Fast Ewald summation efficiently evaluates Coulomb interactions and is widely used in molecular dynamics simulations. It is based on a split into a short-range and a long-range part, where evaluation of the latter is accelerated using the fast Fourier transform (FFT). The accuracy and computational cost depend critically on the mollifier in the kernel split and the window function used in the spreading and interpolation steps that enable the use of the FFT. The first prolate spheroidal wavefunction (PSWF) has optimal concentration in real and Fourier space simultaneously, and is used when defining both a mollifier and a window function. We provide a complete description of the method and derive rigorous error estimates. In addition, we obtain closed-form approximations of the Fourier truncation and aliasing errors, yielding explicit parameter choices for the achieved error to closely match the prescribed tolerance. Numerical experiments confirm the analysis: PSWF-based Ewald summation achieves a given accuracy with significantly fewer Fourier modes and smaller window supports than Gaussian- and B-spline-based approaches, providing a superior alternative to existing Ewald methods for particle simulations.
Fast Ewald Summation with Prolates for Charged Systems in the NPT Ensemble
Center for Computational Mathematics, Flatiron Institute（西蒙斯基金会平顿研究所计算数学中心）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
Accelerating Molecular Dynamics Simulations using Fast Ewald Summation with Prolates
Center for Computational Mathematics, Flatiron Institute（计算数学中心，Flatiron研究所）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
Ball Prolate Spheroidal Wave Functions In Arbitrary Dimensions
Central China Normal University（华中师范大学）；Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）；Nanyang Technological University（南洋理工大学）；Beijing Computational Sciences and Research Center（北京计算科学研究中心）；Wayne State University（韦恩州立大学）
Finite difference method in prolate spheroidal coordinates for freely suspended spheroidal particles in linear flows of viscous and viscoelastic fluids
An $\mathcal{O}\left(1\right)$ algorithm for the numerical evaluation of the Sturm-Liouville eigenvalues of the spheroidal wave functions of order zero
