Advances in Mathematical Physics

Volume 2013 (2013), Article ID 293706, 10 pages

http://dx.doi.org/10.1155/2013/293706

Research Article

Maximum Norm Error Estimates of ADI Methods for a Two-Dimensional Fractional Subdiffusion Equation

Yuan-Ming Wang1,2

¹Department of Mathematics, East China Normal University, Shanghai 200241, China

²Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science, E-Institute of Shanghai Universities, Shanghai Normal University, Shanghai 200234, China

Received 25 June 2013; Accepted 8 July 2013

Academic Editor: Ming Li

Copyright © 2013 Yuan-Ming Wang. 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.

Abstract.

This paper is concerned with two alternating direction implicit (ADI) finite difference methods for solving a two-dimensional fractional subdiffusion equation. An explicit error estimate for each of the two methods is provided in the discrete maximum norm. It is shown that the methods have the same order as their truncation errors with respect to the discrete maximum norm. Numerical results are given to confirm the theoretical analysis results.

1. Introduction

Fractional differential equations and fractional calculus arise in various application problems in science and engineering [1–16]. Various numerical methods have been developed for the computation of fractional differential equations [17–34]. Fractional subdiffusion equations describe a special type of anomalous diffusion [35], and it is a more difficult task to solve this kind of equation numerically.

Numerical works for fractional subdiffusion equations are mostly focused on one-dimensional problems due to the memory effect in fractional derivatives; see, for example, [19, 20, 20–26, 31–33, 36–42]. A two-dimensional anomalous subdiffusion equation was numerically treated in [43, 44], where explicit and implicit finite difference schemes were proposed. Chen et al. [28] extended their work in [43] to a variable-order subdiffusion equation. Liu et al. [45] developed an implicit meshless approach based on the radial basis function for the numerical simulation of a two-dimensional subdiffusion problem. Chen and Liu [18] considered an implicit difference scheme for a three-dimensional fractional advection-diffusion equation, and a Richardson extrapolation was applied to improve the accuracy.

The complexity of the fractional differential equations comes from the involving fractional derivatives that are nonlocal and have the character of history dependence and universal mutuality. This means that the computations would be costly if the implicit schemes were applied, especially for solving multidimensional problems [43, 44]. Some researchers have explored some techniques for reducing this cost. These techniques include the adaptive technique [46] and the matrix transfer technique [47, 48]. It is well known that alternating direction implicit (ADI) methods are unconditionally stable as the traditional implicit methods. On the other hand, they reduce a multidimensional problem to a series of independent one-dimensional problems, and thus the computational complexities and the computational cost can be greatly reduced. Therefore, ADI methods for fractional differential equations have the potential to significantly reduce the computational cost, while maintaining the stability of the numerical methods. The works in [29, 49–53] treated ADI finite difference methods for space fractional diffusion equations, and the work in [54] discussed ADI finite difference methods for fractional diffusion wave equations. Recently, Cui [55] derived an ADI compact finite difference scheme for a two-dimensional fractional subdiffusion equation, where the Grünwald formula is used to approximate the temporal Riemann-Liouville fractional derivative, and the spatial derivatives are approximated by a compact finite difference scheme. Another way to treat fractional subdiffusion problem is to transform the original subdiffusion equation into an equivalent equation by replacing the temporal Riemann-Liouville fractional derivative by the temporal Caputo fractional derivative; see, for example, [40, 42, 56]. An advantage of this approach is that the approximation (see [1, 24, 25, 40, 57, 58]) can be used to deal with the temporal fractional derivative, and so the resulting scheme has the better temporal accuracy than the first order without the Crank-Nicolson technique (see [25, 40–42]). Based on the previous approach, Zhang and Sun [59] constructed two ADI finite difference schemes, called -ADI and BD-ADI schemes, for the following two-dimensional problem of subdiffusion equation with the temporal Caputo fractional derivative:

where , , is the boundary of , is a positive constant, is the two-dimensional Laplacian, and denotes the temporal Caputo fractional derivative operator defined as

The main concern in that paper is the construction of the schemes and error estimates in the discrete -norm. Since -norm error estimates do not provide immediate insight on the phase error occurring during the time evolution, it is more preferable to give error estimates in the discrete maximum norm when we measure computation errors in practice. In this paper, we continue the investigation of the paper [59], by establishing a maximum norm error estimate for the ADI discretizations. It is known that an -norm error estimate does not imply a maximum norm error estimate for two-dimensional problems. We here present a technique of discrete energy analysis in order to obtain an explicit maximum norm error estimate.

