Volume 11, Issue 4
Spectral Deferred Correction Methods for Fractional Differential Equations

Chunwan Lv, Mejdi Azaiez & Chuanju Xu

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 729-751.

Published online: 2018-06

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  • Abstract

In this paper, we propose and analyze a spectral deferred correction method for the fractional differential equation of order α. The proposed method is based on a well-known finite difference method of $(2−α)$-order, see [Sun and Wu, Appl. Numer. Math., 56(2), 2006] and [Lin and Xu, J. Comput. Phys., 225(2), 2007], for prediction of the numerical solution, which is then corrected through a spectral deferred correction method. In order to derive the convergence rate of the prediction-correction iteration, we first derive an error estimate for the $(2−α)$-order finite difference method on some non-uniform meshes. Then the convergence rate of orders $\mathcal{O}(τ^{(2−α)(p+1)})$ and  $\mathcal{O}(τ^{(2−α)+p})$ of the overall scheme is demonstrated numerically for the uniform mesh and the Gauss-Lobatto mesh respectively, where $τ$ is the maximal time step size and $p$ is the number of correction steps. The performed numerical test confirms the efficiency of the proposed method.

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@Article{NMTMA-11-729, author = {}, title = {Spectral Deferred Correction Methods for Fractional Differential Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {11}, number = {4}, pages = {729--751}, abstract = {

In this paper, we propose and analyze a spectral deferred correction method for the fractional differential equation of order α. The proposed method is based on a well-known finite difference method of $(2−α)$-order, see [Sun and Wu, Appl. Numer. Math., 56(2), 2006] and [Lin and Xu, J. Comput. Phys., 225(2), 2007], for prediction of the numerical solution, which is then corrected through a spectral deferred correction method. In order to derive the convergence rate of the prediction-correction iteration, we first derive an error estimate for the $(2−α)$-order finite difference method on some non-uniform meshes. Then the convergence rate of orders $\mathcal{O}(τ^{(2−α)(p+1)})$ and  $\mathcal{O}(τ^{(2−α)+p})$ of the overall scheme is demonstrated numerically for the uniform mesh and the Gauss-Lobatto mesh respectively, where $τ$ is the maximal time step size and $p$ is the number of correction steps. The performed numerical test confirms the efficiency of the proposed method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2018.s03}, url = {http://global-sci.org/intro/article_detail/nmtma/12469.html} }
TY - JOUR T1 - Spectral Deferred Correction Methods for Fractional Differential Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 729 EP - 751 PY - 2018 DA - 2018/06 SN - 11 DO - http://doi.org/10.4208/nmtma.2018.s03 UR - https://global-sci.org/intro/article_detail/nmtma/12469.html KW - AB -

In this paper, we propose and analyze a spectral deferred correction method for the fractional differential equation of order α. The proposed method is based on a well-known finite difference method of $(2−α)$-order, see [Sun and Wu, Appl. Numer. Math., 56(2), 2006] and [Lin and Xu, J. Comput. Phys., 225(2), 2007], for prediction of the numerical solution, which is then corrected through a spectral deferred correction method. In order to derive the convergence rate of the prediction-correction iteration, we first derive an error estimate for the $(2−α)$-order finite difference method on some non-uniform meshes. Then the convergence rate of orders $\mathcal{O}(τ^{(2−α)(p+1)})$ and  $\mathcal{O}(τ^{(2−α)+p})$ of the overall scheme is demonstrated numerically for the uniform mesh and the Gauss-Lobatto mesh respectively, where $τ$ is the maximal time step size and $p$ is the number of correction steps. The performed numerical test confirms the efficiency of the proposed method.

Chunwan Lv, Mejdi Azaiez & Chuanju Xu. (2020). Spectral Deferred Correction Methods for Fractional Differential Equations. Numerical Mathematics: Theory, Methods and Applications. 11 (4). 729-751. doi:10.4208/nmtma.2018.s03
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