Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation

In this study, a classical spectral-finite difference scheme （SFDS） for the three-dimensional （3D） parabolic equation is reduced by using proper orthogonal decomposition （POD） and singular value decomposition （SVD）. First, the 3D parabolic equation is discretized in spatial variables by using spectr...

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Bibliographic Details
Published in:Frontiers of mathematics in China 2015-10, Vol.10 (5), p.1025-1040
Main Authors: An, Jing, Luo, Zhendong, Li, Hong, Sun, Ping
Format: Article
Language:English
Subjects:
Accuracy
Data compression
Decomposition
Linear equations
Load
Mathematical analysis
Mathematics
Mathematics and Statistics
Methods
Numerical analysis
POD
Principal components analysis
Research Article
SFD
Studies
三维
外推
抛物线方程
有限差分法
误差估计
降阶
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Summary:In this study, a classical spectral-finite difference scheme （SFDS） for the three-dimensional （3D） parabolic equation is reduced by using proper orthogonal decomposition （POD） and singular value decomposition （SVD）. First, the 3D parabolic equation is discretized in spatial variables by using spectral collocation method and the discrete scheme is transformed into matrix formulation by tensor product. Second~ the classical SFDS is obtained by difference discretization in time-direction. The ensemble of data are comprised with the first few transient solutions of the classical SFDS for the 3D parabolic equation and the POD bases of ensemble of data are generated by using POD technique and SVD. The unknown quantities of the classical SFDS are replaced with the linear combination of POD bases and a reduced- order extrapolation SFDS with lower dimensions and sufficiently high accuracy for the 3D parabolic equation is established. Third, the error estimates between the classical SFDS solutions and the reduced-order extrapolation SFDS solutions and the implementation for solving the reduced-order extrapolation SFDS are provided. Finally, a numerical example shows that the errors of numerical computations are consistent with the theoretical results. Moreover, it is shown that the reduced-order extrapolation SFDS is effective and feasible to find the numerical solutions for the 3D parabolic equation.
ISSN:1673-3452
1673-3576
DOI:10.1007/s11464-015-0469-8