A High Accuracy Numerical Method Based on Interpolation Technique for Time-Fractional Advection-Diffusion Equations

Chen, Yan; Zhang, Xindong

doi:https://doi.org/10.1155/2024/2740720

Journal of Mathematics

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 2740720 | https://doi.org/10.1155/2024/2740720

A High Accuracy Numerical Method Based on Interpolation Technique for Time-Fractional Advection-Diffusion Equations

Yan Chen¹and Xindong Zhang¹

Academic Editor: Xian-Ming Gu

Received19 Sept 2023

Revised18 Dec 2023

Accepted31 Dec 2023

Published29 Jan 2024

Abstract

In this paper, the time-fractional advection-diffusion equation (TFADE) is solved by the barycentric Lagrange interpolation collocation method (BLICM). In order to approximate the fractional derivative under the definition of Caputo, BLICM is used to approximate the unknown function. We obtain the discrete scheme of the equation by combining BLICM with the Gauss-Legendre quadrature rule. The convergence rate for the TFADE equation of the BLICM is derived, and the accuracy of the discrete scheme can be improved by modifying the number of Gaussian nodes. To illustrate the efficiency and accuracy of the present method, a few numerical examples are presented and compared with the other existing methods.

1. Introduction

The fractional partial differential equation (FPDE) has become more widely used in recent decades and has become an important tool in many areas [1–7]. There are numerous numerical schemes for FPDE, for instance, the finite difference method [8–11], finite element method [12, 13], and the spectral method [14, 15], among others. Compared with other numerical methods, BLICM is a high-precision, high-efficiency method of numerical computation, which does not require a dense computational mesh, and its computational program is very easy to write. In recent years, BLICM has been applied in many fields. Some researches on the barycentric interpolation method can be found in [16–21], among others.

The standard advection-diffusion equation describes the changes in a concentration profile as a result of simultaneous diffusion and advection. If we replace the first-order time derivative by the fractional one, then we can obtain the time-fractional advection-diffusion equation. The fractional advection-diffusion equation (FADE) is presented as a useful approach for the description of transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns [22]. In recent years, several methods have been proposed for solving FADE. Liu et al. [23] considered the space-time fractional advection-diffusion equation by the difference method. Jiang et al. [24] obtained the analytical solutions for the multiterm time-space Caputo–Riesz fractional advection-diffusion equations on a finite domain. Wei et al. [25] studied the time-fractional advection-diffusion equation by the local discontinuous Galerkin method. Tayebi et al. [26] presented a meshless method for solving a two-dimensional variable-order time-fractional advection-diffusion equation. Aghdam et al. [27] developed a numerical method for solving the space-time fractional advection-diffusion equation.

To our knowledge, there are no relevant results using BLICM for solving the TFADE. Based on the aforementioned reasons, the main motivation of this paper is to introduce the BLICM for TFADE. In this paper, we investigate the following TFADE:where , and are the diffusion coefficient and advection coefficient, respectively, is a known function, is the unknown function, and , , and are given continuous functions. For , the Caputo fractional derivative is defined as follows:which is one of the common fractional derivatives and has been applied in many areas. Properties and more details about Caputo’s fractional derivative can be sought out in [28–30].

The remainder of the paper is organised as follows. In Section 2, the basic form of the BLICM is elaborated, and based on this form, an integer-order differential form of BLICM is proposed. Then, a numerical algorithm for the Caputo fractional derivative is proposed in conjunction with the Gauss-Legendre quadrature rule. Finally, the discrete scheme for the TFADE is obtained at the end of this section. The error of the numerical algorithm is theoretically analyzed in Section 3. We provide some numerical examples in Section 4 to demonstrate the effectiveness of this scheme. Lastly, we give the summary in Section 5.

2. A High-Precision Numerical Algorithm for TFADE

2.1. BLICM with the Second Class of Chebyshev Nodes

Barycentric Lagrange interpolation (BLI) is an improvement of Lagrange interpolation. In 2004, Berrut and Trefethen proposed BLI [31].

