分数阶偏微分方程与近场动力学等非局部模型的高保真快速算法与数值分析

Fractional partial differential equations and related nonlocal models provide more appropriate and accurate description for problems with nonlocal behavior such as anomalous diffusion than traditional integer-order partial differential equations do. However, numerical methods for nonlocal models generate dense matrices, and so require O(N2) of memory and (at each time step) O(N3) of computational cost, where N is the number of spatial unknowns. This makes their realistic three-dimensional applications virtually impossible. 1. We were the first internationally to derive fast high-fidelity numerical methods for (one- and multidimensional) space-fractional partial differential equations and related nonlocal models discretized on uniform spatial partitions, which reduce the memory requirement from O(N2) to O(N) and computational cost from O(N3) to O(N log2 N). Three dimensional numerical experiments showed that our fast method reduced the CPU time from almost three months to below 6 seconds while retaining the accuracy of the underlying method. The first objective of this proposal is to derive fast high-fidelity numerical methods for nonlocal problems on more general spatial meshes. 2. We recently showed that in the context of fractional differential equations, the regularity of coefficients and right-hand side (plus the smoothness of the physical domain for multidimensional problems) does not guarantee the regularity of the true solution to the problem. Hence, the high-order numerical methods developed and the high-order convergence rates proved in the literature under the assumption of the sufficiently high regularity of the true solution of the fractional differential equations are unfounded and cannot be guaranteed mathematically by any means. The second objective of this proposal is to develop high-order numerical methods and prove corresponding high-order convergence estimates without any assumption on the true solution of the problem but only in terms of the given data of the problem.

分数阶偏微分方程等非局部模型对反常扩散等兼有非局部特性的问题提供了比整数阶方程更为准确合理的描述。但它们的数值格式会产生稠密矩阵，传统方式求解要求O(N2)的存储量与（每一时间步）O(N3)的计算量，N是空间未知量个数。这使得它们的实际三维应用不可行。 我们通过深入研究数值格式的结构针对一致剖分的情况首次在国际上建立了求解（一维和多维）分数阶方程等非局部问题的高保真快速算法，其存储量为O(N)，计算量为O(Nlog2N)。三维算例显示计算时间从近3个月改进为6秒并保持精度。本项目的研究目标是建立更一般空间网格上非局部问题的高保真快速算法。 我们的研究显示系数和右端（多维问题还包括区域）的正则性不能保证分数阶方程解的正则性。因此，文献上假定精确解光滑所得到的误差分析及高阶算法无法保证。本项目的第二个目标是在不假定解的正则性的条件下建立分数阶方程高阶数值方法及证明相应高阶误收敛性。

在项目执行期间，我们针对分数阶偏微分方程开展了数学分析，并对一类问题的适定性进行了深入研究。我们还对空间分数阶方程以及时间空间分数阶的数值格式研发了快速计算方法。我们也针对相关的非局部模型开展了数学分析、数值分析、以及快速方法方面的研究，并取得了一系列的研究成果

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