A fourth-order approximation of fractional derivatives with its applications☆
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 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 order left and right RL fractional derivatives of the function on are defined as (1) left RL fractional derivative: (2) right RL fractional derivative: If , then and .
Let and denote where 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 equation where and are the left and right RL fractional derivatives with , respectively. The diffusion coefficients and are nonnegative constants with . If , then , and if
Derivation of the difference scheme
In this section, we consider the following two-sided space fractional diffusion equation in two dimensions where , , and , are RL fractional derivatives with , respectively. The diffusion coefficients and , are nonnegative constants with . And if ;
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.
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