An improvement of the Stokesian Dynamics method for many-particle systems is
presented. A direct calculation of the hydrodynamic interaction is used rather than
imposing periodic boundary conditions. The two major difficulties concern the
accuracy and the speed of calculations. The accuracy discussed in this work is not
concerned with the lubrication correction but, rather, focuses on the multipole
expansion which until now has only been formulated up to the so-called FTS version or the
first order of force moments. This is improved systematically by a real-space multipole
expansion with force moments and velocity moments evaluated at the centre of the
particles, where the velocity moments are calculated through the velocity derivatives;
the introduction of the velocity derivatives makes the formulation and its extensions
straightforward. The reduction of the moments into irreducible form is achieved by
the Cartesian irreducible tensor. The reduction is essential to form a well-defined
linear set of equations as a generalized mobility problem. The order of truncation is not
limited in principle, and explicit calculations of two-body problems are shown with
order up to 7. The calculating speed is improved by a conjugate-gradient-type iterative
method which consists of a dot-product between the generalized mobility matrix and
the force moments as a trial value in each iteration. This provides an O(N2) scheme
where N is the number of particles in the system. Further improvement is achieved
by the fast multipole method for the calculation of the generalized mobility problem
in each iteration, and an O(N) scheme for the non-adaptive version is obtained. Real
problems are studied on systems with N = 400 000 particles. For mobility problems
the number of iterations is constant and an O(N) performance is achieved; however
for resistance problems the number of iterations increases as almost N1/2 with a high
accuracy of 10−6 and the total cost seems to be O(N3/2).