非局部微积分方程高阶数值方法研究

Nonlocal (fractional) integro-differential equations are widely used in scientific fields such as medicine, finance, and chemistry et al. In general, there is an integral term with a weakly singular kernel in these equations which usually implies a nonsmooth solution. As a result, the corresponding numerical methods in literature are of lower order. Hence, it is a very important topic to construct high order numerical schemes, in particular for high dimensional cases. This project is to: (1) study high order numerical schemes for the nonlinear time fractional diffusion equation, especially, how to, by employing the two grid method, handle the nonlinear term and the nonsmoothness of the solution in time which cause the lower accuracy of numerical methods;(2) study high-order numerical schemes of the nonlocal integro-differential equation, in particular how to construct an alternate direction implicit scheme of the multi-dimension problems, and analyze the error of the full-discrete numerical scheme by using the Laplace transformation method.

非局部（分数次）微积分方程已被广泛应用于医学、金融和化学等科学领域。此类方程一般带有弱奇异核的积分项或非线性项，解不光滑，采用传统的方法进行数值求解时间方向往往被限制在低阶情形，同时高维方程存在计算困难等问题。因此，对高维非局部微积分方程建立高效的数值格式是一个仍待解决的重要科学问题。本项目拟：（1）研究非线性时间分数阶扩散方程的高阶数值格式；采用两网格方法和有限元方法解决非线性项和非光滑性导致时间方向收敛阶低的问题;（2）研究高维非局部微积方程的高阶数值格式；采用交替方向隐格式解决多维计算困难问题，采用Laplace变换方法对所构造全离散数值格式进行误差估计分析。

研究了三类非光滑时间分数阶积分微分方程的高效数值方法。（1）对非光滑时间分数阶电报方程，时间方向利用交替方向隐式（ADI）差分格式进行离散，空间方向构造了Crank- Nicolson有限差分格式和紧差分格式，给出了稳定性和收敛性分析结果。（2）对非光滑三维时间分数阶回火方程，时间方向构造了BDF2和ADI差分格式，空间方向采用差分法和紧差分法进行离散，证明了全离散数值格式的稳定性和收敛性。（3）对非光滑多维分数阶积分微分方程，时间方向利用ADI数值格式进行离散，空间方向采用有限差分法，给出了全离散数值格式的稳定性和收敛性结果。