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

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.

Automated summary

In this paper, the authors propose effective numerical schemes for the time-fractional Black-Scholes equation. The original equation is converted into an equivalent integral-differential equation and discretized using piecewise linear interpolation for the time-integral term and compact difference formula for the spatial direction. The derived fully discrete compact difference scheme demonstrates second-order accuracy in time and fourth-order accuracy in space, with rigorous proofs of stability and convergence. Furthermore, the authors extend the results to non-uniform meshes for dealing with non-smooth solutions and provide a temporal non-uniform mesh-based compact difference scheme.