POD-(H)DG Method for Incompressible Flow Simulations
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POD-(H)DG Method for Incompressible Flow Simulations Guosheng Fu1
· Zhu Wang2
Received: 17 April 2020 / Revised: 12 August 2020 / Accepted: 30 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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. Keywords HDG · DG · POD · Burgers’ equation · Navier–Stokes equations Mathematics Subject Classification 65N30 · 65N12 · 76S05 · 76D07
1 Introduction Reduced order modeling has been widely used in flow control and optimization problems to alleviate the huge computational cost needed in many-query solutions of the large-scale dynamical systems associated to these problems [2,13,14,19]. To achieve the high computational efficiency, model reduction methods construct from data a numerical surrogate model with the dimension greatly reduced from the original system. To build such a low-dimensional model, one can use non-intrusive approaches such as operator learning [3,18], or intrusive approaches such as projection-based methods [7]. The method to be used in this paper falls into the second category. In particular, we consider the proper orthogonal decomposition (POD) method—one of the most popular snapshot-based model reduction techniques. The
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.
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Guosheng Fu [email protected] Zhu Wang [email protected]
1
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, USA
2
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA 0123456789().: V,-vol
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Journal of Scientific Computing
(2020) 85:24
general POD model reduction methodology splits the overall calculation into offline and online stages. At the offline stage, a handful of reduced basis vectors are determined and a low-dimensional, reduced order model (ROM) is constructed by learning algorithms or by projecting equations to the space spanned by the reduced basis. At the online stage, the ROM is used alternative to the original system for simulations that can be finished in short time or even real time. When the system contains non-polynomial nonlinearities, hyper-reduction has to be used in order to guarantee the online computational complexity to be independent of the dimension of the original system [9,10]. The ROM can be discretized by any
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