  We develop efficient numerical methods for performing many-body Brownian
dynamics simulations of a recently-observed fingering instability in an active
suspension of colloidal rollers sedimented above a wall [M. Driscoll, B.
Delmotte, M. Youssef, S. Sacanna, A. Donev and P. Chaikin, Nature Physics,
2016, doi:10.1038/nphys3970]. We present a stochastic Adams-Bashforth
integrator for the equations of Brownian dynamics, which has the same cost as
but is more accurate than the widely-used Euler-Maruyama scheme, and uses a
random finite difference to capture the stochastic drift proportional to the
divergence of the configuration-dependent mobility matrix. We generate the
Brownian increments using a Krylov method, and show that for particles confined
to remain in the vicinity of a no-slip wall by gravity or active flows the
number of iterations is independent of the number of particles. Our numerical
experiments with active rollers show that the thermal fluctuations set the
characteristic height of the colloids above the wall, both in the initial
condition and the subsequent evolution dominated by active flows. The
characteristic height in turn controls the timescale and wavelength for the
development of the fingering instability.
