Efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential Equations
Year: 2022
Author: George Em Karniadakis, Xuejuan Chen, Zhiping Mao, George Em Karniadakis
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 4 : pp. 876–902
Abstract
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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2022-0012s
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 4 : pp. 876–902
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 27
Keywords: Stiff problem generalized minimal residual geometric mesh refinement long time evolution fractional phase field models.
Author Details
-
Spectral deferred correction method for fractional initial value problem with Caputo–Hadamard derivative
Liu, Xiaoyuan | Cai, MinMathematics and Computers in Simulation, Vol. 226 (2024), Iss. P.323
https://doi.org/10.1016/j.matcom.2024.07.007 [Citations: 0] -
Accurate numerical simulations for fractional diffusion equations using spectral deferred correction methods
Yang, Zhengya | Chen, Xuejuan | Chen, Yanping | Wang, JingComputers & Mathematics with Applications, Vol. 153 (2024), Iss. P.123
https://doi.org/10.1016/j.camwa.2023.11.001 [Citations: 1]