Temporal Integrators for Langevin Equations with Applications to Fluctuating Hydrodynamics and Brownian Dynamics

Thermal fluctuations play a significant role in the dynamics of small scale hydrodynamics systems. These fluctuations can be modeled through the addition of stochastic forcing terms resulting in a system of Langevin equations which require specialized stochastic temporal integrators. We develop a general family of such schemes by adding random forcing to standard predictor corrector solvers in a way that gives second order weak accuracy for the commonly used linearized approximation to the original dynamics. We also construct predictor-corrector methods for integrating the overdamped limit of systems of equations with a fast and slow variable in the limit of infinite separation of the fast and slow timescales. The multiplicative noise in the overdamped equations gives rise to a thermal drift term which we handle using a random finite difference method in a computationaly efficient way. We further develop specialized temporal integrators for the equations of Brownian Dynamics, using a Stokes solver combined with immersed boundary techniques to compute hydrodynamic interactions. This approach is extended to describe the dynamics of immersed rigid bodies by employing normalized quaternions to represent orientation. We test our schemes by performing several numerical experiments and comparing to analytic or existing computational results.