A cure for instabilities due to advection-dominance in POD solution to advection-diffusion-reaction equations. (English) Zbl 07508508
Summary: In this paper, we propose to improve the stabilized POD-ROM introduced in [S. Rubino, ESAIM, Proc. Surv. 64, 121–136 (2018; Zbl 1483.76040)] to deal with the numerical simulation of advection-dominated advection-diffusion-reaction equations. In particular, we propose a three-stage stabilizing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This approach mainly consists in three ingredients: (1) the addition of a “streamline diffusion” stabilization term to the governing projected equations, (2) the modification of the correlation matrix defining the POD modes associated to the advection stabilization term, and (3) an a-posteriori stabilization scheme. Numerical studies are performed to discuss the accuracy and performance of the new method in handling strongly advection-dominated cases.
Keywords:
finite element method; filtered advection stabilization; a-posteriori stabilization; proper orthogonal decomposition; reduced order models; convection-dominated flowsCitations:
Zbl 1483.76040Software:
FreeFem++References:
| [1] | Ahmed, N.; Chacón Rebollo, T.; John, V.; Rubino, S., Analysis of a full space-time discretization of the Navier-Stokes equations by a local projection stabilization method, IMA J. Numer. Anal., 37, 3, 1437-1467 (2017) · Zbl 1407.76059 |
| [2] | Ahmed, N.; Chacón Rebollo, T.; John, V.; Rubino, S., A review of variational multiscale methods for the simulation of turbulent incompressible flows, Arch. Comput. Methods Eng., 24, 1, 115-164 (2017) · Zbl 1360.76105 |
| [3] | Ahmed, N.; Matthies, G., Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent linear convection-diffusion-reaction equations, J. Sci. Comput., 67, 3, 988-1018 (2016) · Zbl 1342.65192 |
| [4] | Ahmed, N.; Rubino, S., Numerical comparisons of finite element stabilized methods for a 2D vortex dynamics simulation at high Reynolds number, Comput. Methods Appl. Mech. Eng., 349, 191-212 (2019) · Zbl 1441.76051 |
| [5] | Akhtar, I.; Borggaard, J.; Hay, A., Shape sensitivity analysis in flow models using a finite-difference approach, Math. Probl. Eng., Article 209780 pp. (2010) · Zbl 1191.76072 |
| [6] | Aleksić, K.; King, R.; Noack, B. R.; Lehmann, O.; Morzyński, M.; Tadmor, G., Nonlinear flow control using a low dimensional Galerkin model, Facta Univ. Ser. Autom. Control Robot., 7, 1, 63-70 (2008) |
| [7] | Azaïez, M.; Ben Belgacem, F., Karhunen-Loève’s truncation error for bivariate functions, Comput. Methods Appl. Mech. Eng., 290, 57-72 (2015) · Zbl 1425.65066 |
| [8] | Azaïez, M.; Ben Belgacem, F.; Chacón Rebollo, T., Error bounds for POD expansions of parameterized transient temperatures, Comput. Methods Appl. Mech. Eng., 305, 501-511 (2016) · Zbl 1425.74446 |
| [9] | Azaïez, M.; Ben Belgacem, F.; Chacón Rebollo, T.; Gómez Mármol, M.; Sánchez Muñoz, I., Error bounds in high-order Sobolev norms for POD expansions of parameterized transient temperatures, C. R. Math. Acad. Sci. Paris, 355, 4, 432-438 (2017) · Zbl 1362.65095 |
| [10] | Azaïez, M.; Chacón Rebollo, T.; Perracchione, E.; Vega, J. M., Recursive POD expansion for the advection-diffusion-reaction equation, Commun. Comput. Phys., 24, 5, 1556-1578 (2018) · Zbl 1475.74125 |
| [11] | Baiges, J.; Codina, R.; Idelsohn, S., Explicit reduced-order models for the stabilized finite element approximation of the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 72, 12, 1219-1243 (2013) · Zbl 1455.76078 |
| [12] | Balajewicz, M.; Tezaur, I.; Dowell, E., Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier-Stokes equations, J. Comput. Phys., 321, 224-241 (2016) · Zbl 1349.76171 |
| [13] | Ballarin, F.; Manzoni, A.; Quarteroni, A.; Rozza, G., Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, Int. J. Numer. Methods Eng., 102, 5, 1136-1161 (2015) · Zbl 1352.76039 |
| [14] | Bergmann, M.; Bruneau, C.-H.; Iollo, A., Enablers for robust POD models, J. Comput. Phys., 228, 2, 516-538 (2009) · Zbl 1409.76099 |
| [15] | Bergmann, M.; Cordier, L., Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models, J. Comput. Phys., 227, 16, 7813-7840 (2008) · Zbl 1388.76073 |
