A Fast Compact Finite Difference Method for Fractional Cattaneo Equation Based on Caputo–Fabrizio Derivative

Qiao, Haili; Liu, Zhengguang; Cheng, Aijie

doi:https://doi.org/10.1155/2020/3842946

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2020 | Article ID 3842946 | https://doi.org/10.1155/2020/3842946

A Fast Compact Finite Difference Method for Fractional Cattaneo Equation Based on Caputo–Fabrizio Derivative

Haili Qiao,¹Zhengguang Liu,²and Aijie Cheng¹

Academic Editor: Eric Feulvarch

Received15 Oct 2019

Accepted16 Jan 2020

Published19 Mar 2020

Abstract

The Cattaneo equations with Caputo–Fabrizio fractional derivative are investigated. A compact finite difference scheme of Crank–Nicolson type is presented and analyzed, which is proved to have temporal accuracy of second order and spatial accuracy of fourth order. Since this derivative is defined with an integral over the whole passed time, conventional direct solvers generally take computational complexity of and require memory of , with M and N the number of space steps and time steps, respectively. We develop a fast evaluation procedure for the Caputo–Fabrizio fractional derivative, by which the computational cost is reduced to operations; meanwhile, only memory is required. In the end, several numerical experiments are carried out to verify the theoretical results and show the applicability of the fast compact difference procedure.

1. Introduction

Fractional diffusion equations have become a strong and forceful tool to describe the phenomenon of anomalous diffusion, and more research works have been obtained in the last decades [1–6]. However, since the fractional derivative is nonlocal and has weak singularity, it is impossible to solve fractional diffusion equations analytically in most cases. Instead, seeking numerical solutions is becoming an indispensable tool for research work about fractional equations.

Different from the traditional derivative of the integer order, the fractional derivative depends on the total information in the correlative region, and this is the so-called nonlocal properties. Just because of this, it consumes computational time extremely to solve fractional equations. We hope to develop effective numerical schemes, which not only have better stability and higher accuracy but also require less storage memory and save computational cost.

About stability and convergence analysis of the numerical schemes for fractional equations, the readers can refers to [7, 8] for spatial fractional order equation, [9–18] for temporal fractional diffusion equations, and [19–22] for space-time-fractional equations. About the complexity, i.e., storage requirement and computation cost of an algorithm, researchers devote themselves to reduce storage requirement and computational time by analyzing the particular structure of coefficient matrices arising from the discretization system or reutilizing the intermediate data skillfully. We call these algorithms fast methods, including fast finite difference methods [23–28], fast finite element methods [29], and fast collocation methods [30, 31]. A fast method for Caputo fractional derivatives is proposed [32, 33]. Lu et al. [34] presented a fast method of approximate inversion for triangular Toeplitz tridiagonal block matrix, which is successfully applied to the fractional diffusion equations. Comparatively, there is less research work about the fast method for temporal fractional derivative than that for spatial fractional operators.

A time-fractional Cattaneo equation is considered with the following form:where ; for one-dimensional case, and for two-dimensional case; is the source term; are the prescribed functions for initial conditions; and is a new Caputo fractional derivative without singular kernel, which is defined in the next section.

Our purpose is to establish a fast finite difference scheme of high order for this equation. We will extract the recursive relation between the time step and the k time step of the finite difference solution. The computational work is significantly reduced from to , and the memory requirement from to , where M and N are the total numbers of points for space steps and time steps, respectively. For improving the accuracy, a compact finite difference scheme is established. Theoretical analysis shows that the fast compact difference scheme has spatial accuracy of fourth order and temporal accuracy of second order. Several numerical experiments are implemented, which verify the effectiveness, applicability, and convergence rate of the proposed scheme.

This paper is organized as follows: some definitions and notations are prepared in Section 2. The compact finite difference scheme is described and then the stability and convergence rates are rigorously analyzed for the scheme in Section 3. The compact finite difference scheme is extended to the case of two space dimensions in Section 4. Fast evaluation and efficient storage are established skillfully in Section 5. Some numerical experiments are carried out in Section 6. In the end, we summarize the major contribution of this paper in Section 7.

