  The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal
fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This
paper examines explicit Eulerian discretizations of the full LLNS equations.
Several CFD approaches are considered (including MacCormack's two-step
Lax-Wendroff scheme and the Piecewise Parabolic Method) and are found to give
good results (about 10% error) for the variances of momentum and energy
fluctuations. However, neither of these schemes accurately reproduces the
density fluctuations. We introduce a conservative centered scheme with a
third-order Runge-Kutta temporal integrator that does accurately produce
density fluctuations. A variety of numerical tests, including the random walk
of a standing shock wave, are considered and results from the stochastic LLNS
PDE solver are compared with theory, when available, and with molecular
simulations using a Direct Simulation Monte Carlo (DSMC) algorithm.
