Abstract
Using the asymmetric discretization technique, an explicit finite difference scheme is constructed for one-dimensional spatial fractional diffusion equations (FDEs). The spatial fractional derivative is approximated by the weighted and shifted Grünwald difference operator. The scheme can be solved explicitly by calculating unknowns in the different nodal-point sequences at the odd time-step and the even time-step. The uniform stability is proven and the error between the discrete solution and analytical solution is theoretically estimated. Numerical examples are given to verify theoretical analysis.
1. Introduction
Fractional operator has been used to solve a certain class of problems arising from the real world [1–7]. Fractional spatial derivatives are suitable to describe anomalous dispersion and diffusion models. In particular, when the second-order derivative is replaced by the Riemann-Liouville fractional derivative of order in space, the fractional equation represents a superdiffusive flow model.
Up to now, several finite difference methods have been presented for the initial-boundary value problem of the one-dimensional spatial fractional diffusion equations (FDEs) [8–15]. Because of the nonlocal property of fractional differential operators, many numerical methods for FDEs have full coefficient matrices which require storage of and computational cost of for a problem of size , where is the number of discrete points in the spatial coordinate. How to save storage space and reduce the number of calculations is essential. It is well known that implicit scheme is more stable, but more complex in calculation than the explicit scheme.
The main purpose of this paper is to construct a semi-implicit finite difference scheme for solving one-dimensional spatial FDE by combining a second-order weighted and shifted Grünwald difference (WSGD) operator [16, 17] with the asymmetric discretization technique introduced by Saulyev [18]. Formally, the obtained finite difference scheme is implicit; however we can obtain the discrete solution of the scheme explicitly by using different nodal-point stencils at odd time-step and even time-step. Instead of solving the linear algebraic system, we compute the unknowns according to the sequences in left-to-right order at the odd time-step and then compute the unknowns according to the sequences in right-to-left order at the even time-step. We prove that the scheme is uniformly stable when , and the error between the numerical and analytical solutions in discrete norm is of order , where is the order of spatial fractional derivative, and and are the space and time mesh sizes. The error estimate here is better than that in [19], where the error in discrete norm is of order . Some examples are presented to show that the numerical results match our theoretical analysis.
This asymmetric technique has been used to construct parallel algorithms in several research results [20–23]. Within the scope of our knowledge, there are only two papers including this paper and the one Rui and Huang presented [19] to analyze the error for the asymmetric technique and get the error estimates between the discrete and analytical solutions. Former papers presented the stability property, mentioned that the truncation error is of order for parabolic problems [20, 21], or presented the asymmetric technique [22, 23] in numerical calculations.
The rest of the paper is organized as follows. In Section 2 the semi-implicit finite difference scheme is presented, and in Section 3 it is proven that the scheme proposed is unconditionally stable for the problem with the order of fractional derivative belonging to . In Section 4 the error estimation is analyzed, and in Section 5 numerical experiments are carried out to verify our theoretical analysis. Finally some concluding remarks are given in the last section.
Throughout the paper, we use to denote a generic constant, which may have different values in different context.
2. The Semi-Implicit Finite Difference Method for FDES
In this section, combining a second-order WSGD operator for the Riemann-Liouville fractional derivatives with the asymmetric discretization technique, the finite difference scheme is constructed for FDEs.
Consider the following one-dimensional space-fractional diffusion equation:where is the fractional order and [8, 9].
2.1. Second-Order Operator for Riemann-Liouville Fractional Derivatives
Now from the shifted Grünwald finite difference formula, a second-order WSGD operator is introduced for Riemann-Liouville derivatives. First, some preliminary knowledge involving the Riemann-Liouville derivatives is given [24].
Definition 1. The order left and right Riemann-Liouville fractional derivatives of the function on are, respectively, defined as follows:
(a) The left Riemann-Liouville fractional derivative: (b) The right Riemann-Liouville fractional derivative: if , and .
The fractional derivative can be approximated by the standard Grüwald formula or the shifted Grüwald formula [8, 9] as follows:where and the coefficients are defined as
Lemma 2 (see [8, 9, 24]). The coefficients satisfy the following properties for ,
Let and be the time increment and spacial mesh size, respectively. Here and are positive integers. For , and , we denote
Based on the weighted and shifted Grünwald difference (WSGD) operators, the following second-order operator is introduced to approximate the Riemann-Liouville fractional derivative [16] at the grid point . For ,
The simplified form of the approximate formula (11) for the Riemann-Liouville fractional derivative iswhere
From Lemma 2 and some straightforward calculations, we obtain the following properties of the coefficient in (12) [16].
Lemma 3 (see [16]). The coefficients satisfy the following properties for ,
Combining (12) with the implicit Euler discretization, the finite difference scheme for solving equations (1)-(3) is introduced.where .
Now, the following semi-implicit finite difference scheme is constructed using the asymmetric technique [18, 19].
For and , one goal is to find such thatThe initial and boundary conditions are as follows:The scheme is an implicit scheme formally. In order to solve it explicitly, we calculate the unknowns according to the following nodal-point sequences. Here different nodal-point stencils are given at levels and (see Figures 1-2).
The specific calculation processes are as follows. Let From (16),and(1) Solve (19) according to the following sequence from time-step to time-step : (cf. Figure 3).
