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, finite difference methods, spectral methods, time fractional diffusion equation.
AB - <p style="text-align: justify;">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.</p>