Elsevier

Journal of Computational Physics

Volume 281, 15 January 2015, Pages 787-805
Journal of Computational Physics

A fourth-order approximation of fractional derivatives with its applications

https://doi.org/10.1016/j.jcp.2014.10.053Get rights and content

Abstract

A new fourth-order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted Grünwald formulae combining the compact technique. The properties of proposed fractional difference quotient operator are presented and proved. Then the new approximation formula is applied to solve the space fractional diffusion equations. By the energy method, the proposed quasi-compact difference scheme is proved to be unconditionally stable and convergent in L2 norm for both 1D and 2D cases. Several numerical examples are given to confirm the theoretical results.

Introduction

During the past decades, fractional calculus has been playing more and more important roles in many fields. The space fractional diffusion equation is one of the popular mathematical models of fluid flow in porous materials, anomalous diffusion transport, chemistry, etc. (see [11], [20], [23], [28]). Though there are some analytical methods like the Fourier transform method, the Laplace transform method, the Mellin transform method, and the Green function method, it is still difficult to evaluate the fractional derivative for most functions, and the exact solution of the fractional diffusion equation is hardly to be given either. Hence, it is essential to develop the efficient numerical methods.

Due to the equivalence of the Riemann–Liouville (RL) derivative and Grünwald Letnikov (GL) derivative under smooth assumptions imposed on the initial value [23], the GL definition is commonly used to approximate the RL derivative with first order. However, when applying the approximation to the space fractional partial differential equation, it leads to an unstable scheme. To overcome this difficulty, an integer shifted (right shift for left side derivative or left shift for right side derivative) Grünwald formula was proposed by Meerschaert and Tadjeran [15] to approximate the space fractional derivative, and successfully applied to solve the advection–dispersion equations. Based on the shifted Grünwald formula, Meerschaert et al. did a series of work for numerically solving the space fractional diffusion equations [14], [16], [34], [35].

Unlike solving the classical partial differential equations by the finite difference method, which leads to a sparse matrix after discretization, the corresponding finite difference equation of the space fractional diffusion equations leads to a full matrix. To increase the efficiency of calculation and solve the high-dimensional problems, many researchers devoted to developing the high-order methods for space fractional diffusion equations. For the time fractional differential equations, the high-order approximations of fractional derivatives have been considered by many authors and some numerical methods and analysis have been proposed, see [5], [6], [43], [44], [45]. For the space fractional differential equations, recently, Tadjeran et al. [35] presented the asymptotic expansion for the error of the shifted Grünwald finite difference formula and studied the second-order convergence of the extrapolation method. Sousa [32] proposed a second-order accuracy scheme to approximate the spatial fractional Caputo derivatives via splines method. However, the author did not give stability and convergence analysis. The main reason is that the matrix generated by spline method is not equipped with special structure, for instance, the strict diagonal dominant, or Toeplitz's like, which will add extra computational cost. Balance between accuracy and computational cost should be taken into consideration. To our knowledge, another version of shifted formula, named fractional shifted Grünwald formula, was proposed by Oldham and Spanier in [20], and further investigated by Tuan and Gorenflo [37]. As a generalization of the symmetric difference operator, the new formula enjoys superior convergence properties and can be regarded as a modified Grünwald definition. Since this formula needs the evaluation at points other than the known values at the equi-spaced grid points, other numerical approximation technique like the Lagrange three-point interpolation is employed in practice computation so as to approximate RL fractional derivatives with the second-order accuracy. When applied to solve the space fractional equations, this formulae will bring a misalignment between the intermediate points and the equi-spaced grid points. To avoid this misalignment, Nasir et al. [18] shifted the point of the derivative to an intermediate point thus realigned the points of the difference formulae back to the grid points and also obtained a second-order scheme. More recently, Deng's group [12], [36] constructed a class of second-order finite difference scheme, termed weighted and shifted Grünwald difference (WSGD) approximations, to the RL space fractional derivatives. Subsequently, adopting the same idea and by compact technique, they attempted to construct higher order scheme, and obtained a third order scheme, named quasi-compact scheme [51]. Based on Lubich's high order operators [13] and allowing to use the points outside of the domain, Deng's group further derived a class of second-, third- and fourth-order difference approximations for the RL space fractional derivatives [3], [4].

