In this paper, a high-order and fast numerical method is investigated for the time-fractional Black-Scholes equation. In order to deal with the typical weak initial singularities of the solution, we construct a finite difference scheme with variable time steps, where the fractional derivative is approximated by the nonuniform Alikhanov formula and the sum-of-exponentials (SOE) technique. In the spatial direction, an average approximation with fourth-order accuracy is employed. The stability and the convergence with second-order in time and fourth-order in space of the proposed scheme are religiously derived by the energy method. Numerical examples are given to demonstrate the theoretical statement.
翻译:在本文中,对时间偏差的黑色-碎块方程式采用高顺序和快速数字方法。为了处理典型的微弱的解决方案初始奇数,我们用可变时间步骤构建了一定的差别方案,其中分数衍生物近似于非单式的Alikhanov公式和特效总和(SOE)技术。在空间方向,采用了具有四级精确度的平均近似法。在宗教上,能源方法得出了拟议方案时间和四级空间的稳定性和与第二级的趋同。提供了数字例子来展示理论说明。