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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

The Black-Scholes model, proposed in 1973 by Black and Scholes [

In this paper, we continue the work of R.H.De Staelen et al. [

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 (

In order to construct the compact finite difference scheme for the problem (

Let

Since the grid function

We also define

It holds (see [

In order to discretize (

Assuming

From Lemma 2 of [

Assuming

According to some simple calculations, the proof follows from Taylor expansions of the function

Since the above lemmas, we then discretize (

From Lemmas

The compact difference scheme (

The compact difference scheme (

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

Letting

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

Suppose

Suppose

Suppose

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

Let

Then it holds

We take the inner product of equation (

Letting

Let

It follows from Theorem

The constraint condition

For demonstrating the efficiency of the compact difference scheme (

Suppose

We first consider a problem, which is governed by equation (

For different

Next, we compute the spatial convergence order of the compact difference scheme (

The errors and the temporal convergence orders of the compact difference scheme (

| | | | | | | |
---|---|---|---|---|---|---|---|

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 |

The errors and the spatial convergence orders of the compact difference scheme (

| | | | | | | |
---|---|---|---|---|---|---|---|

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 |

In this example, we test the error and the convergence order of the compact difference scheme (

Apply the compact difference scheme (

From Table

The errors and the temporal convergence orders of the compact difference scheme (

| | | | | | | |
---|---|---|---|---|---|---|---|

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 |

The errors and the spatial convergence orders of the compact difference scheme (

| | | | | | | |
---|---|---|---|---|---|---|---|

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 |

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

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

The authors declare that they have no conflicts of interest.

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.