Two high-order compact difference schemes with temporal graded meshes for time-fractional Black-Scholes equation

Jie Gu; Lijuan Nong; Qian Yi; An Chen; Jie Gu; Lijuan Nong; Qian Yi; An Chen

doi:10.3934/nhm.2023074

Networks and Heterogeneous Media

2023, Volume 18, Issue 4: 1692-1712. doi: 10.3934/nhm.2023074

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Research article

Two high-order compact difference schemes with temporal graded meshes for time-fractional Black-Scholes equation

1.
College of Science, Guilin University of Technology, Guilin, 541004, China
2.
College of Mathematics and Statistic, Guangxi Normal University, Guilin, 541004, China

Received: 05 August 2023 Revised: 27 September 2023 Accepted: 07 October 2023 Published: 12 October 2023

In this paper, two high-order compact difference schemes with graded meshes are proposed for solving the time-fractional Black-Scholes equation. We first eliminate the convection term in the equivalent form of the considered equation by using exponential transformation, then combine the sixth-order/eighth-order compact difference method with a temporal graded meshes-based trapezoidal formulation for the temporal integral term to obtain the fully discrete high-order compact difference schemes. The stability and convergence analysis of the two proposed schemes are studied by applying Fourier analysis. Finally, the effectiveness of the proposed schemes and the correctness of the theoretical results are verified by two numerical examples.
- time-fractional Black-Scholes equation,
- high-order compact difference scheme,
- graded meshes,
- stability,
- error estimate
Citation: Jie Gu, Lijuan Nong, Qian Yi, An Chen. Two high-order compact difference schemes with temporal graded meshes for time-fractional Black-Scholes equation[J]. Networks and Heterogeneous Media, 2023, 18(4): 1692-1712. doi: 10.3934/nhm.2023074

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Abstract

In this paper, two high-order compact difference schemes with graded meshes are proposed for solving the time-fractional Black-Scholes equation. We first eliminate the convection term in the equivalent form of the considered equation by using exponential transformation, then combine the sixth-order/eighth-order compact difference method with a temporal graded meshes-based trapezoidal formulation for the temporal integral term to obtain the fully discrete high-order compact difference schemes. The stability and convergence analysis of the two proposed schemes are studied by applying Fourier analysis. Finally, the effectiveness of the proposed schemes and the correctness of the theoretical results are verified by two numerical examples.

References

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