We propose a regularization for reduced-order models (ROMs) of the quasi-geostrophic equations (QGE) to increase accuracy when the proper orthogonal decomposition (POD) modes retained to construct the reduced basis are insufficient to describe the system dynamics. Our regularization is based on the so-called BV-$\alpha $ model, which modifies the nonlinear term in the QGE and adds a linear differential filter for the vorticity. To show the effectiveness of the BV-$\alpha $ model for ROM closure, we compare the results computed by a POD-Galerkin ROM with and without regularization for the classical double-gyre wind forcing benchmark. Our numerical results show that the solution computed by the regularized ROM is more accurate, even when the retained POD modes account for a small percentage of the eigenvalue energy. Additionally, we show that, although computationally more expensive than the ROM with no regularization, the regularized ROM is still a competitive alternative to full-order simulations of the QGE.

Revised:

Accepted:

Online First:

Published online:

Michele Girfoglio ^{1};
Annalisa Quaini ^{2};
Gianluigi Rozza ^{1}

@article{CRMECA_2023__351_S1_457_0, author = {Michele Girfoglio and Annalisa Quaini and Gianluigi Rozza}, title = {A linear filter regularization for {POD-based} reduced-order models of the quasi-geostrophic equations}, journal = {Comptes Rendus. M\'ecanique}, pages = {457--477}, publisher = {Acad\'emie des sciences, Paris}, volume = {351}, number = {S1}, year = {2023}, doi = {10.5802/crmeca.183}, language = {en}, }

TY - JOUR AU - Michele Girfoglio AU - Annalisa Quaini AU - Gianluigi Rozza TI - A linear filter regularization for POD-based reduced-order models of the quasi-geostrophic equations JO - Comptes Rendus. Mécanique PY - 2023 SP - 457 EP - 477 VL - 351 IS - S1 PB - Académie des sciences, Paris DO - 10.5802/crmeca.183 LA - en ID - CRMECA_2023__351_S1_457_0 ER -

%0 Journal Article %A Michele Girfoglio %A Annalisa Quaini %A Gianluigi Rozza %T A linear filter regularization for POD-based reduced-order models of the quasi-geostrophic equations %J Comptes Rendus. Mécanique %D 2023 %P 457-477 %V 351 %N S1 %I Académie des sciences, Paris %R 10.5802/crmeca.183 %G en %F CRMECA_2023__351_S1_457_0

Michele Girfoglio; Annalisa Quaini; Gianluigi Rozza. A linear filter regularization for POD-based reduced-order models of the quasi-geostrophic equations. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 457-477. doi : 10.5802/crmeca.183. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.183/

[1] Finite element methods for incompressible viscous flow, Handbook of Numerical Analysis (P. G. Ciarlet; J.-L. Lions, eds.), Volume 9, North-Holland, Amsterdam, 2003 | Zbl

[2] Numerical solution for the two-dimensional Stokes problem through the stream-function vorticity formulation, Functional Analysis and Numerical Analysis, Japan–France Seminar, Tokyo and Kyoto, 1976 (1978), pp. 99-142 | Zbl

[3] Streamline-upwind/Petrov–Galerkin procedures for the vorticity-stream function form of the Navier–Stokes equations, Numerical Methods in Laminar and Turbulent Flow, Proceedings of the Fifth International Conference, Montreal, Canada, July 6–10, 1987, Volume 5, Pineridge Press, Swansea, Wales, 1987, pp. 197-209

[4] Petrov–Galerkin methods on multiply connected domains for the vorticity-stream function formulation of the incompressible Navier–Stokes equations, Int. J. Numer. Methods Fluids, Volume 8 (1988) no. 10, pp. 1269-1290 | DOI | MR | Zbl

[5] Iterative solution of the stream function-vorticity formulation of the Stokes problem, applications to the numerical simulation of incompressible viscous flow, Comput. Methods Appl. Mech. Eng., Volume 87 (1991) no. 2, pp. 117-155 | DOI | MR | Zbl

