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Fast parareal iterations for fractional diffusion equations
Wu, Shu-Lin1; Zhou, Tao2
AbstractNumerical 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.
KeywordParareal algorithm Fractional PDEs Convergence analysis Parameter optimization
Funding ProjectNSFC[11301362] ; NSFC[61573010] ; Project of China Postdoctoral Science Foundation[2015M580777] ; Project of China Postdoctoral Science Foundation[2016T90841] ; NSF of Technology & Education of Sichuan Province[2014JQ0035] ; NSF of Technology & Education of Sichuan Province[15ZA0220] ; NSF of SUSE[2015LX01] ; National Natural Science Foundation of China[91530118] ; National Natural Science Foundation of China[11571351]
WOS Research AreaComputer Science ; Physics
WOS SubjectComputer Science, Interdisciplinary Applications ; Physics, Mathematical
WOS IDWOS:000390511500010
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Document Type期刊论文
Corresponding AuthorZhou, Tao
Affiliation1.Sichuan Univ Sci & Engn, Sch Sci, Zigong, Sichuan, Peoples R China
2.Chinese Acad Sci, AMSS, Inst Computat Math & Sci Engn Comp, LSEC, Beijing, Peoples R China
Recommended Citation
GB/T 7714
Wu, Shu-Lin,Zhou, Tao. Fast parareal iterations for fractional diffusion equations[J]. JOURNAL OF COMPUTATIONAL PHYSICS,2017,329:210-226.
APA Wu, Shu-Lin,&Zhou, Tao.(2017).Fast parareal iterations for fractional diffusion equations.JOURNAL OF COMPUTATIONAL PHYSICS,329,210-226.
MLA Wu, Shu-Lin,et al."Fast parareal iterations for fractional diffusion equations".JOURNAL OF COMPUTATIONAL PHYSICS 329(2017):210-226.
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