approximations and compares supremizer velocity enrichment with a pressure Poisson equation pressure-stabilisation strategy.
Results
Both ROMs capture the main velocity and pressure flow features with sufficient accuracy, while PPE-ROM performs better for long-time integrations than SUP-ROM.
Takeaways & Limitations
Supremizer stabilisation effectively stabilises the reduced system and provides an alternative to other stabilisation methods, while PPE-ROM is preferable for long-time integration.
Takeaways & Limitations
The supremizer approach cannot be rigorously shown to satisfy the inf-sup condition in this setting and relies on heuristic or post-processing checks, especially beyond non-geometric parametrization.
Abstract
from arXiv · showhide
In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier-Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.
1. Introduction
The paper develops POD-Galerkin reduced-order models for parametrized unsteady fluid dynamics, using finite-volume full-order approximations and addressing reduced pressure-instability challenges.
- Motivation: Standard CFD discretizations can become infeasible when many system configurations must be tested or computational cost is constrained.Reduced-order modelling is presented as a way to overcome this limitation.
- Approach: The study applies POD-Galerkin reduction to parametrized, time-dependent fluid-dynamics equations starting from high-dimensional finite-volume approximations.Finite-volume-based reduced-basis models are less exploited than finite-element-based reduced spaces.
- Problem: Galerkin ROMs for incompressible Navier-Stokes equations can suffer from transient ODE instabilities and inf-sup pressure instabilities caused by spurious pressure modes.This work focuses on the second instability type.
- Contributions: Two pressure-stabilisation strategies are compared: supremizer enrichment of the velocity space and a pressure Poisson equation during the online stage.The comparison also considers long-time integration of systems with periodic response.
- Contributions: The supremizer stabilisation technique is introduced for the first time in the context of a finite-volume approximation.The methods are tested on lid-driven cavity and circular-cylinder-flow benchmarks at moderate Reynolds numbers.
2. Mathematical formulation and full-order approximation of the Navier-Stokes Equation
The full-order problem is the parametrized unsteady incompressible Navier-Stokes system, discretized on finite volumes through conservation-law fluxes and assembled algebraically for reduced projection.
- Mathematical formulation: The model seeks velocity u and pressure p for unsteady incompressible parametrized Navier-Stokes equations on Q = Ω×[0, T] in d = 2, 3.The parameter is the kinematic viscosity ν(μ), assumed constant in space, while inlet data are time-independent.
- Mathematical formulation: The boundary is partitioned into inlet, outlet, and physical-wall portions, with prescribed inlet and wall velocities and a traction condition at the outlet.The initial velocity is prescribed at t = 0.
- The finite Volume Approximation: The finite-volume full-order approximation restricts the weak-form solution to finite-dimensional velocity and pressure spaces over a tessellation of convex, non-overlapping cells.Finite-volume trial functions are piecewise constant and discontinuous, unlike continuous piecewise-polynomial finite-element functions.
- The finite Volume Approximation: The discrete problem imposes a residual equation for all finite-volume test functions, and divergence terms are rewritten as boundary fluxes using Gauss’s theorem.This yields the semi-discretized momentum and mass-conservation equations.
- The finite Volume Approximation: Acceleration coefficients form matrix M, while discretized convection, diffusion, pressure gradient, and velocity divergence contribute matrices C, A, B, and P.The nonlinear convective term is linearized using a previously calculated velocity satisfying continuity.
- The finite Volume Approximation: Face-centered velocity and pressure values are reconstructed from cell-centered values using interpolation schemes.The convective term may require upwind, second-order linear upwind, or MUSCL schemes for stable and accurate computation.
- The finite Volume Approximation: The full-order system is solved with a partitioned PIMPLE algorithm combining SIMPLE and PISO procedures.The particular numerical schemes are specified for the numerical experiments.
3. Reduced order model with a POD-Galerkin method
The reduced model uses POD spaces built from parameter- and time-dependent finite-volume snapshots, then projects the Navier–Stokes equations onto reduced velocity and pressure bases. Stability is addressed through supremizer enrichment or a pressure Poisson formulation, with offline treatment of most reduced operators and a cubic-storage limitation for the convective tensor.
- POD basis construction: POD snapshots include both parameter and time dependence, yielding velocity and pressure snapshot matrices for reduced-basis construction.The total snapshot count is Ns = Nr · Nt.
