三维分数阶偏微分方程的高效数值算法
Efficient numerical algorithms for three-dimensional fractional partial differential equations
AI总结：
本文提出局部一维数值方法求解三维分数阶偏微分方程，采用二阶有限差分和Crank-Nicolson格式，证明无条件稳定且时空二阶收敛，并针对Riesz分数阶扩散方程通过降低分裂误差改进算法。
摘要：
本文详细讨论了局部一维数值方法，用于高效求解三维分数阶偏微分方程，包括分数阶对流扩散方程和Riesz分数阶扩散方程。采用二阶有限差分格式离散空间分数阶导数，Crank-Nicolson格式处理时间导数。我们从理论上证明并数值验证了所提出的数值方法在空间和时间方向上均无条件稳定且具有二阶收敛性。特别地，对于Riesz分数阶扩散方程，利用降低分裂误差的思想进一步改进算法，并严格证明了改进格式的无条件稳定性和收敛性，同时进行了数值验证。
英文摘要：
This paper detailedly discusses the locally one-dimensional numerical methods for efficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional diffusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
