High order schemes for the tempered fractional diffusion equations
Abstract
Lévy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering Lévy measure of Lévy flights leads to the tempered stable Lévy processes which combine both the stable and Gaussian trends; and the very large jumps are unlikely and all their moments exist. The probability density functions of the tempered stable Lévy 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 Grünwald difference (temperedWSGD) operators, in space are obtained by using the properties of the tempered fractional calculus and weighting and shifting their first order Grünwald type difference approximations. And the CrankNicolson 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.
¡¡¡¡
Mathematics Subject Classification (2010): 26A33, 65M12, 65M06
Key words: tempered fractional calculus, temperedWSGD operator, superconvergent, stability and convergence
1 Introduction
The probability density function of Lévy flights [20, 23] has a characteristic function of stretched Gaussian form, causing the asymptotic decay as . It produces that the second moment diverges, i.e., . The divergent second moments may not be feasible for some even nonBrownian 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 [21] introduce the truncated Lévy 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 [32], who add a highorder powerlaw factor, and Chechkin et al. [9], who add a nonlinear friction term. Exponentially tempering the probability of large jumps of Lévy flights, i.e., making the Lévy density decay as with , to get finite moments seems to be the most popular one; and the corresponding tempered fractional differential equations are derived [1, 4, 5, 6]. In fact, the powerlaw waiting time can also be exponentially tempered [23]. This paper focused on providing high order schemes for numerically solving tempered fractional diffusion equations, which involve tempered fractional derivatives. In fact, the tempered fractional integral has a long history. Buschman’s earlier work [2] reports the fractional integration with weak singular and exponential kernels; for more detailed discussions, see Srivastava and Buschman’s book [16] and the references therein. The definitions of tempered fractional calculus are much similar to the ones of fractional substantial calculus [3]; but they are introduced from completely different physical backgrounds; e.g., fractional substantial calculus is used to characterize the functional distribution of anomalous diffusion [3]. Mathematically the fractional substantial calculus is timespace coupled operator but the tempered fractional calculus is not; numerically the fractional substantial calculus is discretized in the time direction [10], but here the tempered fractional calculus is treated as space operator.
Tempered fractional calculus is the generalization of fractional calculus, or fractional calculus is the special/limiting case of tempered fractional calculus. Some important progresses have been made for numerically solving the fractional partial differential equations (PDEs), e.g., the finite difference methods are used to simulate the space fractional advection diffusion equations [17, 22, 18]. Recently, it seems that more efforts of the researchers are put on the high order schemes and fast algorithms. Based on the Toeplitzlike structure of the matrix corresponding the finite difference methods of fractional PDEs, Wang et al [36] numerically solve the fractional diffusion equations with the computational cost. Later, Pang and Sun [27] propose a multigrid method to solve the discretized system of the fractional diffusion equation. By introducing the linear spline approximation, Sousa and Li present a second order discretization for the RiemannLiouville fractional derivatives, and establish an unconditionally stable weighted finite difference method for the onedimensional fractional diffusion equation in [33]. Ortigueira [26] gives the “fractional centred derivative” to approximate the Riesz fractional derivative with second order accuracy; and this method is used by Çelik and Duman in [7] to approximate fractional diffusion equation with the Riesz fractional derivative in a finite domain. More recently, by weighting and shifting the Grünwald discretizations, Tian et al [34] propose a class of second order difference approximations, called WSGD operators, to the RiemannLiouville fractional derivatives.
So far, there are limited works addressing the finite difference schemes for the tempered fractional diffusion equations. Baeumera and Meerschaert [1] provide finite difference and particle tracking methods for solving the tempered fractional diffusion equation with the second order accuracy. The stability and convergence of the provided schemes are discussed. Cartea and delCastilloNegrete [4] derive a general finite difference scheme to numerically solve a BlackMertonScholes model with tempered fractional derivatives. Recently, Marom and Momoniat [19] compare the numerical solutions of three kinds of fractional BlackMertonScholes equations with tempered fractional derivatives. And the stability and convergence of the presented schemes are not given. To the best of our knowledge, there is no published work to the high order difference schemes for the tempered fractional diffusion equation. In this paper, with the similar method presented in [22, 1], we first propose the first order shifted Grünwald type approximation for the tempered fractional calculus; then motivated by the idea in [34], we design a series of high order schemes, called the temperedWSGD operators, by weighting and shifting the first order Grünwald type approximations to the tempered fractional calculus. The obtained high order schemes are applied to solve the tempered fractional diffusion equation and the CrankNicolson discretization is used in the time direction. The unconditionally numerical stability and convergence are detailedly discussed; and the corresponding numerical experiments are carried out to illustrate the effectiveness of the schemes.
The remainder of the paper is organized as follows. In Sec. 2, we introduce the definitions of the tempered fractional calculus and derive their first order shifted Grünwald type approximations and the high order discretizations, the temperedWSGD operators. In Sec. 3, the tempered fractional diffusion equation is numerically solved by using the temperedWSGD operators to approximate the space derivative and the CrankNicolson discretization to the time derivative; and the numerical stability and convergence are discussed. The effectiveness and convergence orders of the presented schemes are numerically verified in Sec. 4. And the concluding remarks are given in the last section.
2 Definitions of the tempered fractional calculus and the derivation of the temperedWSGD operators
We first introduce the definitions of the tempered fractional integral and derivative then focus on deriving their high order discretizations, the temperedWSGD operators.
2.1 Definitions and Fourier transforms of the tempered fractional calculus
We introduce the definitions of the tempered fractional calculus and perform their Fourier transforms.
Definition 1 ([2, 5]).
Let be piecewise continuous on or corresponding to the right integral and integrable on any finite subinterval of or corresponding to the right integral, , . Then

