Finite difference methods for the time fractional diffusion equation on non-uniform meshes
Finite difference methods for the time fractional diffusion equation on non-uniform meshes
复制标题
非均匀网格上时间分数扩散方程的有限差分法
DOI：
10.1016/j.jcp.2014.02.008
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发表时间：
2014-05-15
影响因子：
4.1
通讯作者：
Liao, Hong-lin
中科院分区：
文献类型：
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作者：
Zhang, Ya-nan;Sun, Zhi-zhong;Liao, Hong-lin
由于分数阶导数是具有弱奇异核的积分，在均匀网格上离散会导致精度降低。研究了非均匀网格上Caputo导数的有限差分逼近。将该方法应用于求解分数阶扩散方程，得到了一个半离散格式。证明了算法的无条件稳定性和H-1模收敛性。利用紧致差分方法，通过空间离散，构造了一个全离散差分格式。对两类非均匀网格建立了误差估计。数值实验结果与均匀网格下的方法进行了比较，验证了本文方法的有效性。此外，移动局部加密技术的引入，以提高数值解的时间精度。(C)2014 Elsevier Inc. All rights reserved.
Since fractional derivatives are integrals with weakly singular kernel, the discretization on the uniform mesh may lead to poor accuracy. The finite difference approximation of Caputo derivative on non-uniform meshes is investigated in this paper. The method is applied to solve the fractional diffusion equation and a semi-discrete scheme is obtained. The unconditional stability and H-1 norm convergence are proved. A fully discrete difference scheme is constructed with space discretization by compact difference method. The error estimates are established for two kinds of nonuniform meshes. Numerical tests are carried out to support the theoretical results and comparing with the method on uniform grid shows the efficiency of our methods. Moreover, a moving local refinement technique is introduced to improve the temporal accuracy of numerical solution. (C) 2014 Elsevier Inc. All rights reserved.