Let be different interpolation nodes. The value of at point is denoted by ; the interpolation polynomial can be written as the famous Lagrange interpolation polynomial.where

Let and the barycentric weight be defined as . This implies , and can be writen as follows:

By equations (3) and (5), then we can obtain:

For a fixed point , we can deduce that

By equations (6) and (7), we can get the barycentric Lagrange interpolation polynomial (BLIP) denoted by such thatwhere

Similarly, for different interpolation nodes , we can get the BLIP of denoted by such thatwhere andwith .

BLI has good numerical stability as the nodes distribution density is proportional to the function . As mentioned in [31], the simplest node distribution that satisfies the above condition is the Chebyshev node family. In this paper, we choose the second class of Chebyshev nodesand the BLI weight of these nodes are as follows:

The value of at point is , for variable , similar to equation (8), we can get the following BLIP:in which the value of the unknown function on nodes is denoted as . Then, the BLIP of can be represented as follows:

Combining equations (14) with (15), the BLIP of at nodes can be obtained as follows:

2.2. Differential Matrix of BLI

By equations (8) and (10), we get

By equation (16), we can obtain the following ones:

It follows by equation (9) that

For above equation, if multiply on both sides simultaneously, then we can get that

Letby equation (21), the following equations can be obtained:

Fix a node , we observe that

In order to get the differential matrix of BLI, we present some results below which will be applied in our arguments. We will discuss in two cases.

Case 1. .
Since , we can achieve

Case 2. .
It is pointed out that is also barycentic Lagrange interpolation basis function satisfying the property , thus we can get thatwhere denotes the -order derivative of function .
Thus, the BLIP for the -order derivative of on nodes can be obtained as follows:Similarly, we can obtain the BLIP for the -order derivative of on nodes , that isFinally, the BLIP for the -order derivative of on nodes can be obtained in the following form:where the -order differential matrices of BLI have the following forms, which will be needed in Subsection 2.4.

2.3. Calculation Scheme of Caputo Fractional Derivative

We ponder the numerical scheme of Caputo fractional derivative in this subsection. By equation (2), we can infer that

In equation (32), if we replace , by the terms of equation (30) and discretize the domain by nodes in space and nodes in time, then the following holds:

By using Gauss-Legendre quadrature rule [32], we can get the following formula:where is integral point, and are integral weight and the number of point for Gauss–Legendre quadrature rule, respectively. Combining above two discrete formats, we can obtain the discrete scheme of Caputo fractional derivative as follows:

2.4. Discrete Scheme of TFADE

In this subsection, the BLICM is used to approximate equation (1). By equations (18), (19), and (35), we present the following equation:

Let equation (36) hold at nodes , then

Let , combining and , then equation (37) can be written as follows:

Writing equation (38) in the matrix form, we get

Taking all values of and , by equation (39), we can be obtained the following matrix form:where

and are identity matrices of order and , respectively, and .

Let and , then the discrete form of equation (1) can be expressed as follows:and the discrete formats of the initial value conditions are as follows:

3. Error Analysis

The error estimates of BLICM based on the second class of Chebyshev nodes are presented in this section. We first give the following definitions and lemmas for future applications in our arguments.

Definition 3. Let be the function space consisting of the interpolated basis functions defined by equation (9), and be the function space consisting of the interpolated basis functions defined by equation (11).

Definition 4. For , define : and : , they are interpolation operators for and , and satisfying the following equation:Similarly, let , we can define : , it satisfies the following equation:

It is obvious that . Let . Since , by the definition of linear operator, we conclude that is a linear operator. Similarly, and are all linear operators.

Definition 5 (see [33]) (Lebesgue constant). .

Lemma 6 (see [34]). Assuming that interpolation nodes in the interval , , then, for all , there exists , related to , such thatwhere is the interpolation polynomial of .

Lemma 7 (see [35]). When the BLICM at the second class of Chebyshev nodes, its Lebesgue constant satisfies the following equation:

Theorem 8. Let , , and let be the exact solution of equation (1) and be the numerical solution of equation (38) at the nodes with and , thenwhere is the maximum one of and .

Proof. Using the triangle inequality, we achieve thatBy Lemma 6, we obtainApplying (see [36]), the above equation enables us to write the following equation:Similarly, we can getSince and are linear operators, combining Definition 5 and equation (52), we haveCombining these with Lemma 7 we deduce thatLet , by equations (51) and (54) we conclude thatThis completes the proof.
Analysis similar to that in the proof of Theorem 8 in [36], we can get the following error estimate of Caputo fractional derivative.

