高阶方案用于 tempered 分数扩散方程
High order schemes for the tempered fractional diffusion equations
AI总结：
本文提出高阶差分方案用于求解 tempered 分数扩散方程，通过 tempered 分数微积分性质和加权移位方法构造高阶差分算子，并验证其稳定性和收敛性。
摘要：
具有无限矩的跳跃飞行模型用于描述复杂系统中的超扩散。通过指数性地 tempering Lévy 飞行的跳跃测度，得到 tempered 稳定 Lévy 过程，结合 α- 稳定和高斯趋势；非常大的跳跃不太可能且所有矩都存在。 tempered 稳定 Lévy 过程的概率密度函数解 tempered 分数扩散方程。本文聚焦于设计 bounded 域上 tempered 分数扩散方程的高阶差分方案。通过 tempered 分数微积分性质和加权移位其一阶 Grünwald 型差分近似，得到称为 tempered 和加权和移位 Grünwald 差分（tempered-WSGD）的高阶差分近似。在时间方向使用 Crank-Nicolson 离散化。建立了所提数值方案的稳定性和收敛性；并通过数值实验验证理论结果并证明方案的有效性。
英文摘要：
Lévy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering the Levy measure of Lévy flights leads to the tempered stable Lévy processes which combine both the $α$-stable and Gaussian trends; and the very large jumps are unlikely and all their moments exist. The probability density functions of the tempered stable Lévy processes solve the tempered fractional diffusion equation. This paper focuses on designing the high order difference schemes for the tempered fractional diffusion equation on bounded domain. The high order difference approximations, called the tempered and weighted and shifted Grünwald difference (tempered-WSGD) operators, in space are obtained by using the properties of the tempered fractional calculus and weighting and shifting their first order Grünwald type difference approximations. And the Crank-Nicolson discretization is used in the time direction. The stability and convergence of the presented numerical schemes are established; and the numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the schemes.
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