[6] Tuning the mesh of a mixed method for the stream function Vorticity formulation of the Navier–Stokes equations, Numer. Math., Volume 63 (1992), pp. 145-163 | DOI | MR | Zbl

[7] Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, Cambridge, 2006 | DOI

[8] Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic Press, Waltham, MA, 2011

[9] Fundamentals of Geophysical Fluid Dynamics, Cambridge University Press, Cambridge, 2006

[10] Approximate deconvolution large eddy simulation of a stratified two-layer quasi-geostrophic ocean model, Ocean Model., Volume 63 (2012), pp. 1-20 | DOI

[11] A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences, Comput. Math. Appl., Volume 102 (2021), pp. 261-276 | DOI | MR | Zbl

[12] Model reduction for parametrized optimal control problems in environmental marine sciences and engineering, SIAM J. Sci. Comput., Volume 40 (2017), p. B1055-B1079 | DOI | MR | Zbl

[13] Reduced order models for the quasi-geostrophic equations: A brief survey, Fluids, Volume 6 (2020), 16

[14] Advanced Reduced Order Methods and Applications in Computational Fluid Dynamics, SIAM, Philadelphia, 2022 | DOI

[15] Model Order Reduction, De Gruyter, Berlin, Boston, 2020

[16] Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods Eng., Volume 15 (2008), pp. 229-275 | DOI | Zbl

[17] An efficient description of the dynamics of barotropic flow, J. Atmos. Sci., Volume 52 (1995), pp. 915-936 | DOI

[18] A stabilized proper orthogonal decomposition reduced-order model for large scale quasi-geostrophic ocean circulation, Adv. Comput. Math., Volume 41 (2015), pp. 1289-1319 | DOI | Zbl

[19] A dynamic closure modeling framework for model order reduction of geophysical flows, Phys. Fluids, Volume 31 (2019), 046602

[20] Data-driven correction reduced order models for the quasi-geostrophic equations: A numerical investigation, Int. J. Comput. Fluid Dyn., Volume 34 (2020), pp. 147-159 | DOI | MR | Zbl

[21] Data-driven variational multiscale reduced order models, Comput. Methods Appl. Mech. Eng., Volume 373 (2021), 113470 | MR | Zbl

[22] Data-driven filtered reduced order modeling of fluid flows, SIAM J. Sci. Comput., Volume 40 (2018), p. B834-B857 | DOI | MR | Zbl

[23] Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison, Comput. Methods Appl. Mech. Eng., Volume 237–240 (2012), pp. 10-26 | DOI | MR

[24] A hybrid approach for model order reduction of barotropic quasi-geostrophic turbulence, Fluids, Volume 3 (2018) no. 4, 86 | DOI

[25] Nonintrusive reduced order modeling framework for quasigeostrophic turbulence, Phys. Rev. E, Volume 100 (2019), 053306 | MR

[26] Extreme learning machine for reduced order modeling of turbulent geophysical flows, Phys. Rev. E, Volume 97 (2018), 042322

[27] Low-order stochastic mode reduction for a realistic barotropic model climate, J. Atmos. Sci., Volume 62 (2005), pp. 1722-1745 | DOI | MR

[28] Dispersive-dissipative eddy parameterization in a barotropic model, J. Phys. Oceanogr., Volume 31 (2001), pp. 2525-2531 | DOI

[29] Modeling mesoscale turbulence in the barotropic double-gyre circulation, J. Phys. Oceanogr., Volume 33 (2003), pp. 2355-2365 | DOI

[30] Numerical study of a regularized barotropic vorticity model of geophysical flow, Numer. Methods Partial Diff. Equ., Volume 31 (2015), pp. 1492-1514 | DOI | MR | Zbl