Theorem 9. Let , then the following error estimate for the BLICM of Caputo fractional derivative holds:where is the maximum one of and , and are constants independent of and , and are represented as the lengths of the interval in two dimensions.

4. Numerical Examples

This section demonstrates the superiority of BLICM in solving TFADE through some examples. All numerical results are implemented on the AMD Ryzen 5 5600H Windows 10 system by using MATLAB R2022b. The space-time discrete scheme equation (42) is a system of linear algebraic equations , which can be solved as (“” is the built-in function in MATLAB). By the way, the space-time discrete linear system equation (42) is very related to the so-called all-at-once linear system, which can be solved by the parallel iterative method [37] in order to improve the computational effectiveness. But that is outside the scope of this paper and we shall not pursue that here.

The absolute error and the relative error in all examples are defined as follows:where and denote the exact value and numerical value on , respectively. The convergence order is defined as , where is the current error and is the previous error.

Example 1. Let , , , , , and the forcing function is in equation (1). The exact solution is .
Taking 900 Gaussian nodes and using equation (42) to solve the Example 1. Table 1 shows the relative error and convergence order of Example 1 for and 0.7, respectively. Moreover, the maximum relative error with nodes by our method is about . As and , Figures 1 and 2 show the absolute error under the different types of nodes. From Figures 1 and 2, we can find that the second class of Chebyshev nodes are generally more accurate than equidistant nodes, which shows that Chebyshev nodes are more suitable for BLICM.

(a)

(b)

(a)

(b)

Example 2. Let , , , , , , and the forcing function is in equation (1). The exact solution is .
By equation (42), the results of Example 2 are obtained in Table 2 and Figure 3. Table 2 shows the relative error and convergence order of Example 2 for and 0.7 with . As , , and , the absolute errors of Example 2 are shown in Figure 3. By Figure 3 we can find that the fluctuation of error size and distribution is small when is taken at different values, which shows the excellent stability of BLICM.

(a)

(b)

(c)

(d)

Example 3. Let , , , , , and the forcing function is in equation (1). The exact solution is .
This example can be found in [38]. The results of Example 3 are as follows. Table 3 shows the relative errors for different with . Table 4 reports the relative errors for and compare the present results with the results obtained by the method in [38] (see Example 2 in [38]). We perceive from this table that the results obtained by the proposed method are more accurate than the results in [38]. For different , Figure 4 shows the variability of relative error for Example 3 with different number of Gaussian nodes, it implies that the stability of the relative errors becomes better with the increasing of the number of nodes. For and , the absolute errors for Example 3 are shown in Figure 5 with and . The numerical results of the third example also confirm the theoretical prediction and verify the effectiveness of the proposed method.

(a)

(b)

(a)

(b)

(c)

(d)

Example 4. Let , , , , , , and the forcing function is in equation (1). The exact solution is .
In this example, we want to test the problem of which the exact solution is not smooth enough. The results of Example 4 are shown in Table 5 for different . The main purpose of this example is to verify the effectiveness of the proposed method. We can find that the performance of the proposed method will get worse for given problem, but it is still effective.

5. Conclusion

In this paper, we investigate the numerical algorithm for solving TFADE by using BLICM. Discrete scheme of TFADE is given by combining BLICM with Gauss-Legendre quadrature rule. Theoretical analysis and numerical results show that the discrete scheme constructed in our paper has high numerical convergence speed and accuracy. A comparison of the obtained results with exact solutions and other existing methods reveals that our method is more accurate and efficient for TFADE. The proposed method can be extended to solve problems of integer and noninteger orders in high dimensions. In our future work, we will include the problems of high dimensions and nonlinear fractional partial differential equations.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Xinjiang Uygur Autonomous Region Graduate Student Research Innovation Program (Grant no. XJ2023G231), Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant nos. 2022D01E13 and 2022D01B113), NSFC (Grant no. 11861068), and the Scientific Research Foundation for Outstanding Young Teachers of Xinjiang Normal University (Grant no. XJNU202112).

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Copyright © 2024 Yan Chen and Xindong Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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