Some researchers have paid attention to the matrices generated when solving the space fractional diffusion equations by the finite difference method and found that the matrices have the structure of Toeplitz type. Based on the fact, the fast finite difference solver has been derived, which has significantly reduced storage requirement and computational cost while retaining the same accuracy as the existing numerical methods, see [22], [38], [39].

As a powerful tool in the scientific computing and applications, the finite element method [8], [9], [26], [40], [49], [50] and the spectral method [1] are very essential numerical methods for solving the space fractional diffusion equations. Some other popular numerical techniques, such as the matrix approach [24], [25], [46], the tau approach [27], the variational iteration method [10], [19], the homotopy analysis method [7], [17] etc., have also been extensively investigated.

In this paper, we focus on the high order approximation of the RL fractional derivative and its application on solving the space fractional diffusion equations. Like approximating the integer order derivatives by extrapolation method [31], [33], asymptotic expansions for the truncation errors of the approximation of fractional derivatives are core and play vital roles in constructing the high order approximation of fractional derivatives. The essential idea of the high order approximation is to vanish the low order leading terms in asymptotic expansions for the truncation errors by the weighted average. So this motivates us to focus our attention on the fractional asymptotic expansions for the truncation errors. The shift Grünwald formulae [35] provide us a direction. By carefully weighting the Grünwald approximation formulae with different shifts and combining the compact technique, we propose a new fourth-order approximation for the RL fractional derivatives, which is one of main contributions of this paper. Compared to the existing fourth-order approximations proposed in [3], [4], where the high-order approximations are achieved by weighting and shifting high-order Lubich's operators, the derivation of the our proposed approximation is straightforward and easy due to the fact that we only used the first-order Lubich's operator, i.e., Grünwald difference operator. Unlike using the points beyond the spatial domain as that in [3], [4], we apply the new fourth-order approximation to solve the space fractional diffusion equations. The proposed quasi-compact difference scheme is proved, both for 1D and 2D cases, to be unconditionally stable and have the fourth-order accuracy in spatial direction and second-order accuracy in temporal direction in L2 norm. This is another contribution of our work. And it is should be pointed out that our analysis method is new and novel.

The reminder of this paper is organized as follows. In Section 2, we derive a fourth-order approximation to the RL fractional derivatives. The quasi-compact difference scheme for the 1D space fractional diffusion equations is derived and several properties of the proposed operators are listed and proved in Section 3. The convergence and stability of the proposed scheme is also proved by a prior estimate in this section. In Section 4, we extend the quasi-compact difference scheme to the 2D case and prove its convergence and stability. To show effectiveness of the algorithm, we perform the numerical experiments to verify the theoretical results in Section 5. Finally, we conclude the paper with some remarks in the last section.

Section snippets

A fourth-order approximation of spatial fractional derivatives

We start with the definitions of the RL fractional derivatives.

Define 2.1

(See [23].) The α(n1<α<n) order left and right RL fractional derivatives of the function f(x) on [a,b] are defined as

(1) left RL fractional derivative:Dxαaf(x)=1Γ(nα)dndxnaxf(ξ)(xξ)αn+1dξ;

(2) right RL fractional derivative:Dbαxf(x)=(1)nΓ(nα)dndxnxbf(ξ)(ξx)αn+1dξ; If α=n, then Dxαaf(x)=dndxnf(x) and Dbαxf(x)=(1)ndndxnf(x).

Let fL1(R) and denoteCn+α(R)={f|+(1+|k|)n+α|fˆ(k)|dk<}, where fˆ(k)=eikxdx is the Fourier

One dimensional space fractional diffusion equation

This section mainly focuses on the fourth-order quasi-compact finite difference scheme for the following one dimensional two-sided space fractional diffusion equationut(x,t)=K1+Dxαau(x,t)+K1Dbαxu(x,t)+p(x,t),(x,t)(a,b)×(0,T],u(a,t)=ϕa(t),u(b,t)=ϕb(t),t(0,T],u(x,0)=u0(x),x[a,b], where Dxαa and Dbαx are the left and right RL fractional derivatives with 1<α2, respectively. The diffusion coefficients K1+ and K1 are nonnegative constants with (K1+)2+(K1)20. If K1+0, then ϕa(t)0, and if K1