The outline of the paper is as follows. In Section 2, we derive ADI finite difference schemes for (1) and present our main results of the maximum norm error estimates. The proof of the main error results is given in Section 3. In Section 4, we give some numerical results demonstrating the accuracy of the schemes in the discrete maximum norm. Section 5 contains some concluding remarks.

2. ADI Schemes and Maximum Norm Error Estimates

We partition with nonisotropic uniform mesh sizes and in the and directions, respectively. The integers and . The mesh points (). Let and be the sets of mesh points lying in and on , respectively, and let . For any grid function , we denote

For a positive integer , we let be the time step. Define and

For the temporal approximation, we introduce the operators

where and .

Using the Taylor expansion and the approximation of (see [25, 58, 59]), we have

where

There exists a positive constant independent of , , and the time level such that

Substituting (6) into (1), we obtain

2.1. Construction of -ADI and BD-ADI Schemes

In order to construct an ADI scheme, we add the term to (9). This yields

where

It was shown in [59] that for a positive constant independent of , , and the time level . Thus, there exists a positive constant independent of , , and the time level such that

Denote by the finite difference approximation to , and let . After multiplying (10) by and then dropping the term , we derive a finite difference scheme as follows:

By introducing the intermediate variable , we obtain the following -ADI scheme (see [59]):

Adding the term to (9), we have

where

Since for a positive constant independent of , , and the time level (see [59]), there exists a positive constant independent of , , and the time level such that

By (15), we obtain the following finite difference scheme:

It is equivalent to the following BD-ADI scheme (see [59]):

2.2. Maximum Norm Error Estimates

For any grid function , we define its maximum norm by

Let be the value of the solution of (1) at the mesh point , and let be the solution of the -ADI scheme (14) or the BD-ADI scheme (19). We now present our main results of the maximum norm estimate for the error in the following two theorems. Their proofs will be given in the next section.

Theorem 1. Assume that the solution of (1) is sufficiently smooth, and let be the solution of the -ADI scheme (14). Then

where and .

Theorem 2. Assume that the solution of (1) is sufficiently smooth, and let be the solution of the BD-ADI scheme (19). Then

where and .

Theorems 1 and 2 show that the ADI difference solution from (14) or (19) converges to the analytical solution of (1) in the discrete maximum norm. We also see from the estimates (12), (17), (21), and (22) that the -ADI scheme (14) and the BD-ADI scheme (19) have the same order as their truncation error with respect to the discrete maximum norm.

3. Proof of the Main Results

Let be the set of all grid functions defined in and vanishing on . For arbitrary , we define the following inner products:

For any , we introduce the following norms:

Using a simple calculation, we have that for arbitrary ,

Before proving Theorems 1 and 2, we first introduce the following embedding theorem from [60, page 281].

Lemma 3. For any , one has

Proof of Theorem 1. Let . Then by (10) and (13),

where

This implies that for each . Taking the inner product of the first equation in (27) and , we get

It follows from (25) that

Similarly,

Since

we have from Cauchy-Schwarz inequality that

Substituting (30) and (33) into (29) gives the following:

where

This implies that

Since and by (12), , the estimate (21) follows from (36) and (26) immediately.

Proof of Theorem 2. The proof follows from the similar argument as that in the proof of Theorem 1 and we give a sketch. Let . By (15) and (18),

where is defined by (28). Taking the inner product of the first equation in (37) and , we get

Since

we obtain that

where

Therefore

The estimate (22) follows from the previous inequality, (17), and (26) immediately.

4. Numerical Results

In this section, we give some numerical results to demonstrate the accuracy of the -ADI scheme (14) and the BD-ADI scheme (19) with respect to the discrete maximum norm. Some detailed numerical comparisons of these two ADI schemes with the implicit scheme proposed in [43] can be found in [59]. The dependence of the solution of (1) on the anomalous diffusion exponent was also exhibited in [59] through some numerical results.

