The Landau-Lifshitz Navier-Stokes ͑LLNS͒ equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This paper examines explicit Eulerian discretizations of the full LLNS equations. Several computational fluid dynamics approaches are considered ͑including MacCormack's two-step Lax-Wendroff scheme and the piecewise parabolic method͒ and are found to give good results for the variance of momentum fluctuations. However, neither of these schemes accurately reproduces the fluctuations in energy or density. We introduce a conservative centered scheme with a third-order Runge-Kutta temporal integrator that does accurately produce fluctuations in density, energy, and momentum. A variety of numerical tests, including the random walk of a standing shock wave, are considered and results from the stochastic LLNS solver are compared with theory, when available, and with molecular simulations using a direct simulation Monte Carlo algorithm.

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Abstract. This paper describes the extension of a recently developed numerical solver for the LandauLifshitz Navier-Stokes (LLNS) equations to binary mixtures in three dimensions. The LLNS equations incorporate thermal fluctuations into macroscopic hydrodynamics by using white-noise fluxes. These stochastic PDEs are more complicated in three dimensions due to the tensorial form of the correlations for the stochastic fluxes and in mixtures due to couplings of energy and concentration fluxes (e.g., Soret effect). We present various numerical tests of systems in and out of equilibrium, including timedependent systems, and demonstrate good agreement with theoretical results and molecular simulation.Mathematics Subject Classification. 35R60, 60H10, 60H35, 82C31, 82C80.