Advances in Mathematical Physics
Volume 2019 (2019), Article ID 3263589, 9 pages
https://doi.org/10.1155/2019/3263589

Research Article

An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

Lei Ren and Lei Liu

School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu 476000, China

Correspondence should be addressed to Lei Ren; renqihan@hotmail.com

Received 11 June 2019; Accepted 25 July 2019; Published 21 August 2019

Academic Editor: Andrei D. Mironov

Copyright © 2019 Lei Ren and Lei Liu. 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.

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the - approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

1. Introduction

The Black-Scholes model, proposed in 1973 by Black and Scholes [1] and Merton [2], gives a theoretical estimate of the price of European-style options. Until Now, some of Black-Scholes models involving the fractional derivatives have emerged. In [3], Wyss priced a European call option by a time-fractional Black-Scholes model. In [4], Liang et al. derive a biparameter fractional Black-Merton-Scholes equation and obtain the explicit option pricing formulas for the European call option and put option, individually. An explicit closed-form analytical solution for barrier options under a generalized time-fractional Black-Scholes model by using eigenfunction expansion method together with the Laplace transform is derived in [5]. In [6], a discrete implicit numerical scheme with a spatially second-order accuracy and a temporally order accuracy is constructed; the stability and convergence of the proposed numerical scheme are analysed using Fourier analysis. In [7], H.Zhang et al. use some numerical technique to price a European double-knock-out barrier option, and then the characteristics of the three fractional Black-Scholes models are analysed through comparison with the classical Black-Scholes model. More recently, a numerical scheme of fourth-order in space and in time is derived in [8]; the solvability and convergence of the proposed numerical scheme are proved rigorously using a Fourier analysis. Some computationally efficient numerical methods have been proposed for solving fractional differential equation, for example, which include finite difference methods, finite element methods, finite volume methods, spectral methods, and meshless methods [9–26].

In this paper, we continue the work of R.H.De Staelen et al. [8]. The class of equations is given by

with the following boundary (barrier) and final conditions

and its initial condition

where is the risk free rate, is the dividend rate, and is the volatility of the returns. The functions and are the rebates paid when the corresponding barrier is hit. The terminal playoff of the option is . The fractional derivative in (1) is a Caputo derivative defined as

As described in [8], we consider the transform problem of (1)

The rest of the paper is organized as follows: in Section 2, an efficient implicit numerical scheme with second-order accuracy in time and fourth-order accuracy in space is constructed. The analysis of the stability and convergence are presented in Section 3. In Section 4, numerical examples are given to illustrate the accuracy of the presented scheme and to support our theoretical results. Concluding remarks are given in the last section.

2. Construction of the Compact Finite Difference Scheme

In order to simplify the computation and analysis of the following compact finite difference scheme for Black-Scholes model, we use an indirect approach by introducing a suitable transformation.

According to some simple calculations, we transform equation (5) into

where

It is clear that is a solution of (5) if and only if is a solution of (6).

In order to construct the compact finite difference scheme for the problem (5), we consider the above equivalent form (6).

Let be the time step and be the spatial step, where are positive integers.

Since the grid function , we then define difference operators as follows:

We also define

where , and

Lemma 1. It holds (see [27])

In order to discretize (6) into a compact finite difference system, we introduce the following lemmas.

Lemma 2. Assuming , we have

where .

Proof. From Lemma 2 of [9], we can obtain the proof of lemma.

Lemma 3. Assuming . When , we obtain

Proof. According to some simple calculations, the proof follows from Taylor expansions of the function at the point for and .

Since the above lemmas, we then discretize (6) into a compact finite difference scheme. In order to analyse, we define

We also define the grid functions as follows:

For the second-order spatial derivative , we adopt the following fourth-order compact approximation (see [28])

We consider equation (6) at the point ; we can obtain

From Lemmas 2 and 3, we have

where

We apply to equation (18); then we have

where

and

If we omit , then we have the compact finite difference scheme:

3. Stability and Convergence of the Proposed Compact Difference Scheme

Theorem 4. The compact difference scheme (23) is uniquely solvable.

Proof. The compact difference scheme (23) can be written in matrix form

where

The tridiagonal coefficient matrix yields

It is easy to see that the tridiagonal coefficient matrix is strictly diagonally dominant. Therefore, the coefficient matrix is nonsingular and hence invertible.

Next, we consider the stability and convergence analysis of the compact difference scheme (23).

Letting , for grid functions , we define the inner product and norm as follows:

According to simple calculations, we obtain

In order to analyse, we introduce the discrete inner product and norm:

Based on above inner product and norm, we have the following lemmas.

Lemma 5 (see [29]). Suppose , we obtain

Lemma 6 (see [27]). Suppose , we obtain

Lemma 7 (see [9]). Suppose , we obtain

In the next, we then analyse the stability and convergence of the scheme (23).

Theorem 8 (stability). Let be the solution of the compact difference scheme (23) with . Assume that one of the conditions holds for some positive constant .
Then it holds

Proof. We take the inner product of equation (23) with yield

Using Lemma 7,

When for some positive constant , we have from the Cauchy-Schwarz inequality and Lemmas 6 that

By (35) and the Cauchy-Schwarz inequality,

Substituting (38) into (35) leads to

The above inequality can be rewritten as

Since by the definition of ,

we have from (40) that

Letting

and assuming , we obtain

and we have the needed estimates.

