Staggered Schemes for Fluctuating Hydrodynamics

TL;DR

This paper introduces second-order accurate numerical schemes for fluctuating hydrodynamics that preserve fluctuation-dissipation balance and accurately reproduce equilibrium covariances, validated against experimental data.

Contribution

The authors develop and verify second-order spatial and temporal discretizations for fluctuating hydrodynamics on staggered grids, ensuring correct equilibrium covariances and applicability to experimental fluctuation data.

Findings

Numerical schemes reproduce equilibrium covariances to second and third order.

Simulations match experimental measurements of giant fluctuations.

Results show good agreement between compressible and incompressible models.

Abstract

We develop numerical schemes for solving the isothermal compressible and incompressible equations of fluctuating hydrodynamics on a grid with staggered momenta. We develop a second-order accurate spatial discretization of the diffusive, advective and stochastic fluxes that satisfies a discrete fluctuation-dissipation balance, and construct temporal discretizations that are at least second-order accurate in time deterministically and in a weak sense. Specifically, the methods reproduce the correct equilibrium covariances of the fluctuating fields to third (compressible) and second (incompressible) order in the time step, as we verify numerically. We apply our techniques to model recent experimental measurements of giant fluctuations in diffusively mixing fluids in a micro-gravity environment [A. Vailati et. al., Nature Communications 2:290, 2011]. Numerical results for the static spectrum of non-equilibrium concentration fluctuations are in excellent agreement between the compressible and incompressible simulations, and in good agreement with experimental results for all measured wavenumbers.