论文标题
本质周期性的快速多极法：近似最佳绿色功能
Natively Periodic Fast Multipole Method: Approximating the Optimal Green Function
论文作者
论文摘要
快速多极方法（FMM）使用周期性的边界条件“本地”，如果它使用周期性的绿色函数来计算每个FMM OCT-Tree节点相互作用区域中的多极扩展。可以为这种方法定义“最佳”绿色函数，该方法导致数值解决方案，该溶液在多数极端高阶的极限下收敛到等效的粒子网溶液。可以得出最佳绿色函数的离散功能方程，但实际上并不有用，因为尚不清楚其解决方案的方法。取而代之的是，本文为最佳绿色函数提供了一个近似值，该函数在LMAX NORM中准确地比1E-3更好，而L2 Norm中的1E-4对于实际有用的多极计数。这样的近似最佳绿色功能提供了一种实用的方法，可以“本地”实现定期边界条件，而无需计算晶格总和或依靠混合FMM-PM方法。
The Fast Multipole Method (FMM) obeys periodic boundary conditions "natively" if it uses a periodic Green function for computing the multipole expansion in the interaction zone of each FMM oct-tree node. One can define the "optimal" Green function for such a method that results in the numerical solution that converges to the equivalent Particle-Mesh solution in the limit of sufficiently high order of multipoles. A discrete functional equation for the optimal Green function can be derived, but is not practically useful as methods for its solution are not known. Instead, this paper presents an approximation for the optimal Green function that is accurate to better than 1e-3 in LMAX norm and 1e-4 in L2 norm for practically useful multipole counts. Such an approximately optimal Green function offers a practical way for implementing FMM with periodic boundary conditions "natively", without the need to compute lattice sums or to rely on hybrid FMM-PM approaches.
