Volume 12, Issue 2
Improved Error Estimates of a Finite Difference/Spectral Method for Time-Fractional Diffusion Equations

Chunwan Lv & Chuanju Xu

Int. J. Numer. Anal. Mod., 12 (2015), pp. 384-400

Published online: 2015-12

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  • Abstract
In this paper, we first consider the numerical method that Lin and Xu proposed and analyzed in [Finite difference/spectral approximations for the time-fractional diffusion equation, JCP 2007] for the time-fractional diffusion equation. It is a method basing on the combination of a finite different scheme in time and spectral method in space. The numerical analysis carried out in that paper showed that the scheme is of (2 - α)-order convergence in time and spectral accuracy in space for smooth solutions, where α is the time-fractional derivative order. The main purpose of this paper consists in refining the analysis and providing a sharper estimate for both time and space errors. More precisely, we improve the error estimates by giving a more accurate coefficient in the time error term and removing the factor in the space error term, which grows with decreasing time step. Then the theoretical results are validated by a number of numerical tests.
  • Keywords

Error estimates finite difference methods spectral methods time fractional diffusion equation

  • AMS Subject Headings

65M12 65M06 65M70 35S10

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@Article{IJNAM-12-384, author = {Chunwan Lv and Chuanju Xu}, title = {Improved Error Estimates of a Finite Difference/Spectral Method for Time-Fractional Diffusion Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2015}, volume = {12}, number = {2}, pages = {384--400}, abstract = {In this paper, we first consider the numerical method that Lin and Xu proposed and analyzed in [Finite difference/spectral approximations for the time-fractional diffusion equation, JCP 2007] for the time-fractional diffusion equation. It is a method basing on the combination of a finite different scheme in time and spectral method in space. The numerical analysis carried out in that paper showed that the scheme is of (2 - α)-order convergence in time and spectral accuracy in space for smooth solutions, where α is the time-fractional derivative order. The main purpose of this paper consists in refining the analysis and providing a sharper estimate for both time and space errors. More precisely, we improve the error estimates by giving a more accurate coefficient in the time error term and removing the factor in the space error term, which grows with decreasing time step. Then the theoretical results are validated by a number of numerical tests.}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/495.html} }
TY - JOUR T1 - Improved Error Estimates of a Finite Difference/Spectral Method for Time-Fractional Diffusion Equations AU - Chunwan Lv & Chuanju Xu JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 384 EP - 400 PY - 2015 DA - 2015/12 SN - 12 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/495.html KW - Error estimates KW - finite difference methods KW - spectral methods KW - time fractional diffusion equation AB - In this paper, we first consider the numerical method that Lin and Xu proposed and analyzed in [Finite difference/spectral approximations for the time-fractional diffusion equation, JCP 2007] for the time-fractional diffusion equation. It is a method basing on the combination of a finite different scheme in time and spectral method in space. The numerical analysis carried out in that paper showed that the scheme is of (2 - α)-order convergence in time and spectral accuracy in space for smooth solutions, where α is the time-fractional derivative order. The main purpose of this paper consists in refining the analysis and providing a sharper estimate for both time and space errors. More precisely, we improve the error estimates by giving a more accurate coefficient in the time error term and removing the factor in the space error term, which grows with decreasing time step. Then the theoretical results are validated by a number of numerical tests.
Chunwan Lv & Chuanju Xu. (1970). Improved Error Estimates of a Finite Difference/Spectral Method for Time-Fractional Diffusion Equations. International Journal of Numerical Analysis and Modeling. 12 (2). 384-400. doi:
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