A direct O(N log2 N) finite difference method for fractional diffusion equations
A direct O(N log2 N) finite difference method for fractional diffusion equations
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DOI：
10.1016/j.jcp.2010.07.011
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发表时间：
2010-10
期刊：
影响因子：
--
通讯作者：
Hong Wang;Kaixin Wang;Treena Sircar
中科院分区：
文献类型：
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作者：
Hong Wang;Kaixin Wang;Treena Sircar
分数扩散方程对表现出无法通过二阶扩散方程精确建模的异常扩散的现象进行建模。由于分数阶微分算子的非局部性质，数值方法具有完整的系数矩阵，需要 O(N2) 的存储和 O(N3) 的计算成本，其中 N 是网格点的数量。在本文中，我们开发了一种用于分数扩散方程的快速有限差分方法，该方法仅需要 O(N) 的存储和 O(Nlog2N) 的计算成本，同时保留与常规有限差分方法相同的精度和近似性质。数值实验显示了该方法的实用性。
Fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods have full coefficient matrices which require storage of O(N2) and computational cost of O(N3) where N is the number of grid points. In this paper we develop a fast finite difference method for fractional diffusion equations, which only requires storage of O(N) and computational cost of O(Nlog2N) while retaining the same accuracy and approximation property as the regular finite difference method. Numerical experiments are presented to show the utility of the method.