Frontiers of Mathematics in China

, Volume 10, Issue 5, pp 1025–1040 | Cite as

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

Research Article

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 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.

Keywords

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

MSC

65N12 65N30 65M60 

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Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceGuizhou Normal UniversityGuiyangChina
  2. 2.School of Mathematics and PhysicsNorth China Electric Power UniversityBeijingChina
  3. 3.School of Mathematical SciencesInner Mongolia UniversityHohhotChina

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