A hierarchical matrix approach for computing hydrodynamic interactions

Abstract

For simulations of large numbers of small, spherical particles in a Stokes flow, the long-range hydrodynamic interactions approximated by the Rotne–Prager–Yamakawa (RPY) kernel can be summed rapidly using, for example, the fast multipole method (FMM) or the particle-mesh Ewald (PME) method. In this paper, we develop new fast methods for computing these sums using the H2 hierarchical matrix representation, for open and for periodic boundary conditions. To the best of our knowledge, the method for infinite periodic sums using the H2 hierarchical matrix representation is the first such method developed. We also consider a more general RPY kernel that handles polydisperse particle radii, and show analytically and experimentally that the proxy surface method for efficiently constructing the H2 hierarchical matrix representation remains effective in this case. Numerical tests demonstrate the well-controlled accuracy of the H2 summation methods and their linear-scaling computation and storage cost. We find that the H2 matrix approach has lower cost for computing the summations compared to FMM and PME, but higher precomputation cost (required for each particle configuration). This precomputation cost can be amortized over several summations when computing Brownian displacements or forces in Brownian and Stokesian dynamics simulations with very large numbers of particles.