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

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(Np2) summations needed to calculate the mutual viscous interactions between all Np 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.