Waveform relaxation for fractional sub-diffusion equations
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Waveform relaxation for fractional sub-diffusion equations Jun Liu1 · Yao-Lin Jiang2 · Xiao-Long Wang3 · Yan Wang1 Received: 23 August 2019 / Accepted: 13 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We report a new kind of waveform relaxation (WR) method for general semi-linear fractional sub-diffusion equations, and analyze the upper bound for the iteration errors. It indicates that the WR method is convergent superlinearly, and the convergence rate is dependent on the order of the time-fractional derivative and the length of the time interval. In order to accelerate the convergence, we present the windowing WR method. Then, we elaborate the parallelism based on the discrete windowing WR method, and present the corresponding fast evaluation formula. Numerical experiments are carried out to verify the effectiveness of the theoretic work. Keywords Waveform relaxation · Fractional sub-diffusion equations · Superlinear convergence · Windowing technique · Parallelism Mathematics Subject Classification (2010) 49M20 · 35K57 · 65M15
1 Introduction In recent years, fractional differential equations have attracted more and more researchers’ attention, since they have inherent advantages to describe the various processes and phenomenon with memory and hereditary properties [1, 2]. Fractional sub-diffusion equation is one of the most typical equations and it is often used to model the viscoelastic anomalous diffusion in disordered media [3], nanoprecipitate
Jun Liu
[email protected] 1
College of Science, China University of Petroleum (East China), Qingdao, 266580, Shandong, China
2
Department of Mathematical Sciences, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, China
3
School of Science, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, China
Numerical Algorithms
growth in solid solutions [4], diffusion process in magnetic resonance imaging [5], and so on. Because of the nonlocal properties, the sub-diffusion equations are more complicated than the classical integer-order differential equations, and it is usually difficult to obtain the analytic solutions of the sub-diffusion equations. Various numerical schemes for such equations are emerging [6–8]. The waveform relaxation (WR) method, as an iterative method for time-dependent equations, produces a series of functions with respect to the time variable, to approximate the solution of the equations. The WR method is firstly proposed for the simulation of large circuits [9], and it is able to decouple a complicated large system into a series of weakly coupled small sub-systems, which can be solved simultaneously. It has been proved in [10, 11] that the WR method has superlinear convergence for general nonlinear ordinary differential equations (ODEs) on bounded domain, and is of linear convergence for linear ODEs on unbounded domain. The WR method has been widely used to solve differential algebra equations (DAEs) [12, 13], integral-differential-algebraic equations (IDAEs) [14], and delay-differen
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