Wavelets and fast summations for particle simulations of gravitational flows of miscible drops

Abstract

For sedimentation of miscible drops through quiescent liquid of the same viscosity, a recent paper [J. Fluid. Mech. 447 (2001) 299] has shown the effectiveness of computer simulations based upon a swarm of point forces in tracking coalescence, mixing and rupture. Robustness of the approach was offset by the slow O( N p 2) summations needed to calculate the mutual viscous interactions between all N p particles. Motivated by applications of wavelets to linear operators, this paper develops a conceptually simple scheme for dramatically accelerating the simulations. After lumping the particles together into N local clusters (three-dimensional “pixels”), a Haar discrete wavelet transform (DWT) is used to “compress” the “bitmapped” six-dimensional image of pixel–pixel hydrodynamic interactions. Depending upon the criterion of accuracy, the numerically observed scaling of the operation count seems to be either O( N) or O[N( log N) α] . The DWT also works without modification for hydrodynamic wall effects, where the kernel is not purely translational and therefore fast convolutions (FFT) cannot be used.