the left RiemannLiouville tempered fractional integral of order is defined to be

the right RiemannLiouville tempered fractional integral of order is defined to be
Definition 2 ([13, 15, 28]).
For , let be times continuously differentiable on or corresponding to the right derivative and its times derivative be integrable on any subinterval of or corresponding to the right derivative. Then

the left RiemannLiouville fractional derivative:

the right RiemannLiouville fractional derivative:
Definition 3 ([1, 5]).
For , let be times continuously differentiable on or corresponding to the right derivative and its times derivative be integrable on any subinterval of or corresponding to the right derivative, . Then

the left RiemannLiouville tempered fractional derivative:

the right RiemannLiouville tempered fractional derivative:
If , then the left and right RiemannLiouville tempered fractional derivatives and reduce to the left and right RiemannLiouville fractional derivatives and defined in Definition 2.
Definition 4.
Remark 1.
In the above definitions, the ‘a’ can be extended to ‘’ and ‘b’ to ‘’. In the following analysis, we assume that is defined on and whenever necessary can be smoothly zero extended to or or even . Then ; ; ; and .
Lemma 1 ([2, 16, 1]).
Let and its times derivative belong to , . Then the Fourier transforms of the left and right RiemannLiouville tempered fractional integrals are
(2.3) 
and
(2.4) 
and the Fourier transforms of the left and right RiemannLiouville tempered fractional derivatives are
(2.5) 
and
(2.6) 
and the Fourier transforms of the variants of the left and right RiemannLiouville tempered fractional derivatives give
(2.7) 
and
(2.8) 
where the Fourier transform of is defined by
2.2 Discretizations of the tempered fractional calculus
In this subsection, we derive the Grünwald type discretizations for the tempered fractional calculus. The standard Grünwald discretization generally yields an unstable finite difference scheme when it is used to solve the time dependent fractional PDEs [22]. To remedy this defect, Meerschaert et al introduce a shifted Grünwald formula. The similar numerical unstability also happens for the time dependent tempered fractional PDEs; so the shift for the Grünwald type discretizations of the tempered fractional derivative is also necessary.
Lemma 2.
Let , and its Fourier transform belong to ; and . Defining the shifted Grünwald type difference operator
(2.9) 
then
(2.10) 
where denotes the normalized Grünwald weights.
Remark 3.
The point is the superconvergent point of the approximation to , i.e., with (the deriving process is similar to the one given in [25]).
Remark 4.
Proof.
The proof is similar to the one given in [1]. Taking Fourier transform on both sides of (2.9), we obtain
(2.12)  
where
(2.13) 
Denoting
from (2.12) and (2.5) there exists
With the condition , and using the RiemannLebesgue Lemma, it yields
where the property of the Fourier transforms for the left RiemannLiouville tempered fractional derivatives (2.5) is used. ∎
Lemma 3.
Let , and its Fourier transform belong to ; and . Define the tempered shifted Grünwald type difference operator
(2.14) 
Then
(2.15) 
Remark 5.
The point is the superconvergent point of the approximation to , i.e., with (the deriving process is similar to the one given in [25]).
Remark 6.
Proof.
The approximation accuracy of the classic difference operator can be improved by adding the band of discretization stencils [11]. And then the computational cost increases accordingly. However, because of the nonlocal property of the fractional operator, even for the first order discretizations, the stencil covers the whole interval. Without introducing new computational cost, we can improve the approximation accuracy of the discretized fractional operators by modifying the Grünwald type weights. The improved discretized tempered fractional operators are called tempered weighted and shifted Grünwald difference (temperedWSGD) operators.
Theorem 4.
Let , and its Fourier transform belong to ; and define the left temperedWSGD operator by
(2.17) 
where and are determined by (2.21)(2.24). Then, for any integer , there exists
(2.18) 
uniformly for .
Let , and its Fourier transform belong to ; and define the right temperedWSGD operator by
(2.19) 
where and are determined by (2.21)(2.24). Then, for any integer , there is
(2.20) 
uniformly for .

For , are real numbers and satisfy the linear system
(2.21) 
For , are real numbers and satisfy
(2.22) 
For , are real numbers and the following hold
(2.23) 
For , are real numbers and the following hold
(2.24)
Proof.
The standard Fourier transforms are again used here. Performing the Fourier transform on the left hand of (2.17), we obtain
(2.25)  
where or , By a simple Taylor’s expansion, we get
(2.26)  
Denoting , in view of (2.26), (2.5), and (2.21)(2.24), we have
(2.27) 
Due to , there exists
By the similar arguments we can prove (2.20). ∎
Remark 7.
Remark 8.
If , and its Fourier transform belong to ; and . Defining the shifted Grünwald type difference operator
(2.30) 
then
(2.31) 
where denotes the normalized Grünwald weights.
If , and its Fourier transform belong to ; and . Define the shifted Grünwald type difference operator
(2.32) 
Then
(2.33) 
Remark 9.
To get the discretizations, including the first and high orders, of the left and right RiemannLiouville tempered fractional integrals of order :