Derivation of the difference scheme

In this section, we consider the following two-sided space fractional diffusion equation in two dimensionsut(x,y,t)=K1+Dxαau(x,y,t)+K1Dbαxu(x,y,t)+K2+Dyβcu(x,y,t)+K2Ddβyu(x,y,t)+p(x,y,t),(x,y,t)Ω×(0,T],u(x,y,t)=ϕ(x,y,t),(x,y,t)Ω×(0,T],u(x,y,0)=u0(x,y),(x,y)Ω¯, where Ω=(a,b)×(c,d), Dxαa, Dbαx and Dyαc, Ddαy are RL fractional derivatives with 1<α,β2, respectively. The diffusion coefficients Ki+ and Ki, i=1,2 are nonnegative constants with (Ki+)2+(Ki)20. And ϕ(a,y,t)0 if K1+0; ϕ(x,c,t)

Numerical examples

In this section, several numerical examples are presented to show the efficiency and accuracy of the schemes.

Conclusion

In this paper, we have derived a fourth-order approximation to the RL fractional derivative by combining the weighted and shifted Grünwald formulae and the compact technique. The approximation has been successfully applied to solve the one- and two-dimensional space fractional diffusion equations. Unconditional stability and convergence of all proposed fourth-order quasi-compact scheme have been proved by the energy method. The spatial convergence order of our scheme can reach four if the

Acknowledgement

We express our gratitude to the referees for their many valuable suggestions that improved this article.

References (52)

  • E. Sousa

    Numerical approximations for fractional diffusion equations via splines

    Comput. Math. Appl.

    (2011)
  • C. Tadjeran et al.

    A second-order accurate numerical method for the two-dimensional fractional diffusion equation

    J. Comput. Phys.

    (2007)
  • C. Tadjeran et al.

    A second-order accurate numerical approximation for the fractional diffusion equation

    J. Comput. Phys.

    (2006)
  • H. Wang et al.

    A super fast-preconditioned iterative method for steady-state space-fractional diffusion equations

    J. Comput. Phys.

    (2013)
  • Q. Yang et al.

    Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

    Appl. Math. Model.

    (2010)
  • Y.X. Zhang et al.

    Improved matrix transform method for the Riesz space fractional reaction dispersion equation

    J. Comput. Appl. Math.

    (2014)
  • H. Zhang et al.

    Galerkin finite element approximation of symmetric space-fractional partial differential equations

    Appl. Math. Comput.

    (2010)
  • Z.G. Zhao et al.

    Fractional difference/finite element approximations for the time–space fractional telegraph equation

    Appl. Math. Comput.

    (2012)
  • Y.Y. Zheng et al.

    A note on the finite element method for the space fractional advection diffusion equation

    Comput. Math. Appl.

    (2010)
  • M.H. Chen et al.

    Fourth order accurate scheme for the space fractional diffusion equations

    SIAM J. Numer. Anal.

    (2014)
  • M.H. Chen et al.

    Fourth order difference approximations for space Riemann–Liouville derivatives based on weighted and shifted Lubich difference operators

    Commun. Comput. Phys.

    (2014)
  • C. Chen et al.

    Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation

    SIAM J. Sci. Comput.

    (2010)
  • M. Dehghan et al.

    Solving nonlinear fractional partial differential equations using the homotopy analysis method

    Numer. Methods Partial Differ. Equ.

    (2010)
  • V.J. Ervin et al.

    Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation

    SIAM J. Numer. Anal.

    (2007)
  • V.J. Ervin et al.

    Variational formulation for the stationary fractional advection dispersion equation

    Numer. Methods Partial Differ. Equ.

    (2006)
  • A. Kilbas et al.

    Theory and Applications of Fractional Differential Equations

    (2006)
  • Cited by (0)

    The research is supported by National Natural Science Foundation of China (No. 11271068) and by the Fundamental Research Funds for the Central Universities and the Research and Innovation Project for College Graduates of Jiangsu Province (Grant No.: KYLX_0081).

    View full text