In our numerical computations, we take an equal mesh size in each of the space directions; that is, . We compute the discrete maximum norm error of the numerical solution by

and its convergence orders by

where represents the value of the exact analytic solution of (1) at .

Example 4. We consider the problem (1) in the domain , and let . Assume that the solution of this problem is . It can be checked that the corresponding known functions are given by

In Table 1, we present the maximum norm error and the temporal convergence order of the numerical solution by the -ADI scheme (14) and the BD-ADI scheme (19). We see that the -ADI scheme has the temporal accuracy of in the discrete maximum norm, and the best temporal accuracy is attained at . It is also seen that the BD-ADI scheme generates the temporal accuracy of in the discrete maximum norm, and it gets the best temporal accuracy when . These observations coincide well with the theoretical analysis.

Table 1: The maximum norm error and the temporal accuracy of for Example 4.

-ADI scheme ()
BD-ADI scheme ()

error
order

error
order

1/20
1/2

0.860979
1/3

1.266265

1/40

0.882911

1.285881

1/80

0.910189

1.301046

1/160

0.936057

1.319085

1/320

0.959731

1.351071

1/640

1/20
2/3

1.260363
1/2

1.463622

1/40

1.288661

1.474131

1/80

1.303399

1.488745

1/160

1.307319

1.514438

1/320

1.297645

1.535827

1/640

1/20
3/4

1.174836
2/3

1.207510

1/40

1.197125

1.242740

1/80

1.210140

1.264770

1/160

1.216775

1.276115

1/320

1.217031

1.274583

1/640

Table 2 gives the maximum norm error and the spatial convergence order of the numerical solution by the -ADI scheme (14) and the BD-ADI scheme (19). As expected from the theoretical analysis, these two schemes have the second-order spatial accuracy.

Table 2: The maximum norm error and the spatial accuracy of for Example 4.

-ADI scheme ()
BD-ADI scheme ()

error
order
error
order

1.994343

2.001730

1.989441

2.000950

1.960585

2.002088

1.844870

2.007753

Example 5. We consider the subdiffusion equation

with boundary and initial conditions

where and denotes the Riemann-Liouville fractional derivative operator defined as

Operating Riemann-Liouville fractional derivative operator on both sides of (46), we obtain the equivalent problem of the subdiffusion equation with the temporal Caputo fractional derivative [40, 42, 59] as follows:

We now solve the aforementioned problem by the -ADI scheme (14) and the BD-ADI scheme (19). Tables 3 and 4 give the maximum norm error and the convergence orders and of the numerical solution . It is easily seen that the numerical results confirm the theoretical analysis results.

Table 3: The maximum norm error and the temporal accuracy of for Example 5.

-ADI scheme ()
BD-ADI scheme ()

error
order

error
order

1/20
1/2

0.835782
1/3

1.247710

1/40

0.906370

1.295329

1/80

0.947443

1.315075

1/160

0.970534

1.325682

1/320

0.983962

1.335982

1/640

1/20
2/3

1.192172
1/2

1.423830

1/40

1.262779

1.463914

1/80

1.297640

1.481554

1/160

1.317329

1.496663

1/320

1.333601

1.524750

1/640

1/20
3/4

1.063599
2/3

1.112230

1/40

1.128142

1.171085

1/80

1.161548

1.211600

1/160

1.180856

1.239224

1/320

1.191836

1.255314

1/640

Table 4: The maximum norm error and the spatial accuracy of for Example 5.

-ADI scheme ()
BD-ADI scheme ()

error
order
error
order

1.901804

1.901896

1.985610

1.974325

2.072000

2.032967

2.263270

2.158045

5. Conclusions

We have studied two ADI finite difference methods for a two-dimensional fractional subdiffusion equation. An explicit error estimate for each of the two methods has been provided in the discrete maximum norm. It has been shown that the methods have the same order as their truncation errors with respect to the discrete maximum norm. The maximum norm error estimates presented here are more preferable for measuring computation errors in practice, compared to the -norm error estimates in [59]. Numerical results have confirmed the theoretical analysis results.

Acknowledgments

This work was supported in part by E-Institutes of Shanghai Municipal Education Commission no. E03004 and Shanghai Leading Academic Discipline Project no. B407.

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