We present a reduced order method based on proper orthogonal decomposition for the viscous Burgers’ equation and the incompressible Navier–Stokes equations discretized using an implicit-explicit hybrid discontinuous Galerkin/discoutinuous Galerkin (IMEX HDG/DG) scheme. A novel closure model, which can be easily computed offline, is introduced. Numerical results are presented to test the proposed POD model and the closure model.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Data Availibility Statement
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request
Antonietti, P.F., Pacciarini, P., Quarteroni, A.: A discontinuous Galerkin reduced basis element method for elliptic problems. ESAIM: Math. Model. Numer. Anal. 50, 337–360 (2016)
Antoulas, A.: Approximation of large-scale dynamical systems. In: Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), vol. 6. Philadelphia, PA (2005)
Antoulas, A., Beattie, C., Güğercin, S.: Interpolatory methods for model reduction. Society for Industrial and Applied Mathematics (SIAM). Philadelphia, PA (2020)
Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32, 797–823 (1995)
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, 1136–1161 (2015)
Bell, J.B., Colella, P., Glaz, H.M.: A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85, 257–283 (1989)
Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57, 483–531 (2015)
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)
Carlberg, K., Farhat, C., Cortial, J., Amsallem, D.: The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242, 623–647 (2013)
Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764 (2010)
Cockburn, B.: Static condensation, hybridization, and the devising of the HDG methods. In: Barrenechea, G., Brezzi, F., Cangiani, A., Georgoulis, E. (eds.) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 114, pp. 129–177. Springer, Cham (2016)
Fareed, H., Singler, J.R., Zhang, Y., Shen, J.: Incremental proper orthogonal decomposition for PDE simulation data. Comput. Math. Appl. 75, 1942–1960 (2018)
Gunzburger, M.D.: Perspectives in flow control and optimization. In: Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), vol. 5. Philadelphia, PA (2003)
Hesthaven, J.S., Rozza, G., Stamm, B., et al.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations, vol. 590. Springer, Cham (2016)
Lehrenfeld, C.: Hybrid Discontinuous Galerkin methods for solving incompressible flow problems. Diploma Thesis, MathCCES/IGPM, RWTH Aachen (2010)
Lehrenfeld, C., Schöberl, J.: High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput. Methods Appl. Mech. Eng. 307, 339–361 (2016)
Nigro, P.S.B., Anndif, M., Teixeira, Y., Pimenta, P.M., Wriggers, P.: An adaptive model order reduction by proper snapshot selection for nonlinear dynamical problems. Comput. Mech. 57, 537–554 (2016)
Peherstorfer, B., Willcox, K.: Data-driven operator inference for nonintrusive projection-based model reduction. Comput. Methods Appl. Mech. Eng. 306, 196–215 (2016)
Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations: An Introduction, vol. 92. Springer, Cham (2015)
San, O., Iliescu, T.: Proper orthogonal decomposition closure models for fluid flows: Burgers equation. Int. J. Numer. Anal. Model. Ser. B 5, 217–237 (2014)
Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R.: Benchmark computations of laminar flow around a cylinder. In: Hirschel, E.H. (ed.) Flow Simulation with High-Performance Computers II. Notes on Numerical Fluid Mechanics (NNFM), vol. 48, pp. 547–566 (1996)
Schöberl, J.: C\(++\)11 Implementation of finite elements in NGSolve. ASC Report 30/2014, Vienna University of Technology, Institute for Analysis and Scientific Computing (2014)
Shen, J., Singler, J.R., Zhang, Y.: HDG-POD reduced order model of the heat equation. J. Comput. Appl. Math. 362, 663–679 (2019)
Sirovich, L.: Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Math. 45, 561–571 (1987)
Uzunca, M., Karasözen, B.: Energy stable model order reduction for the Allen–Cahn equation. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds.) Model Reduction of Parametrized Systems, vol. 17, pp. 403–419. Springer (2017)
Wang, Z.: Nonlinear model reduction based on the finite element method with interpolated coefficients: semilinear parabolic equations. Numer. Methods Part. Differ. Equ. 31, 1713–1741 (2015)
Yano, M.: Discontinuous Galerkin reduced basis empirical quadrature procedure for model reduction of parametrized nonlinear conservation laws. Adv. Comput. Math. 45, 2287–2320 (2019)
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Guosheng Fu gratefully acknowledges the partial support of this work from U.S. National Science Foundation through Grant DMS-2012031. Zhu Wang gratefully acknowledges the partial support of this work from U.S. National Science Foundation through Grant DMS-1913073 and Office of the Vice President for Research at the University of South Carolina through an ASPIRE grant.
About this article
Cite this article
Fu, G., Wang, Z. POD-(H)DG Method for Incompressible Flow Simulations. J Sci Comput 85, 24 (2020). https://doi.org/10.1007/s10915-020-01328-4
- Burgers’ equation
- Navier–Stokes equations
Mathematics Subject Classification