  In this work it is shown how the immersed boundary method of (Peskin2002) for
modeling flexible structures immersed in a fluid can be extended to include
thermal fluctuations. A stochastic numerical method is proposed which deals
with stiffness in the system of equations by handling systematically the
statistical contributions of the fastest dynamics of the fluid and immersed
structures over long time steps. An important feature of the numerical method
is that time steps can be taken in which the degrees of freedom of the fluid
are completely underresolved, partially resolved, or fully resolved while
retaining a good level of accuracy. Error estimates in each of these regimes
are given for the method. A number of theoretical and numerical checks are
furthermore performed to assess its physical fidelity. For a conservative
force, the method is found to simulate particles with the correct Boltzmann
equilibrium statistics. It is shown in three dimensions that the diffusion of
immersed particles simulated with the method has the correct scaling in the
physical parameters. The method is also shown to reproduce a well-known
hydrodynamic effect of a Brownian particle in which the velocity
autocorrelation function exhibits an algebraic tau^(-3/2) decay for long times.
A few preliminary results are presented for more complex systems which
demonstrate some potential application areas of the method.
