论文标题
时间分数扩散PDE的数值保护定律
Numerical conservation laws of time fractional diffusion PDEs
论文作者
论文摘要
本文的第一部分介绍了足够的条件，以确定及时的任意分数顺序扩散方程的保护定律。满足这些条件的离散类似物的数值方法具有近似连续性的保护定律。在本文的第二部分中，我们提出了一种将空间中有限差异方法与光谱积分器结合在一起的方法。时间集成器已经在文献中应用于以$α\ in（0,1）$的订单$α\的caputo分数导数求解时间分数方程。它在这里被概述了任意顺序的近似Caputo和Riemann-Liouville的分数衍生物。我们将该方法应用于Riemann-Liouville分数衍生物的范围和超扩散方程，并得出其保护法。最后，我们提出了一系列数值实验，以显示该方法的收敛性及其保护特性。
The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have conservation laws that approximate the continuous ones. In the second part of the paper, we propose a method that combines a finite difference method in space with a spectral integrator in time. The time integrator has already been applied in literature to solve time fractional equations with Caputo fractional derivative of order $α\in(0,1)$. It is here generalised to approximate Caputo and Riemann-Liouville fractional derivatives of arbitrary order. We apply the method to subdiffusion and superdiffusion equations with Riemann-Liouville fractional derivative and derive its conservation laws. Finally, we present a range of numerical experiments to show the convergence of the method and its conservation properties.