| [16] | Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext (2011), Springer: Springer New York · Zbl 1220.46002 |
| [17] | Burkardt, J.; Gunzburger, M.; Lee, H.-C., POD and CVT-based reduced-order modeling of Navier-Stokes flows, Comput. Methods Appl. Mech. Eng., 196, 1-3, 337-355 (2006) · Zbl 1120.76323 |
| [18] | Burman, E., Consistent SUPG-method for transient transport problems: stability and convergence, Comput. Methods Appl. Mech. Eng., 199, 17-20, 1114-1123 (2010) · Zbl 1227.76047 |
| [19] | Caiazzo, A.; Iliescu, T.; John, V.; Schyschlowa, S., A numerical investigation of velocity-pressure reduced order models for incompressible flows, J. Comput. Phys., 259, 598-616 (2014) · Zbl 1349.76050 |
| [20] | Chacón-Rebollo, T.; Domínguez-Delgado, A., On the self-stabilization of Galerkin finite element solution of incompressible fluid flows, (Computational Science for the 21st Century (1997), Wiley), 221-230 · Zbl 0919.76047 |
| [21] | Chacón Rebollo, T.; Gómez Mármol, M.; Girault, V.; Sánchez Muñoz, I., A high order term-by-term stabilization solver for incompressible flow problems, IMA J. Numer. Anal., 33, 3, 974-1007 (2013) · Zbl 1426.76234 |
| [22] | Chapelle, D.; Gariah, A.; Sainte-Marie, J., Galerkin approximation with proper orthogonal decomposition: new error estimates and illustrative examples, ESAIM: Math. Model. Numer. Anal., 46, 4, 731-757 (2012) · Zbl 1273.65125 |
| [23] | Couplet, M.; Sagaut, P.; Basdevant, C., Intermodal energy transfers in a proper orthogonal decomposition-Galerkin representation of a turbulent separated flow, J. Fluid Mech., 491, 275-284 (2003) · Zbl 1063.76570 |
| [24] | DeCaria, V.; Iliescu, T.; Layton, W.; McLaughlin, M.; Schneier, M., An artificial compression reduced order model (2019), arXiv preprint |
| [25] | Dehghan, M., Numerical solution of the three-dimensional advection-diffusion equation, Appl. Math. Comput., 150, 1, 5-19 (2004) · Zbl 1038.65074 |
| [26] | Dehghan, M., Weighted finite difference techniques for the one-dimensional advection-diffusion equation, Appl. Math. Comput., 147, 2, 307-319 (2004) · Zbl 1034.65069 |
| [27] | Dehghan, M.; Abbaszadeh, M., The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations, J. Comput. Phys., 351, 478-510 (2017) · Zbl 1380.65301 |
| [28] | Dehghan, M.; Abbaszadeh, M., An upwind local radial basis functions-differential quadrature (RBF-DQ) method with proper orthogonal decomposition (POD) approach for solving compressible Euler equation, Eng. Anal. Bound. Elem., 92, 244-256 (2018) · Zbl 1403.65109 |
| [29] | Galletti, B.; Bruneau, C. H.; Zannetti, L.; Iollo, A., Low-order modelling of laminar flow regimes past a confined square cylinder, J. Fluid Mech., 503, 161-170 (2004) · Zbl 1116.76344 |
| [30] | Giere, S.; Iliescu, T.; John, V.; Wells, D., SUPG reduced order models for convection-dominated convection-diffusion-reaction equations, Comput. Methods Appl. Mech. Eng., 289, 454-474 (2015) · Zbl 1425.65111 |
| [31] | Graham, W. R.; Peraire, J.; Tang, K. Y., Optimal control of vortex shedding using low-order models. I. Open-loop model development, Int. J. Numer. Methods Eng., 44, 7, 945-972 (1999) · Zbl 0955.76026 |
| [32] | Hay, A.; Borggaard, J.; Akhtar, I.; Pelletier, D., Reduced-order models for parameter dependent geometries based on shape sensitivity analysis, J. Comput. Phys., 229, 4, 1327-1352 (2010) · Zbl 1329.76058 |
| [33] | Hecht, F., New development in freefem++, J. Numer. Math., 20, 3-4, 251-265 (2012) · Zbl 1266.68090 |
| [34] | Hesthaven, J. S.; Rozza, G.; Stamm, B., Certified Reduced Basis Methods for Parametrized Partial Differential Equations, SpringerBriefs in Mathematics (2016), Springer/BCAM Basque Center for Applied Mathematics/BCAM SpringerBriefs: Springer/BCAM Basque Center for Applied Mathematics/BCAM SpringerBriefs Cham/Bilbao · Zbl 1329.65203 |
| [35] | Holmes, P.; Lumley, J. L.; Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge Monographs on Mechanics (1996), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0890.76001 |
| [36] | Iliescu, T.; Wang, Z., Variational multiscale proper orthogonal decomposition: convection-dominated convection-diffusion-reaction equations, Math. Comput., 82, 283, 1357-1378 (2013) · Zbl 1336.76017 |