- POD basis construction: The POD optimization is equivalent to an eigenvalue problem for correlation matrices formed from the snapshots.Velocity and pressure bases are selected according to the decay of their eigenvalues.
- Reduced Galerkin system: Reduced velocity and pressure coefficients are obtained by Galerkin projection, while most reduced matrices are precomputed offline.The nonlinear convective contribution is handled online through a stored third-order tensor.
- Computational limitation: The convective third-order tensor scales cubically with the number of basis functions, creating high storage costs for richer reduced spaces.The paper notes that N < 20 is used here and identifies EIM-DEIM and Gappy-POD as potentially more affordable alternatives.
- Supremizer enrichment: The supremizer strategy augments the reduced velocity space with basis functions computed from pressure snapshots to target the reduced inf-sup condition.The resulting velocity space is generally non-orthogonal, and the supremizer functions are obtained through separate POD treatment and supremizer problems.
- Pressure Poisson equation: The pressure Poisson strategy replaces the incompressibility constraint with a pressure Poisson equation derived from the divergence of the momentum equation.This formulation requires sufficient solution smoothness and introduces a pressure Neumann boundary condition.
4. Numerical Experiments
The two stabilised POD-Galerkin ROMs are tested on lid-driven cavity and circular-cylinder benchmarks using finite-volume high-fidelity data. Both capture the main flow features, while their relative accuracy, long-time behaviour, and computational costs differ.
- Benchmarks: The benchmarks comprise a non-parametric lid-driven cavity flow and a parametrised circular-cylinder flow at moderate Reynolds numbers.The cylinder case also tests long-time integration beyond the snapshot-generation window.
- Stabilisation design: Equal pressure and supremizer-space dimensions do not automatically guarantee the inf-sup condition in the approximated supremizer enrichment.For the cylinder case, equal numbers of pressure and supremizer modes were experimentally associated with inaccurate results.
- Long-time integration: During long cylinder integrations, the SUP-ROM develops numerical instabilities and a non-physical flow pattern, whereas the PPE-ROM remains regular but exhibits a phase shift and slightly longer vortex-shedding period.Both ROM relative errors increase over time, and the SUP-ROM shows oscillatory growth in total kinetic energy.
- Accuracy comparison: The SUP-ROM gives better pressure results but worse velocity results than the PPE-ROM in the cavity and cylinder tests.The authors attribute the velocity difference to additional supremizer modes that are unnecessary for velocity representation and can pollute the POD velocity space.
- Computational cost: Both ROMs achieve considerable speed-up, but the SUP-ROM is less efficient because its additional supremizer modes create a larger reduced dynamical system.The cylinder offline stage uses six processors, while its online stage remains serial on one processor.
5. Conclusions and perspectives
The work compares two pressure-stabilisation strategies for finite-volume POD-Galerkin ROMs of parametrised unsteady Navier–Stokes equations and finds both effective, with PPE-ROM performing better in long-time integrations.
- The study compares supremizer enrichment and pressure Poisson equation stabilisation for POD-Galerkin ROMs based on finite-volume approximations.
- Supremizer stabilisation is introduced in a finite-volume context and effectively stabilises the resulting reduced system.
- The pressure Poisson equation approach adds a pressure boundary condition that previous works neglected.
- PPE-ROM demonstrates better performance than SUP-ROM for long-time integrations.
- Future work targets higher Reynolds numbers, turbulent flows, long-time stabilisation, and efficient geometrical parametrisation.
Appendix A. List of abbreviations and symbols
The appendix defines the paper’s principal abbreviations and records symbols for stability, POD bases, domains, parameters, and full- and reduced-order unknown counts.
- HF denotes High Fidelity, POD denotes Proper Orthogonal Decomposition, and ROM denotes Reduced Order Model.
- PPE-ROM denotes a ROM with pressure Poisson equation stabilisation, while SUP-ROM denotes a ROM with pressure supremizer stabilisation.
- β is the inf-sup stability constant, η_i is the i-th POD basis function for supremizers, and χ_i is the i-th POD basis function for pressure.
- p and u denote full-order pressure and velocity unknown counts, while N_r^p and N_r^u denote their reduced-order counterparts.