2. Some Notations and Definitions

We provide some definitions which will be used in the following analysis.

First, let us recall the usual Caputo fractional derivative of order α with respect to time variable t, which is given by

By replacing the kernel function with the exponential function and with , Caputo and Fabrizio [35] proposed the following definition of fractional time derivative.

Definition 1 (see [35]). Let , , , then the new Caputo derivative of the fractional order is defined aswhere is a normalization function satisfying . When the function u does not belong to , then this derivative can be reformulated as

Definition 2 (see [36]). The above new Caputo derivative of order can also be reformulated as

Definition 3 (see [35, 36]). Let , if , and , the fractional time derivative of order is defined byParticularly, for , we have

Remark 1. The Caputo–Fabrizio (CF) operator was proposed with a nonsingular kernel for describing material heterogeneities that do not exhibit power-law behavior [35].

Remark 2. An open discussion is ongoing about the mathematical construction of the CF operator. Ortigueira and Tenreiro Machado [37] indicated that the CF fractional derivative is neither a fractional operator nor a derivative operator, the authors of [38, 39] showed that this operator cannot describe dynamic memory, and Giusti [40] indicated that this operator can be expressed as an infinite linear combination of Riemann–Liouville integrals with integer powers. As responses to these criticisms, Atangana and Gómez-Aguilar [41] pointed out the need to account for a fractional calculus approach without an imposed index law and with nonsingular kernels. Furthermore, Hristov [42] indicated that the CF operator is not applicable for explaining the physical examples in [37, 40]; instead, he suggested that the CF operator can be used for the analysis of materials that do not follow a power-law behavior. The authors of [43] believe that models with CF operators produce a better representation of physical behaviors than do integer-order models, providing a way to model the intermediate (between elliptic and parabolic or between parabolic and hyperbolic) behaviors.
To obtain the accuracy of the fourth order in spatial directions, the following lemma is necessary.

Lemma 1 (see [44]). Denote . If , then, it holds that

3. Compact Finite Difference Scheme for One-Dimensional Fractional Cattaneo Equation

In order to construct the finite difference schemes, the interval is divided into subintervals with , and is discretized with , where and are the spatial grid size and temporal step size, respectively. Denote , , then becomes a discretization of the practical computational domain . The values of the function u at the grid points are denoted as , and the approximate solution at the point is denoted as .

Denote . We also introduce the following notations for any mesh function :and define the average operator

It is easy to see thatwhere I is the identical operator. We also denote for vector , and .

For any two gird functions , the discrete inner products and norms are defined as

By summation by parts, it is easy to see that

For the average operator , define

Additionally, let be the space of grid function defined on . For any function , a difference operator is introduced as follows:

3.1. Compact Finite Difference Scheme

We will consider the time-fractional Cattaneo equation equipped with the Caputo–Fabrizio derivative. Vivas-Cruz et al. [43] gave the theoretical analysis of a model of fluid flow in a reservoir with the Caputo–Fabrizio operator. They proved that this model cannot be used to describe nonlocal processes since it can be represented as an equivalent differential equation with a finite number of integer-order derivatives.

The finite difference methods usually lead to stencils through the whole history passed by the solution which consume too much computational work. In this paper, we will establish a high-order finite difference scheme and propose a procedure to reduce the computational cost. In [43], the authors proposed a recurrence formula of discretized CF operator and obtained an algorithm which can be considered a stencil with a one-step expression without the need of integrals over the whole history. It seems that the procedure in our paper and the algorithm in [43] are different in approach but equally satisfactory in result.

For obtaining effective approximation with high order, we introduce the numerical discretization for the fractional Cattaneo equation by means of compact finite difference methods.

At the node , the differential equation takes the following form:

The approximation of the fractional derivative is given by [45]with truncation error , and

Furthermore, by Lemma 1 and equation (15), the space and time derivative are approximated by

Substituting (17) and (19)∼(20) into (16), we getand there exists a constant C, depending on the function u and its derivatives such that

By the initial and boundary value conditions, we have

A compact finite difference scheme can be established by omitting the truncation term and replacing the exact solution in equation (21) with numerical solution :

3.2. Stability Analysis and Optimal Error Estimates

3.2.1. Stability Analysis

The following Lemma about is useful for the analysis of stability.

