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

doi:10.1007/s11464-015-0469-8

Front. Math. China

2015, Vol. 10

Issue (5) : 1025-1040 https://doi.org/10.1007/s11464-015-0469-8

RESEARCH ARTICLE

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

Jing AN¹,Zhendong LUO^2,^*(

),Hong LI³,Ping SUN¹

1. School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China
2. School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
3. School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

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Abstract

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 reducedorder 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.

Keywords Singular value decomposition (SVD) proper orthogonal decomposition (POD) bases spectral-finite difference scheme (SFDS) error estimation parabolic equation

Corresponding Author(s): Zhendong LUO

Issue Date: 24 June 2015

Cite this article:

Jing AN,Zhendong LUO,Hong LI, et al. Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation[J]. Front. Math. China, 2015, 10(5): 1025-1040.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-015-0469-8
https://academic.hep.com.cn/fmc/EN/Y2015/V10/I5/1025

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