Fast parareal iterations for fractional diffusion equations

作     者：Wu, Shu-Lin Zhou, Tao 

作者机构：Sichuan Univ Sci & Engn Sch Sci Zigong Sichuan Peoples R China Chinese Acad Sci AMSS Inst Computat Math & Sci Engn Comp LSEC Beijing Peoples R China 

出 版 物：《JOURNAL OF COMPUTATIONAL PHYSICS》 (计算物理学杂志)

年 卷 期：2017年第329卷

页      面：210-226页

核心收录：

学科分类：07[理学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 0702[理学-物理学] 

基　　金：NSFC [11301362, 61573010] Project of China Postdoctoral Science Foundation [2015M580777, 2016T90841] NSF of Technology & Education of Sichuan Province [2014JQ0035, 15ZA0220] NSF of SUSE [2015LX01] National Natural Science Foundation of China [91530118, 11571351] 

主　　题：Parareal algorithm Fractional PDEs Convergence analysis Parameter optimization 

摘      要：Numerical methods for fractional PDEs is a hot topic recently. This work is concerned with the parareal algorithm for system of ODEs u (t) + Au(t) = f that arising from semi-discretizations of time-dependent fractional diffusion equations with nonsymmetric Riemann-Liouville fractional derivatives. The spatial semi-discretization of this kind of fractional derivatives often results in a coefficient matrix A with spectrum sigma(A) satisfying sigma(A)subset of S(eta) := {lambda is an element of C : R(lambda) =eta, F(lambda) is an element of R}, where eta 0 is a measure of dissipativity of the differential equations. To accelerate the parareal algorithm, we propose a scaled model u (t) + 1/alpha Au(t) = f (with alpha 0) to serve the coarse grid correction, which is an important component of our parareal algorithm. Given eta and alpha, we derive a sharp bound of the convergence factor of the parareal iterations. Moreover, by minimizing such a bound we get optimized scaling factor alpha(opt). It is shown that, compared to alpha =1 (i.e., the classical implementation pattern of the coarse grid correction), the optimized scaling factor significantly improves the convergence rate. Numerical examples are presented to support the theoretical finding. (C) 2016 Elsevier Inc. All rights reserved.