两边空间分数阶对流-扩散方程的一种加权显式有限差分方法

A Weighted Explicit Finite Difference Method for Two-Sided Space Fractional Advection Diffusion Equation

考虑两边空间分数阶对流-扩散方程的初边值问题,基于Grünwald公式和移位Grünwald-Letnikov公式,提出一种加权显式有限差分解法.利用傅里叶变换和特征值法,得到差分格式的稳定性.然后使用最大模估计法证明在相同的条件下,所提出的差分格式是收敛的.最后通过数值例子说明所提出的差分格式是可靠和有效的,并对方程的数值解与精确解进行比较,验证了文中的理论结果.

Based on Grünwald formula and the shifted Grünwald-Letnikov formula,a weighted explicit finite difference method is proposed to solve initial boundary value problems of two-sided space fractional advection diffusion equation. Their stability is analyzed by means of Fourier transform and eigenvalue analysis. Using the technique of maximum norm analysis,it is proved that the scheme convergent under the same condition. Illustrative example is included to demonstrate the validity and applicability of the scheme,and a comparison between the exact analytical and the numerical prediction is made to demonstrate the theoretical results.