TY - JOUR T1 - Improved Error Estimates of a Finite Difference/Spectral Method for Time-Fractional Diffusion Equations 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, finite difference methods, spectral methods, 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 based 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 $(2-\alpha)$-order convergence in time and spectral accuracy in space for smooth solutions, where $\alpha$ 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.