Lemma 2 (see [45]). For the definition of , and , , are held.

Multiplying on both sides of equation (24) and summing up with respect to i from 1 to , the following equation is obtained:

Observing equation (13), we have

By the triangle inequality and Lemma 2, we obtain

Combining equation (25) with (26)∼(27), we get

Let

Summing up with respect to k from 0 to leads to

The initial condition implies that , and then

Theorem 1. For scheme (24), we have the following stable conclusion:

3.2.2. Optimal Error Estimates

Combining equations (21) and (23) with (24), we get an error equation as follows:where and , .

Multiplying on both sides of equation (33) and summing up with respect to i from 1 to , we get

By the triangle inequality and Lemma 2, we obtain

Combining equation (34) with (35), we have

By the definition of Q in stability analysis, the inequality (36) can be rearranged as

Summing up with respect to k from 0 to , we get

Observing that the initial error implies . Then, we have

Theorem 2. Suppose that the exact solution of the fractional Cattaneo equation is smooth sufficiently, then there exists a positive constant C, independent of , and such thatwhere .

4. Compact Finite Difference Scheme in Two Dimensions

In this section, the following fractional Cattaneo equation in two dimensions will be considered:where , is the source term, are the given functions, and is defined by the new Caputo fractional derivative without singular kernel.

In order to construct the finite difference schemes, the rectangle is discretized with and ., and the time interval is discretized with , where , , and are the spatial grid and temporal step sizes, respectively. Denote and , then is a discretization of the physical computational domain . denotes the values of function u at the grid points, and denotes the values of the numerical solution at the point .

Denote and . For any mesh function , we use the following notations:and define the average operator

It is clear thatWe also denote . It is easy to see that .

For any gird function , the discrete inner product and norms are defined as follows:

For the average operator , define

4.1. Compact Finite Difference Scheme

At the node , the differential equation is rewritten as

For the approximation of the time-fractional derivative, we have the following approximation [45]:where the truncation error and

Furthermore, we also have

Substituting (48) and (50)∼(52) into (47) leads toand there exists a constant C, depending on the function u and its derivatives such that

By the initial and boundary conditions, we have

Omitting the truncation error and replacing the true solution with numerical solution , a compact finite difference scheme can be obtained as follows:

4.2. Stability Analysis and Optimal Error Estimates

4.2.1. Stability Analysis

Definition 4 (see [46]). For any gird function , define the normThe lemmas below is useful in the subsequent analysis of stability.

Lemma 3 (see [46]). For any gird function , the following equation is held:

Lemma 4 (see [46]). For any gird function , the following equation is held:where .

Multiplying on both sides of equation (56) and summing up w.r.t. from 1 to and from 1 to , respectively, the following equation is obtained:

Observing Lemma 4, we have

By the triangle inequality and Lemma 2, we obtain

Combining equation (60) with (61)∼(63),we get

Let

Summing up with respect to k from 0 to , we get

Noting that , we have . It follows that

Theorem 3. For the compact finite difference scheme (56), the following stability inequality holds:

Similar to the stability, the convergence can also be analyzed.

Theorem 4. Suppose that the exact solution of the fractional Cattaneo equation is sufficiently smooth, then there exists a positive constant C independent of , and such thatwhere and .

5. Efficient Storage and Fast Evaluation of the Caputo–Fabrizio Fractional Derivative

Since time-fractional derivative operator is nonlocal, the traditional direct method for numerically solving the fractional Cattaneo equations generally requires total memory units and computational complexity, where N and M are the total number of time steps and space steps, respectively.

In this section, we develop a fast solution method for the finite difference scheme of the time-fractional Cattaneo equation.

Letthen

So we have

Remark. We find that at the k-th level, only operations are needed to compute the k-th level since the -th level is known at that point. Thus, the total operations are reduced from to , and the memory requirement decreases from to . We conclude that that this fast method significantly reduces the total computational cost from to and the memory requirement from to .

