A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation
A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation
复制标题
三维时变空间分数扩散方程的快速有限差分法及其高效实现
DOI：
10.1016/j.jcp.2013.06.040
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发表时间：
2013-11
影响因子：
4.1
通讯作者：
Ning Du
中科院分区：
文献类型：
--
作者：
Hong Wang;Ning Du
分数阶扩散方程模拟了二阶扩散方程无法精确模拟的反常扩散现象。由于分数阶微分算子的非局部性质，空间分数阶扩散方程的数值方法产生了结构复杂的稠密甚至全系数矩阵。传统上，这些方法都是用高斯消去法求解的，这需要每时间步O（N3）的计算工作和O（N2）的内存来存储，其中N是离散化中空间网格点的数量。这些方法的显著计算工作量和存储要求使得三维空间分数阶扩散方程的数值模拟在计算上过于昂贵。本文通过对有限差分法系数矩阵的结构分析，将系数矩阵精细地分解为稀疏矩阵和结构化稠密矩阵的组合，发展了三维空间中含时分数阶扩散方程隐式有限差分离散的一种高效、忠实的求解方法.快速方法具有每次迭代的O（N log N）的计算工作量和O（N）的内存需求，同时保持与高斯消去法求解的基本有限差分法相同的精度。三维空间分数阶扩散方程的数值实验表明了快速方法的有效性。
Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the non-local property of fractional differential operators, numerical methods for space-fractional diffusion equations generate dense or even full coefficient matrices with complicated structures. Traditionally, these methods were solved with Gaussian elimination, which requires computational work of O (N 3) per time step and O (N 2) of memory to store where N is the number of spatial grid points in the discretization. The significant computational work and memory requirement of these methods makes a numerical simulation of three-dimensional space-fractional diffusion equations computationally prohibitively expensive. In this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time-dependent space-fractional diffusion equations in three space dimensions, by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices. The fast method has a computational work count of O (N log N) per iteration and a memory requirement of O (N), while retaining the same accuracy as the underlying finite difference method solved with Gaussian elimination. Numerical experiments of a three-dimensional space-fractional diffusion equation show the utility of the fast method.
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影响因子：
3.7
作者：
X. Li;Chuanju Xu;许传炬
通讯作者：
X. Li;Chuanju Xu;许传炬
DOI：
10.2307/2003862
发表时间：
1961-11
期刊：
--
影响因子：
--
作者：
R. Varga;J. Gillis
通讯作者：
R. Varga;J. Gillis
DOI：
10.1016/j.jcp.2009.02.011
发表时间：
2009-06
期刊：
J. Comput. Phys.
影响因子：
--
作者：
E. Sousa
通讯作者：
E. Sousa
影响因子：
1.9
作者：
Langlands, T. A. M.;Henry, B. I.;Wearne, S. L.
通讯作者：
Wearne, S. L.
影响因子：
3.9
作者：
Ervin, Vincent J.;Roop, John Paul
通讯作者：
Roop, John Paul