Fast Ewald summation for Green's functions of Stokes flow in a half-space

TL;DR

This paper introduces a fast Ewald summation method for Green's functions of Stokes flow in a half-space, combining Ewald decomposition and FFTs to efficiently compute long-range interactions, with demonstrated computational advantages.

Contribution

It extends existing free-space Green's function summation techniques to half-space geometries using a novel combination of Ewald decomposition and FFTs.

Findings

The method accelerates the evaluation of half-space Green's functions.

It achieves greater computational savings compared to direct summation.

The approach effectively handles long-range interactions in Stokes flow half-spaces.

Abstract

Recently, Gimbutas et al derived an elegant representation for the Green's functions of Stokes flow in a half-space. We present a fast summation method for sums involving these half-space Green's functions (stokeslets, stresslets and rotlets) that consolidates and builds on the work by Klinteberg et al for the corresponding free-space Green's functions. The fast method is based on two main ingredients: The Ewald decomposition and subsequent use of FFTs. The Ewald decomposition recasts the sum into a sum of two exponentially decaying series: one in real-space (short-range interactions) and one in Fourier-space (long-range interactions) with the convergence of each series controlled by a common parameter. The evaluation of short-range interactions is accelerated by restricting computations to neighbours within a specified distance, while the use of FFTs accelerates the computations in Fourier-space thus accelerating the overall sum. We demonstrate that while the method incurs extra costs for the half-space in comparison to the free-space evaluation, greater computational savings is also achieved when compared to their respective direct sums.