In this paper, we propose third-order semi-discretized schemes in space based on the tempered weighted and shifted Grunwald difference (tempered-WSGD) operators for the tempered fractional diffusion equation. We also show stability and convergence analysis for the fully discrete scheme based a Crank–Nicolson scheme in time. A third-order scheme for the tempered Black–Scholes equation is also proposed and tested numerically. Some numerical experiments are carried out to confirm accuracy and effectiveness of these proposed methods.

中文翻译：

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关于分数阶偏微分方程的高阶格式

在本文中，我们针对回火分数扩散方程，基于回火加权和移位的格伦瓦尔德差（tempered-WSGD）算子，提出了空间中的三阶半离散方案。我们还展示了基于Crank-Nicolson方案的完全离散方案的稳定性和收敛性分析。还提出了经过调解的Black-Scholes方程的三阶方案，并进行了数值测试。进行了一些数值实验，以确认这些建议方法的准确性和有效性。