| [37] | Iliescu, T.; Wang, Z., Are the snapshot difference quotients needed in the proper orthogonal decomposition?, SIAM J. Sci. Comput., 36, 3, A1221-A1250 (2014) · Zbl 1297.65092 |
| [38] | Iliescu, T.; Wang, Z., Variational multiscale proper orthogonal decomposition: Navier-Stokes equations, Numer. Methods Partial Differ. Equ., 30, 2, 641-663 (2014) · Zbl 1452.76048 |
| [39] | John, V.; Knobloch, P., On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. I. A review, Comput. Methods Appl. Mech. Eng., 196, 17-20, 2197-2215 (2007) · Zbl 1173.76342 |
| [40] | John, V.; Novo, J., Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations, SIAM J. Numer. Anal., 49, 3, 1149-1176 (2011) · Zbl 1233.65065 |
| [41] | Kalashnikova, I.; Barone, M. F., 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., 83, 10, 1345-1375 (2010) · Zbl 1202.74123 |
| [42] | Kalashnikova, I.; Barone, M. F., Efficient non-linear proper orthogonal decomposition/Galerkin reduced order models with stable penalty enforcement of boundary conditions, Int. J. Numer. Methods Eng., 90, 11, 1337-1362 (2012) · Zbl 1242.76239 |
| [43] | Knobloch, P.; Lube, G., Local projection stabilization for advection-diffusion-reaction problems: one-level vs. two-level approach, Appl. Numer. Math., 59, 12, 2891-2907 (2009) · Zbl 1180.65139 |
| [44] | Kunisch, K.; Volkwein, S., Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90, 1, 117-148 (2001) · Zbl 1005.65112 |
| [45] | Li, R.; Wu, Q.; Zhu, S., Proper orthogonal decomposition with SUPG-stabilized isogeometric analysis for reduced order modelling of unsteady convection-dominated convection-diffusion-reaction problems, J. Comput. Phys., 387, 280-302 (2019) · Zbl 1452.76094 |
| [46] | Lucia, D. J.; Beran, P. S., Projection methods for reduced order models of compressible flows, J. Comput. Phys., 188, 1, 252-280 (2003) · Zbl 1041.76056 |
| [47] | Rahman, S. M.; Ahmed, S. E.; San, O., A dynamic closure modeling framework for model order reduction of geophysical flows, Phys. Fluids, 31, Article 046602 pp. (2019) |
| [48] | Rubino, S., A streamline derivative POD-ROM for advection-diffusion-reaction equations, (SMAI 2017—\( 8^{\operatorname{e}}\) Biennale Française des Mathématiques Appliquées et Industrielles. SMAI 2017—\( 8^{\operatorname{e}}\) Biennale Française des Mathématiques Appliquées et Industrielles, ESAIM Proc. Surveys, vol. 64 (2018), EDP Sci.: EDP Sci. Les Ulis), 121-136 · Zbl 1483.76040 |
| [49] | Rubino, S., Numerical analysis of a projection-based stabilized POD-ROM for incompressible flows, SIAM J. Numer. Anal., 58, 4, 2019-2058 (2020) · Zbl 1450.65129 |
| [50] | Sagaut, P., Large Eddy Simulation for Incompressible Flows, Scientific Computation (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1091.76001 |
| [51] | Scott, R. L.; Zhang, S., Finite element interpolation of non-smooth functions satisfying boundary conditions, Math. Comput., 54, 190, 483-493 (1990) · Zbl 0696.65007 |
| [52] | Singler, J. R., New POD error expressions, error bounds, and asymptotic results for reduced order models of parabolic PDEs, SIAM J. Numer. Anal., 52, 2, 852-876 (2014) · Zbl 1298.65140 |
| [53] | Sirovich, L., Turbulence and the dynamics of coherent structures. I. Coherent structures, Q. Appl. Math., 45, 3, 561-571 (1987) · Zbl 0676.76047 |
| [54] | Stabile, G.; Rozza, G., Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations, Comput. Fluids, 173, 273-284 (2018) · Zbl 1410.76264 |
| [55] | Tadmor, G.; Lehmann, O.; Noack, B. R.; Cordier, L.; Delville, J.; Bonnet, J.-P.; Morzyński, M., Reduced-order models for closed-loop wake control, Philos. Trans. R. Soc. Lond. A, Math. Phys. Eng. Sci., 369, 1940, 1513-1524 (2011) · Zbl 1219.76020 |
| [56] | Volkwein, S., Model reduction using proper orthogonal decomposition (2011), University of Konstanz, available at |
| [57] | Wang, Z.; Akhtar, I.; Borggaard, J.; Iliescu, T., Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison, Comput. Methods Appl. Mech. Eng., 237, 240, 10-26 (2012) · Zbl 1253.76050 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
