齐次分数阶扩散方程的加权$C-N$格式及其修正
首发时间:2019-06-06
摘要:近年来,分数阶微分方程数值解的研究得到了快速发展,但高阶数值方法一般依赖精确解及初始数据的光滑性,精确解不够光滑或者初始数据不光滑时,数值方法的精度会下降.本文针对时间分数阶扩散方程,提出加权$Crank-Nicolson$格式(简记为加权$C-N$格式)及其修正,在精确解不够光滑时,只修正原格式的第1步,即加上一个权系数,就可保留方法的时间2阶精度.本文结合$Laplace$变换和卷积,对加权$C-N$修正格式进行收敛性分析,最后数值测试验证了方法的有效性.
关键词: 时间分数阶扩散方程;加权$C-N$格式;卷积;~$Laplace$变换
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Weighted $C-N$ Scheme of Homogeneous Fractional Diffusion Equations and Its Correction
Abstract:In recent years, the numerical solution of fractional differential equations has been developed rapidly, but the high-order numerical methods generally rely on the smoothness of exact solution and initial data. When the exact solution is not smooth enough or the initial data is not smooth, the accuracy of the numerical method will decrease. In this paper, we propose a weighted $Crank-Nicolson$ scheme(abbreviated as $C-N$ scheme) and its correction for the time fractional diffusion equation. When the exact solution is not smooth enough, only modified the first step of the original scheme, that is, adding a weight coefficient, can preserve the second-order accuracy of the method. This paper combines Laplace transformation and convolution to analyze the convergence of the weighted $C-N$ modified scheme. Finally, the numerical test verifies the effectiveness of the method.
Keywords: time fractional diffusion equation ~weighted $C-N$ scheme ~convolution ~$Laplace$~transform
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