Fast Ewald summation for Green's functions of Stokes flow in a half-space
This work provides an efficient algorithm for simulating Stokes flow in half-space geometries, which is important for microfluidics and biological fluid dynamics, but is an incremental extension of existing free-space methods.
The authors present a fast Ewald summation method for Green's functions of Stokes flow in a half-space, achieving O(N log N) complexity and demonstrating computational savings compared to direct summation, with speedups of up to 100x for large N.
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.