一种快速的半离散Kansa方法用于求解二维时空分数扩散
A fast semi-discrete Kansa method to solve the two-dimensional spatiotemporal fractional diffusion
AI总结：
本文提出一种快速半离散Kansa方法，用于求解二维时空分数扩散方程，通过分析节点分布和数量对精度和收敛率的影响，提高了计算效率。
摘要：
异常扩散现象在地下水溶质迁移、土壤水分入渗和沉积物运动等过程中普遍存在。时间与空间分数导数对流-扩散方程（FADE）已被广泛用于表征上述异常扩散过程。然而，将时间与空间FADE模型应用于描述现实世界质量传输过程时，其在长时间范围和大不规则域情况下的计算效率较低。本文提出了一种新算法，利用Kansa方法对向量空间分数导数项进行离散化，然后对得到的时间分数普通系统采用解析方法。通过一维和二维案例中的数值例子，分析了节点分布模式和节点数量对精度和收敛率的影响。为了测试所提方法的应用潜力，我们提供了连续和离散情况下二维时间与空间FADE模型的数值结果。结果显示，与以往耗时的粒子蒙特卡罗方法相比，所提方法能很好地模拟异质和各向异性的介质中的溶质羽流。
英文摘要：
Anomalous diffusion is a common phenomenon observed in underground solute transport, soil water infiltration and sediment movement, etc. Time and space fractional derivative advection-dispersion equation (FADE) has been widely employed as the governing equation to characterize above mentioned anomalous diffusion related processes. However, a main problem in application of time and space FADE model to describe the real-world mass transport processes, is its low computation efficiency for long-time range and large irregular domain cases. This study offers a new algorithm in which the Kansa method is used for vector space fractional derivative term discretization and then analytical approach for resulted time fractional ordinary system. The influence of node distribution mode and node numbers on accuracy and convergence rate are analysed through the numerical examples in one and two dimensional cases. To test the application potentials of present methods, we offer the numerical results of two dimensional time and space FADE models in continuous and discrete cases. It shows that the solute plumes in heterogeneous and anisotropic media, can be well simulated by using present method compared with the previous time-consumed particle Monte-Carlo methods.
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