  Including the effect of thermal fluctuations in traditional computational
fluid dynamics requires developing numerical techniques for solving the
stochastic partial differential equations of fluctuating hydrodynamics. These
Langevin equations possess a special fluctuation-dissipation structure that
needs to be preserved by spatio-temporal discretizations in order for the
computed solution to reproduce the correct long-time behavior. In particular,
numerical solutions should approximate the Gibbs-Boltzmann equilibrium
distribution, and ideally this will hold even for large time step sizes. We
describe finite-volume spatial discretizations for the fluctuating Burgers and
fluctuating incompressible Navier-Stokes equations that obey a discrete
fluctuation-dissipation balance principle just like the continuum equations. We
develop implicit-explicit predictor-corrector temporal integrators for the
resulting stochastic method-of-lines discretization. These stochastic
Runge-Kutta schemes treat diffusion implicitly and advection explicitly, are
weakly second-order accurate for additive noise for small time steps, and give
a good approximation to the equilibrium distribution even for very strong
fluctuations. Numerical results demonstrate that a midpoint predictor-corrector
scheme is very robust over a broad range of time step sizes.
