@Article{NMTMA-15-4,
author = {Karniadakis, George, Em and Xuejuan, Chen and Zhiping, Mao and Karniadakis, George, Em},
title = {Efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential Equations},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2022},
volume = {15},
number = {4},
pages = {876--902},
abstract = {<p style="text-align: justify;">We develop an efficient and accurate spectral deferred correction (SDC)
method for fractional differential equations (FDEs) by extending the algorithm in
[14] for classical ordinary differential equations (ODEs). Specifically, we discretize
the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm
(GMRES). We first derive the correction matrix of the SDC method for FDEs and
analyze the convergence region of the SDC method. We then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear
ODEs as well as fractional phase field models, demonstrating that the accelerated
SDC method is much more efficient than the original SDC method, especially for
stiff problems. Furthermore, we resolve the issue of low accuracy arising from the
singularity of the solutions by using a geometric mesh, leading to highly accurate
solutions compared to uniform mesh solutions at almost the same computational
cost. Moreover, for long-time integration of FDEs, using the geometric mesh leads
to great computational savings as the total number of degrees of freedom required
is relatively small.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2022-0012s},
url = {https://global-sci.com/article/90276/efficient-and-accurate-numerical-methods-using-the-accelerated-spectral-deferred-correction-for-solving-fractional-differential-equations}
}