Two-Grid Based Adaptive Proper Orthogonal Decomposition Method for Time Dependent Partial Differential Equations
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Two-Grid Based Adaptive Proper Orthogonal Decomposition Method for Time Dependent Partial Differential Equations Xiaoying Dai1,2 · Xiong Kuang1,2 · Jack Xin3 · Aihui Zhou1,2 Received: 23 June 2019 / Revised: 25 April 2020 / Accepted: 23 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for the numerical solution obtained in the fine grid. Our new method is cheap and easy to be implement. We apply our new method to the solution of time-dependent advection–diffusion equations with the Kolmogorov flow and the ABC flow. The numerical results show that our method is more efficient than the existing POD methods. Keywords Proper orthogonal decomposition · Galerkin projection · Error indicator · Adaptive · Two grid
1 Introduction Time dependent partial differential equations play an important role in scientific and engineering computing. Many physical phenomena are described by time dependent partial differential equations, for example, the seawater intrusion [3], the heat transfer [12], fluid equations [6,57]. The design and analysis of high efficiency numerical schemes for time dependent partial differential equations has always been an active research topic. For the spatial discretization of the time dependent partial differential equations, some classical discretization methods, for example, the finite element method [9], the finite dif-
This work was supported by the National Key Research and Development Program of China under Grant 2019YFA0709601, the National Natural Science Foundation of China under Grants 91730302 and 11671389, the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences under Grant QYZDJ-SSW-SYS010, and the NSF Grant IIS-1632935 and DMS-1854434.
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Xiaoying Dai [email protected]
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LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
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School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
3
Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA 0123456789().: V,-vol
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Journal of Scientific Computing
(2020) 84:47
ference method [30], the plane wave method [27], can be used. However, for many complex systems, these classical discretization methods will usually result in discretized systems with millons or even billions of degree of freedom, especially when the spatial dimension is equal to or larger than three. Therefore, if we use these classical discretization methods to deal with the spatial discretization at each time interval, the computational cost will be very huge [5,10,47]. We realize that many model reduction methods have been developed to reduce the degrees of freedom
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