At small scales, the Navier–Stokes equations traditionally used for modeling fluid flow 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 adds 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 PDEs with stochastic flux terms.
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 based on studying discrete equilibrium structure factors (spectra) as a function of wavenumber and frequency. Within this framework we then discuss the construction of explicit discretizations for miscible fluid mixtures. Finally, we present results on giant fluctuations in diffusively-mixing fluids and discuss the role of fluctuations in the construction of hybrid atomistic/continuum algorithms.