求解含谱分数阶拉普拉斯算子的时空分数阶扩散问题的双快速算法
Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian
AI总结：
针对含谱分数阶拉普拉斯算子的高维时空分数阶扩散问题，提出结合空间半离散格式与快速时间步进L1格式的双快速算法，证明其稳定性与最优时间收敛阶，并通过数值算例验证有效性。
摘要：
本文提出一种高效简洁的双快速算法，用于求解含谱分数阶拉普拉斯算子的高维时空分数阶扩散问题。我们首先建立时空分数阶扩散方程的半离散格式，该格式采用线性有限元或四阶紧致差分方法，结合矩阵传递技术来逼近谱分数阶拉普拉斯算子。随后，我们引入一种用于时间离散的快速时间步进L1格式。所提格式可精确计算矩阵的分数次幂，并利用离散正弦变换在每个时间层执行矩阵-向量乘法，无需借助任何迭代方法，可显著降低计算成本与内存需求。此外，我们针对分级时间网格上基于快速时间步进L1格式的全离散格式进行了稳定性与收敛性分析。分析表明，当分级网格因子取$ω=(2-α)/α$时，可获得最优时间收敛阶$\mathcal{O}(N^{-(2-α)})$，其中$N$为时间网格数。最后，我们通过大量数值算例验证了理论分析结果以及所提格式的有效性。
英文摘要：
This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our analysis shows that the choice of graded mesh factor $ω=(2-α)/α$ shall give an optimal temporal convergence $\mathcal{O}(N^{-(2-α)})$ with $N$ denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme.
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