In this paper, we propose third-order semi-discretized schemes in space based on the tempered weighted and shifted Gr\"unwald 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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关于调和分数式偏微分方程的高阶格式

在本文中，我们提出了基于回火分数扩散方程的回火加权和移位 Gr\"unwald 差分 (tempered-WSGD) 算子的空间三阶半离散化方案。我们还展示了完全的稳定性和收敛性分析。基于时间上的Crank--Nicolson方案的离散方案，还提出了一种用于回火Black-Scholes方程的三阶方案并进行了数值测试，并进行了一些数值实验，以验证这些方法的准确性和有效性。