At small scales, the Navier-Stokes equations traditionally used for fuid modeling break down and thermal fluctuations play an important role in the dynamics. Landau and Lifshitz proposed a modified version of the Navier-Stokes equations, referred to as the fluctuating Navier-Stokes equations (FNS) that incorporates stochastic flux terms designed to incorporate the effect of fluctuations. These stochastic fluxes are constructed so that the FNS equations are consistent with equilibrium fluctuations from statistical mechanics. Here we describe the development and analysis of finite-volume methods for solving the equations of fluctuating hydrodynamics as well as some simplified model problems. A key element in the construction of the numerical methods is designing discretizations that satisfy a discrete fluctuation-dissipation principle. We introduce a systematic approach to characterize the behavior of these types of methods based on studying discrete equilibrium structure factors (spectra) as a function of wavenumber and frequency. Finally, we present numerical results illustrating the impact of fluctuations in non-equilibrium settings.
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