An O(N^2) Approximation for Hydrodynamic Interactions in Brownian   Dynamics Simulations
  In the Ermak-McCammon algorithm for Brownian Dynamics, the hydrodynamic interactions (HI) between N spherical particles are described by a 3N x 3N diffusion tensor. This tensor has to be factorized at each timestep with a runtime of O(N^3), making the calculation of the correlated random displacements the bottleneck for many-particle simulations. Here we present a faster algorithm for this step, which is based on a truncated expansion of the hydrodynamic multi-particle correlations as two-body contributions. The comparison to the exact algorithm and to the Chebyshev approximation of Fixman verifies that for bead-spring polymers this approximation yields about 95% of the hydrodynamic correlations at an improved runtime scaling of O(N^2) and a reduced memory footprint. The approximation is independent of the actual form of the hydrodynamic tensor and can be applied to arbitrary particle configurations. This now allows to include HI into large many-particle Brownian dynamics simulations, where until now the runtime scaling of the correlated random motion was prohibitive. 
Tihamer Geyer and Uwe Winter