[31] Improving numerical accuracy in a regularized barotropic vorticity model of geophysical flow, Int. J. Numer. Anal. Model. Ser. B, Volume 5 (2014), pp. 317-338 | MR | Zbl

[32] A novel large eddy simulation model for the quasi-geostrophic equations in a finite volume setting, J. Comput. Appl. Math., Volume 418 (2023), 114656 | DOI | MR | Zbl

[33] A POD-Galerkin reduced order model for the Navier–Stokes equations in stream function-vorticity formulation, Comput. Fluids, Volume 244 (2022), 105536 | DOI | MR | Zbl

[34] Four-gyre circulation in a barotropic model with double-gyre wind forcing, J. Phys. Oceanogr., Volume 30 (2000), pp. 1461-1471 | DOI

[35] On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment, Int. J. Numer. Methods Eng., Volume 83 (2010), pp. 1345-1375 | DOI | MR | Zbl

[36] Dissipation of energy in isotropic turbulence, Dokl. Akad. Nauk SSSR, Volume 32 (1941), pp. 19-21 | MR | Zbl

[37] A tensorial approach to computational continuum mechanics using object-oriented techniques, Comput. Phys., Volume 12 (1998) no. 6, pp. 620-631 | DOI

[38] Model order reduction, Encyclopedia of Computational Mechanics (E. Stein; R. de Borst; T. J. R. Hughes, eds.), John Wiley & Sons, Ltd, Hoboken (NJ), USA, 2004

[39] A short review on model order reduction based on proper generalized decomposition, Arch. Comput. Methods Eng., Volume 18 (2011), pp. 395-404 | DOI

[40] Proper General Decomposition (PGD) for the resolution of Navier–Stokes equations, J. Comput. Phys., Volume 230 (2011), pp. 1387-1407 | DOI | MR | Zbl

[41] Reduced Basis Methods for Partial Differential Equations, Springer International Publishing, Switzerland, 2016

[42] Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., Volume 40 (2002), pp. 492-515 | DOI | MR | Zbl

[43] Regularized reduced order models for a stochastic Burgers equation, Int. J. Numer. Anal. Model., Volume 15 (2016), pp. 594-607 | MR

[44] An evolve-then-filter regularized reduced order model for convection-dominated flows, Int. J. Numer. Methods Fluids, Volume 84 (2017), pp. 598-615 | DOI | MR

[45] Approximate deconvolution reduced order modeling, Comput. Methods Appl. Mech. Eng., Volume 313 (2016), pp. 512-534 | DOI | MR

[46] A Leray regularized ensemble-proper orthogonal decomposition method for parameterized convection-dominated flows, IMA J. Numer. Anal., Volume 40 (2019) no. 2, pp. 886-913 | DOI | MR | Zbl

[47] An evolve-filter-relax stabilized reduced order stochastic collocation method for the time-dependent Navier–Stokes equations, SIAM/ASA J. Uncertain. Quantif., Volume 7 (2019), pp. 1162-1184 | DOI | MR | Zbl

[48] Pressure stabilization strategies for a LES filtering Reduced Order Model, Fluids, Volume 6 (2021), 302 | DOI

[49] A POD-Galerkin reduced order model for a LES filtering approach, J. Comput. Phys., Volume 436 (2021), 110260 | DOI | MR | Zbl

[50] Consistency of the full and reduced order models for evolve-filter-relax regularization of convection-dominated, marginally-resolved flows, Int. J. Numer. Methods Eng., Volume 32 (2021), pp. 3148-3178

[51] Extension and comparison of techniques to enforce boundary conditions in finite volume POD-Galerkin reduced order models for fluid dynamic problems, 2019 (preprint) | arXiv

[52] Approximate deconvolution large eddy simulation of a barotropic ocean circulation model, Ocean Model., Volume 40 (2011), pp. 120-132 | DOI

[53] The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers, Dokl. Akad. Nauk SSSR, Volume 30 (1941), pp. 299-303

*Cited by Sources: *

Comments - Policy