齐次分数阶扩散方程的加权C-N格式及其修正

Weighted C-N Scheme of HomogeneousFractional Di?usion Equations and Its Correction

摘要: 对于齐次时间分数阶扩散方程，在精确解不光滑时，数值方法精度会下降。针对这种情况，本文提出加权Crank-Nicolson格式(简记为加权C-N格式)及其修正格式，在精确解不光滑时，修正原格式的第1步后，可恢复方法的时间2阶精度。本文接着给出详细的收敛性分析，并且数值算例验证了方法的有效性。

Abstract: For the homogeneous time fractional diffusion equation, the accuracy of the numerical method will decrease when the exact solution is not smooth enough. In this case, we consider a weighted Crank-Nicolson scheme (masked as weighted C-N scheme) and its correction. After correcting the first step of the weighted C-N scheme, the second-order time accuracy can be restored. Then we give the detailed convergence analysis, and numerical examples verify the effectiveness of the scheme.