6. Numerical Experiments

In this section, we carry out several numerical experiments to check the effectiveness of the proposed scheme. The convergence rate and CPU consumption are all compared in the simulations. We take the space-time domain for one-dimensional case and for two-dimensional case. These simulations are implemented in Matlab, and the numerical experiments are run on a computer with 4 GB memory. The time-fractional Cattaneo equation of the following forms is considered.

Example 1. We provide the exact solution , and for different α, we have different accordingly:In Tables 1 and 2, we take and to examine the discrete -norm (-norm) errors and corresponding spatial and temporal convergence rates, respectively. We list the errors and convergence rates (order) of the proposed compact finite difference (CD) scheme, which is almost for different α. Additionally, Table 3 shows the CPU time (CPU) consumed by direct compact (DCD) scheme and fast compact difference (FCD) scheme, respectively. It is obvious that the FCD scheme has a significantly reduced CPU time over the DCD scheme. For instance, when , we choose and and observe that the FCD scheme consumes only 94 seconds, while the DCD scheme consumes 3692 seconds. We can find that the performance of the FCD scheme will be more conspicuous as the time step size decreases.
In Figure 1, we set and and change the total number of time steps N to plot out the CPU time (in seconds) of the FCD scheme and DCD scheme. We can observe that the CPU time increases almost linearly with respect to N for the FCD scheme, while the DCD scheme scales like .
Tables 4 and 5 show the discrete errors and convergence rates of the compact finite difference scheme for Example 1. The space rates are almost for fixed , and the time convergence rates are always for fixed . We can conclude that the numerical convergence rates of our scheme approach almost to .

(a)

(b)

Example 2. The example is described byNote that the exact solution of the above problem is , where and γ is a constant, and for different α, we have different accordingly.
We apply the fast compact difference scheme to discretize the equation. In Figure 2, we set and plot exact and numerical solutions at time for Example 2 with different . For , we also plot exact and numerical solutions with the different γ in Figure 3. In Figure 4, for and , we vary the total number of time steps N to plot out the CPU time (in seconds) of the FCD scheme and DCD scheme. The numerical experiments verified our theoretical results. In Table 6, by equating and fixing , we compute the discrete error and convergence rates with different fractional derivative orders α and different γ. It shows that the compact finite difference scheme has space accuracy of fourth order and temporal accuracy of second order. We set and fix , and the discrete error and convergence rates with different α and are displayed in Table 7. These numerical convergence rates are almost approaching as Example 1.

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(c)

(d)

(a)

(b)

Example 3. If the exact solution is given by , we have different for different α accordingly:In Figure 5, and are fixed, and the total number of time steps N vary to plot out the CPU time (in seconds) of the FCD procedure and DCD procedure, and it presents an approximately linear computation complexity for FCD procedure. We set in Table 8, and in Table 9, the discrete error, discrete error, and convergence rates with different derivative orders α are presented. The fourth-order space accuracy and second-order temporal accuracy can be observed clearly.

(a)

(b)

7. Conclusion

In this paper, we develop and analyze a fast compact finite difference procedure for the Cattaneo equation equipped with time-fractional derivative without singular kernel. The time-fractional derivative is of Caputo–Fabrizio type with the order of . Compact difference discretization is applied to obtain a high-order approximation for spatial derivatives of integer order in the partial differential equation, and the Caputo–Fabrizio fractional derivative is discretized by means of Crank–Nicolson approximation. It has been proved that the proposed compact finite difference scheme has spatial accuracy of fourth order and temporal accuracy of second order. Since the fractional derivatives are history dependent and nonlocal, huge memory for storage and computational cost are required. This means extreme difficulty especially for a long-time simulation. Enlightened by the treatment for Caputo fractional derivative [32], we develop an effective fast evaluation procedure for the new Caputo–Fabrizio fractional derivative for the compact finite difference scheme. Several numerical experiments have been carried out to show the convergence orders and applicability of the scheme.

Inspired by the work [43], the topic about modelling and numerical solutions of porous media flow equipped with fractional derivatives is very interesting and challenging and will be our main research direction in the future.

Data Availability

All data generated or analyzed during this study are included in this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant nos. 91630207 and 11971272.

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