Compact Difference Schemes with Temporal Uniform/Non-Uniform Meshes for Time-Fractional Black–Scholes Equation

Fractal and Fractional

article

Jie Gu¹  Lijuan Nong²  Qian Yi²  An Chen¹ 

[1] College of Science, Guilin University of Technology;College of Mathematics and Statistic, Guangxi Normal University

关键词: time-fractional Black–Scholes equation;    non-uniform meshes;    compact difference scheme;    stability;    error estimate;   

DOI  :  10.3390/fractalfract7040340

学科分类：社会科学、人文和艺术（综合）

来源: mdpi

【 摘 要 】

In this paper, we are interested in the effective numerical schemes of the time-fractional Black–Scholes equation. We convert the original equation into an equivalent integral-differential equation and then discretize the time-integral term in the equivalent form using the piecewise linear interpolation, while the compact difference formula is applied in the spatial direction. Thus, we derive a fully discrete compact difference scheme with second-order accuracy in time and fourth-order accuracy in space. Rigorous proofs of the corresponding stability and convergence are given. Furthermore, in order to deal effectively with the non-smooth solution, we extend the obtained results to the case of temporal non-uniform meshes and obtain a temporal non-uniform mesh-based compact difference scheme as well as the numerical theory. Finally, extensive numerical examples are included to demonstrate the effectiveness of the proposed compact difference schemes.

【 授权许可】

CC BY