  We develop a Fluctuating Immersed Boundary (FIB) method for performing
Brownian dynamics simulations of confined particle suspensions. Unlike
traditional methods which employ analytical Green's functions for Stokes flow
in the confined geometry, the FIB method uses a fluctuating finite-volume
Stokes solver to generate the action of the response functions "on the fly".
Importantly, we demonstrate that both the deterministic terms necessary to
capture the hydrodynamic interactions among the suspended particles, as well as
the stochastic terms necessary to generate the hydrodynamically-correlated
Brownian motion, can be generated by solving the steady Stokes equations
numerically only once per time step. This is accomplished by including a
stochastic contribution to the stress tensor in the fluid equations consistent
with fluctuating hydrodynamics. We develop novel temporal integrators that
account for the multiplicative nature of the noise in the equations of Brownian
dynamics and the strong dependence of the mobility on the configuration for
confined systems. Notably, we propose a random finite difference approach to
approximating the stochastic drift proportional to the divergence of the
configuration-dependent mobility matrix. Through comparisons with analytical
and existing computational results, we numerically demonstrate the ability of
the FIB method to accurately capture both the static (equilibrium) and dynamic
properties of interacting particles in flow.
