  The efficient computation of the overdamped, random motion of micron and
nanometre scale particles in a viscous fluid requires novel methods to obtain
the hydrodynamic interactions, random displacements and Brownian drift at
minimal cost. Capturing Brownian drift is done most efficiently through a
judiciously constructed time-integration scheme that automatically accounts for
its contribution to particle motion. In this paper, we present a generalised
drift-correcting (gDC) scheme that accounts for Brownian drift for suspensions
of rigid particles with arbitrary shape. The scheme seamlessly integrates with
fast methods for computing the hydrodynamic interactions and random increments
and requires a single full mobility solve per time-step. As a result, the gDC
provides increased computational efficiency when used in conjunction with
grid-based methods that employ fluctuating hydrodynamics to obtain the random
increments. Further, for these methods the additional computations that the
scheme requires occur at the level of individual particles, and hence lend
themselves naturally to parallel computation. We perform a series of
simulations that demonstrate the gDC obtains similar levels of accuracy as
compared with the existing state-of-the-art. In addition, these simulations
illustrate the gDC's applicability to a wide array of relevant problems
involving Brownian suspensions of non-spherical particles, such as the
structure of liquid crystals and the rheology of complex fluids.
