Efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential Equations

作     者：Xuejuan Chen Zhiping Mao George Em Karniadakis 

作者机构：School of ScienceJimei UniversityXiamen 361021China School of Mathematical SciencesFujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamenChina Division of Applied MathematicsBrown UniversityProvidenceRI 02912USA 

出 版 物：《Numerical Mathematics(Theory,Methods and Applications)》 (高等学校计算数学学报（英文版）)

年 卷 期：2022年第15卷第4期

页      面：876-902页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：Z.Mao was supported by the Fundamental Research Funds for the Central Universities(Grant 20720210037) G.E.Karniadakis was supported by the MURI/ARO on Fractional PDEs for Conservation Laws and Beyond:Theory,Numerics and Applications(Grant W911NF-15-1-0562) X.Chen was supported by the Fujian Provincial Natural Science Foundation of China(Grants 2022J01338,2020J01703) 

主　　题：Stiff problem generalized minimal residual geometric mesh refinement long time evolution fractional phase field models 

摘      要：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 *** 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 ***,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 ***,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.