Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations

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.Source: §1 · p. 1

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.Source: §1 · p. 1

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.Source: §1 · p. 1

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.Source: §1 · p. 1
• 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.Source: §1 · p. 2

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.Source: §2 · p. 2
• 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.Source: §2 · p. 2

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.Source: §2.1 · p. 2
• 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.Source: §2.1 · p. 2
• 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.Source: §2.1 · p. 3
• 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.Source: §2.1 · p. 3
• 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.Source: §2.1 · p. 3

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.Source: §3 · p. 3
• 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.Source: §3 · p. 4

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.Source: §3 · p. 4

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.Source: §3 · p. 4

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.Source: §3.3 · pp. 5–6

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.Source: §3.4 · p. 6

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.Source: §4 · p. 7, §4.2 · p. 7

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.Source: §4.2 · p. 9

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.Source: §4.2 · pp. 10, 12

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.Source: §4.1 · p. 7, §4.2 · p. 10, §4.3 · pp. 12–13

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.Source: §4.3 · p. 13

• The study compares supremizer enrichment and pressure Poisson equation stabilisation for POD-Galerkin ROMs based on finite-volume approximations.Source: §5 · p. 13
• Supremizer stabilisation is introduced in a finite-volume context and effectively stabilises the resulting reduced system.Source: §5 · p. 13
• The pressure Poisson equation approach adds a pressure boundary condition that previous works neglected.Source: §5 · p. 13
• PPE-ROM demonstrates better performance than SUP-ROM for long-time integrations.Source: §5 · p. 13
• Future work targets higher Reynolds numbers, turbulent flows, long-time stabilisation, and efficient geometrical parametrisation.Source: §5 · p. 13

• HF denotes High Fidelity, POD denotes Proper Orthogonal Decomposition, and ROM denotes Reduced Order Model.Source: Appendix A · p. 14
• PPE-ROM denotes a ROM with pressure Poisson equation stabilisation, while SUP-ROM denotes a ROM with pressure supremizer stabilisation.Source: Appendix A · p. 14
• 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.Source: Appendix A · p. 14
• p and u denote full-order pressure and velocity unknown counts, while N_r^p and N_r^u denote their reduced-order counterparts.Source: Appendix A · p. 14