Letting , we get the following error equation:

Since the above error equation (45), we now obtain the following convergence results.

Theorem 9 (convergence). Let denote the value of the solution of (23) at the mesh point and let be the solution of the compact difference scheme (23). Then when , it holds

where

Proof. It follows from Theorem 8 that

Applying (22), we get

The estimate (46) is proved.

Remark 10. The constraint condition in Theorems 8 and 9 is only for the analysis of the stability and convergence of the compact difference scheme (23). This condition is easily verifiable for practical problems.

4. Numerical Experiment

For demonstrating the efficiency of the compact difference scheme (23), we make two numerical experiments of it.

Suppose be the value of the solution of the problem (1)–(3) at the mesh point . From (22), we can see that

where is a positive constant independent. In order to check this accuracy of the compact difference scheme, we compute the following norm errors:

The temporal convergence order and the spatial convergence order are denoted by

Example 1. We first consider a problem, which is governed by equation (1) in with and

The boundary and initial conditions are given by (2) and (3) with

It is easy to check that is the solution of this problem.
For different , we let the spatial step . Table 1 gives the errors and the temporal convergence orders of the computed solution for and different time step . From the table, we can see that the computed solution has the second-order temporal accuracy. For comparison, the corresponding temporal convergence orders given in [8] has only order; thus it is far less accurate than the compact difference scheme (23) given in this paper.
Next, we compute the spatial convergence order of the compact difference scheme (23). Table 2 presents the errors and the spatial convergence orders . The table demonstrates that the compact difference scheme (23) has the fourth-order spatial accuracy.

Table 1: The errors and the temporal convergence orders of the compact difference scheme (23) for Example 1  .

1/4
1/10
3.7361e–05

3.7314e–05

6.2608e–05

1/20
9.3611e–06
1.9968
9.3492e–06
1.9968
1.5684e–05
1.9970

1/40
2.3429e–06
1.9984
2.3399e–06
1.9984
3.9251e–06
1.9985

1/80
5.8603e–07
1.9992
5.8528e–07
1.9992
9.8176e–07
1.9993

1/160
1.4654e–07
1.9997
1.4635e–07
1.9997
2.4549e–07
1.9997

1/320
3.6634e–08
2.0001
3.6587e–08
2.0001
6.1372e–08
2.0000

1/2
1/10
6.7788e–05

6.7702e–05

1.1393e–04

1/20
1.6994e–05
1.9960
1.6972e–05
1.9960
2.8555e–05
1.9964

1/40
4.2543e–06
1.9980
4.2489e–06
1.9980
7.1480e–06
1.9981

1/80
1.0643e–06
1.9990
1.0630e–06
1.9990
1.7882e–06
1.9991

1/160
2.6617e–07
1.9995
2.6583e–07
1.9995
4.4718e–07
1.9996

1/320
6.6548e–08
1.9999
6.6463e–08
1.9999
1.1181e–07
1.9999

3/4
1/10
8.8226e–05

8.8110e–05

1.4950e–04

1/20
2.2098e–05
1.9973
2.2069e–05
1.9973
3.7435e–05
1.9976

1/40
5.5299e–06
1.9986
5.5226e–06
1.9986
9.3672e–06
1.9987

1/80
1.3832e–06
1.9993
1.3813e–06
1.9993
2.3429e–06
1.9993

1/160
3.4587e–07
1.9996
3.4542e–07
1.9996
5.8585e–07
1.9997

1/320
8.6475e–08
1.9999
8.6361e–08
1.9999
1.4647e–07
1.9999

Table 2: The errors and the spatial convergence orders of the compact difference scheme (23) for Example 1  .

1/4
1/2
1.1190e–04

6.4607e–05

9.1369e–05

1/4
5.4155e–06
4.3690
4.2667e–06
3.9205
5.4429e–06
4.0693

1/8
2.9041e–07
4.2209
2.6840e–07
3.9907
3.3922e–07
4.0041

1/16
1.7125e–08
4.0840
1.6758e–08
4.0015
2.1165e–08
4.0024

1/32
1.0264e–09
4.0604
1.0207e–09
4.0372
1.2889e–09
4.0375

1/2
1/2
1.0340e–04

5.9701e–05

8.4430e–05

1/4
5.0151e–06
4.3659
3.9472e–06
3.9189
4.9995e–06
4.0779

1/8
2.6907e–07
4.2202
2.4841e–07
3.9900
3.1142e–07
4.0048

1/16
1.5831e–08
4.0871
1.5487e–08
4.0036
1.9400e–08
4.0047

1/32
9.2534e–10
4.0967
9.2011e–10
4.0731
1.1538e–09
4.0716

3/4
1/2
9.3459e–05

5.3959e–05

7.6309e–05

1/4
4.5477e–06
4.3611
3.5734e–06
3.9165
4.4770e–06
4.0912

1/8
2.4420e–07
4.2190
2.2506e–07
3.9889
2.7865e–07
4.0060

1/16
1.4335e–08
4.0905
1.4014e–08
4.0053
1.7372e–08
4.0036

1/32
8.2043e–10
4.1270
8.1562e–10
4.1028
1.0150e–09
4.0971

Example 2. In this example, we test the error and the convergence order of the compact difference scheme (23). Consider equation (1) in the domain with and

