High order schemes for the tempered fractional diffusion equations
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High order schemes for the tempered fractional diffusion equations Can Li1,2 · Weihua Deng3
Received: 4 February 2014 / Accepted: 3 October 2015 © Springer Science+Business Media New York 2015
Abstract L´evy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering L´evy measure of L´evy flights leads to the tempered stable L´evy processes which combine both the α-stable and Gaussian trends; and the very large jumps are unlikely and all their moments exist. The probability density functions of the tempered stable L´evy processes solve the tempered fractional diffusion equation. This paper focuses on designing the high order difference schemes for the tempered fractional diffusion equation on bounded domain. The high order difference approximations, called the tempered and weighted and shifted Gr¨unwald difference (tempered-WSGD) operators, in space are obtained by using the properties of the tempered fractional calculus and weighting and shifting their first order Gr¨unwald type difference approximations. And the Crank-Nicolson discretization is used in the time direction. The stability and convergence of the presented numerical schemes are established; and the numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the schemes.
Communicated by: Jan Hesthaven Weihua Deng
[email protected] Can Li [email protected] 1
Department of Applied Mathematics, School of Sciences, Xi’an University of Technology, Xi’an, Shaanxi 710054, People’s Republic of China
2
Beijing Computational Science Research Center, Beijing 100084, People’s Republic of China
3
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People’s Republic of China
C. Li, W. Deng
Keywords Tempered fractional calculus · Tempered-WSGD operator · Superconvergent · Stability and convergence Mathematics Subject Classifications (2010) 26A33 · 34A08 · 65M06 · 65M12
1 Introduction The probability density function of L´evy flights [19, 22] has a characteristic funcα tion e−Dα |k| t (0 < α < 2) of stretched Gaussian form, causing the asymptotic decay as |x|−1−α . It produces that the second moment diverges, i.e., x 2 (t) = ∞. The divergent second moments may not be feasible for some even non-Brownian physical processes of practical interest which take place in bounded domains and involve observables with finite moments. To overcome the divergence of the variance, many techniques are adopted. By simply discarding the very large jumps, Mantegna and Stanley [20] introduce the truncated L´evy flights and show that the obtained stochastic process ultraslowly converges to a Gaussian. From the point of view of an experimental study, because of the limited time, no expected Gaussian behavior can be observed. Two other modifications to achieve finite second moments are proposed by Sokolov et al. [29], who add a high-order power-law factor,
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