时间分数阶偏微分方程的差分方法

Fractional partial differential equations have been applied in many fields of sciences and engineering successfully. The time fractional partial differential equation is a class of important problems. In recent years, the numerical methods of time fractional partial differential equations and corresponding numerical analysis have been developed. However, there also exist many problems need to be solved. The main goal of this program is to study the time splitting methods for the high-dimensional problems in the regular domain, and develop finite difference methods under the Non Cartesian coordinate system. We hope to make considerable progress on the numerical method and corresponding numerical analysis of time fractional partial differential equations.

分数阶偏微分方程已成功应用于科学和工程中的诸多领域,其中时间分数阶偏微分方程是重要的一类问题.近年来,其数值解法和相应的算法分析都得到了发展,但是仍然存在一些困难和需要进一步解决的问题.本项目的主要目的是在规则区域上研究高维问题的时间分裂算法;在非直角坐标系下建立相应的有限差分格式并做算法分析.通过本项目的研究,我们力求在时间分数阶偏微分方程的算法设计和算法分析上有所创新.

本项目已顺利完成预期目标。取得研究成果均已发表在有影响力的国际期刊J comput phys, J sci Comput上, 成果如下：.1. 在非一致网格下对时间分数阶扩散方程建立差分格式，证明了对任意时间网格下，该格式是无条件稳定的。.2. 对二维时间分数阶扩散方程建立了紧交替方向隐式差分格式，证明了格式的稳定性和收敛性.