The boundary and initial conditions are given by (2) and (3) with

It is clear that is the exact analytical solution of this problem.
Apply the compact difference scheme (23) to solve the above problem. Table 3 presents the errors and the temporal convergence orders ; we can see that the computed solution has the second-order temporal accuracy.
From Table 4, we can obtain the errors and the spatial convergence orders . These numerical results demonstrate that the accuracy of the compact difference scheme (23) is fourth-order.

Table 3: The errors and the temporal convergence orders of the compact difference scheme (23) for Example 2  .

1/4
1/10
9.5681e–05

9.5622e–05

1.3266e–04

1/20
2.3985e–05
1.9961
2.3971e–05
1.9961
3.3254e–05
1.9961

1/40
6.0044e–06
1.9981
6.0007e–06
1.9981
8.3244e–06
1.9981

1/80
1.5021e–06
1.9990
1.5012e–06
1.9990
2.0825e–06
1.9991

1/160
3.7564e–07
1.9996
3.7541e–07
1.9996
5.2078e–07
1.9996

1/320
9.3916e–08
1.9999
9.3859e–08
1.9999
1.3020e–07
1.9999

1/2
1/10
1.7283e–04

1.7272e–04

2.4033e–04

1/20
4.3358e–05
1.9950
4.3331e–05
1.9950
6.0286e–05
1.9951

1/40
1.0858e–05
1.9975
1.0852e–05
1.9975
1.5097e–05
1.9976

1/80
2.7169e–06
1.9987
2.7153e–06
1.9987
3.7774e–06
1.9988

1/160
6.7952e–07
1.9994
6.7910e–07
1.9994
9.4474e–07
1.9994

1/320
1.6991e–07
1.9998
1.6980e–07
1.9998
2.3622e–07
1.9998

3/4
1/10
2.2075e–04

2.2061e–04

3.0894e–04

1/20
5.5326e–05
1.9964
5.5291e–05
1.9964
7.7418e–05
1.9966

1/40
1.3849e–05
1.9981
1.3841e–05
1.9981
1.9378e–05
1.9983

1/80
3.4646e–06
1.9991
3.4624e–06
1.9991
4.8476e–06
1.9991

1/160
8.6641e–07
1.9995
8.6587e–07
1.9995
1.2123e–06
1.9996

1/320
2.1663e–07
1.9998
2.1650e–07
1.9998
3.0311e–07
1.9998

Table 4: The errors and the spatial convergence orders of the compact difference scheme (23) for Example 2  .

1/4
1/2
1.2909e–04

7.4532e–05

1.0540e–04

1/4
6.2469e–06
4.3691
4.9219e–06
3.9206
6.2789e–06
4.0693

1/8
3.3499e–07
4.2210
3.0961e–07
3.9907
3.9130e–07
4.0042

1/16
1.9745e–08
4.0846
1.9323e–08
4.0021
2.4403e–08
4.0031

1/32
1.1750e–09
4.0707
1.1685e–09
4.0475
1.4745e–09
4.0488

1/2
1/2
1.1929e–04

6.8871e–05

9.7399e–05

1/4
5.7849e–06
4.3660
4.5533e–06
3.9189
5.7675e–06
4.0779

1/8
3.1036e–07
4.2203
2.8654e–07
3.9901
3.5923e–07
4.0050

1/16
1.8246e–08
4.0883
1.7849e–08
4.0048
2.2357e–08
4.0061

1/32
1.0507e–09
4.1182
1.0447e–09
4.0946
1.3056e–09
4.0980

3/4
1/2
1.0782e–04

6.2247e–05

8.8031e–05

1/4
5.2457e–06
4.3613
4.1220e–06
3.9166
5.1648e–06
4.0912

1/8
2.8166e–07
4.2191
2.5959e–07
3.9891
3.2142e–07
4.0062

1/16
1.6515e–08
4.0921
1.6146e–08
4.0070
2.0016e–08
4.0053

1/32
9.2416e–10
4.1595
9.1874e–10
4.1354
1.1316e–09
4.1447

5. Concluding Remarks

In this paper, a high-order compact finite difference method for a class of time-fractional Black-Scholes equations is presented and analysed. We apply the - approximation formula to the Caputo derivative; then we construct a fourth-order compact finite difference approximation for the spatial derivative. We have analysed the solvability, stability, and convergence of the constructed scheme and provided the optimal error estimates. The constructed scheme has the second-order temporal accuracy and the fourth-order spatial accuracy, which improves the temporal accuracy of the method given in [8].

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by National Natural Science Foundation of China No. 11401363, the Education Foundation of Henan Province No. 19A110030, the Foundation for the Training of Young Key Teachers in Colleges and Universities in Henan Province No. 2018GGJS134.

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