Once we find for , combining with and , can be obtained explicitly.
(2) Solve (20) according to the following sequence from time to time : (cf. Figure 4).
Similarly, can be derived explicitly.
In conclusion, the discrete solutions of (19) and (20) can be obtained explicitly one by one instead of solving the linear algebraic system.
3. Stability of the Semi-Implicit Finite Difference Scheme
In this section, it is proven that the presented scheme (16) is uniformly stable as .
Suppose the coefficient of the diffusion term is a constant. Then is a constant.
Equation (21) can be written as the following linear system:where for nonnegative integer , The matrix entries and for and are defined by
Using the Fourier method, the stability of the presented scheme (16) with the initial value is proven. In general, we suppose and , then (21) becomesLet where is a nonnegative integer.
SoandThenwhereHere
According to Lemma 3,
Lemma 4. If and in (21) with ,
Proof. First, inequality is proved. It is obvious that , so From Lemmas 2 and 3, we haveSo . FurthermoreIn the same way, is obtained. On combining with (37) and (31), we haveThis completes the proof.
DefineThen the linear system (22) can be written as follows:Through simple computations,Then we have
Lemma 5. The eigenvalues of matrices and are nonnegative.
Proof. Let be an eigenvalue of matrix . According to Gerschgorim theorem [25], for every eigenvalue , it holdswhere is the radius and By inequality (36), it is obvious thatFrom (43), we derive that . The eigenvalues of matrix B which are nonnegative can be proven in the same way.
The above two lemmas are sewn together to yield the following theorem that guarantees the stability of the presented scheme (16).
Theorem 6. Suppose is a constant function. If , the scheme is uniformly stable about the initial value and right-hand side term with and
Proof. Using Lemma 5, we obtain andBy (42),From (41) and Lemma 4,Combining (49), (50) with (39),This completes the proof.
4. Error Estimate for the Finite Difference Scheme
In this section, the error estimate is considered for the presented scheme (16). Suppose is a constant.
Let be the exact solution of (1)~(3) at , then we have
Using Taylor expansion, at time-step .
So, (52) can be written as follows:
Since ,
In the same way,at time-step .
Let
Theorem 7. Let be the exact solution of (1)~(3) and be the solution of (16). Suppose is a constant function. When and are sufficiently small, there exists a positive constant independent of and such that for ,
Proof. We define as follows at the odd time-step Similarly, we define the even time-step According to (56), (55), and (16),Using Taylor expansion,Through the estimates for the in [9], there exist two constants and such thatCombining (12) with (63), we knowHere and are constant.
Using (62) and (64), we haveHere and are constant.
Since , there exists a positive constant independent of such thatSo (61) can be written as follows:It is obvious that Set Using Theorem 6,
Remark 8. From (55) the truncation error of the scheme (16) is . Here it is proven that the error between the discrete and the analytical solutions is bounded by by using the asymmetric discretisation technique.
5. Numerical Experiments
In this section, numerical examples of one-dimensional cases are presented to show the effectiveness and convergence orders of the scheme (16).
We consider the following space-fractional diffusion problem:with and .
In order to describe the numerical results, the discrete -norms, the discrete -norms, and the corresponding convergence rates are defined as follows: where , , and , are space mesh sizes.
The discretizing mesh sizes are given according to the following rules.
Case 1. Set , where is a nonnegative integer. By Theorem 7, the convergence order is . We get different values of to verify the theoretical analysis.
Case 2. Set and take the convergence order is .
Two numerical experiments are given with different values of , , and the source function . The errors, corresponding convergence rates, and CPU times are listed in the Tables 1–8. The results in Tables 1–8 are in agreement with the theoretical analysis. In addition, these tables show that, even with very small time-steps, the CPU execution times stay very small as well.
The results in Tables 9–12 are listed by using the finite difference scheme developed by Rui and Huang [19] for Examples 1 and 2. We set and have the same spatial grid sizes from to , since their error estimates between the numerical and analytical solutions are optimal under the condition for . Comparing our results in Tables 1–8 with those in Tables 9–12, one can see that the error estimate produced by our method is better than that in [19] in the sense that their convergence rate is . Furthermore, the time-steps in Tables 1–8 are smaller than that in Tables 9–12 with the spatial grid step staying the same. However the CPU execution times increase by a few seconds, which can prove the validity of our presented scheme.
Example 1. The diffusion coefficient , the analytical solution , and source coefficient are as follows:
The numerical results are listed in Tables 1–4.
Example 2. The diffusion coefficient , the analytical solution , and source coefficient are as follows:
The numerical results are listed in Tables 5–8.
6. Conclusion
In this paper, a uniformly stable explicit finite difference scheme is presented to solve one-dimensional space-fractional diffusion equation. The scheme is constructed combining a WSGD operator with an asymmetric technique. It can be solved explicitly by using different nodal-point stencils at odd and even time-steps. Results of two numerical experiments show that the error estimates are optimal when and , where is a nonnegative integer. There may exist super-convergence of the proposed semi-implicit finite difference method, which will be studied in the future.
Data Availability
The presented data used to support the findings of this study are included with the submitting article.
Conflicts of Interest
The author declares that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the Natural Science Foundation of Ningxia (no. NZ17260). I would like to express thanks to Professor Hongxing Rui (Shandong University) for his careful guidance sincerely.