时间分数阶扩散偏微分方程的数值守恒律
Numerical conservation laws of time fractional diffusion PDEs
摘要：
本文第一部分介绍了任意分数阶扩散方程在时间上守恒律的充分条件。满足这些条件的离散模拟的数值方法具有近似连续守恒律的守恒律。在本文的第二部分，我们提出了一种方法，结合了有限差分法在空间上的谱积分在时间上。时间积分器已经在文献中应用于求解具有$α\in(0,1)$阶Caputo分数阶导数的时间分数阶方程。它在这里推广到近似Caputo和Riemann-Liouville分数阶导数的任意阶。我们将该方法应用于具有Riemann-Liouville分数阶导数的亚扩散和超扩散方程，并得到了其守恒律。最后，我们提出了一系列的数值实验，以显示该方法的收敛性和其守恒性质。
英文摘要：
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.
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