Multiscale temporal integrators for fluctuating hydrodynamics

TL;DR

This paper introduces new multiscale temporal integrators for fluctuating hydrodynamics, combining predictor-corrector schemes with stochastic terms to accurately simulate complex fluid behaviors across different regimes.

Contribution

It develops second-order accurate predictor-corrector integrators for Langevin equations in fluctuating hydrodynamics, including explicit and implicit schemes, and applies them to nonequilibrium fluctuation phenomena.

Findings

Successfully simulated giant concentration fluctuations in microgravity.

Demonstrated the importance of inertia effects at small wavenumbers.

Validated integrators against experimental data.

Abstract

Following on our previous work [S. Delong and B. E. Griffith and E. Vanden-Eijnden and A. Donev, Phys. Rev. E, 87(3):033302, 2013], we develop temporal integrators for solving Langevin stochastic differential equations that arise in fluctuating hydrodynamics. Our simple predictor-corrector schemes add fluctuations to standard second-order deterministic solvers in a way that maintains second-order weak accuracy for linearized fluctuating hydrodynamics. We construct a general class of schemes and recommend two specific schemes: an explicit midpoint method, and an implicit trapezoidal method. We also construct predictor-corrector methods for integrating the overdamped limit of systems of equations with a fast and slow variable in the limit of infinite separation of the fast and slow timescales. We propose using random finite differences to approximate some of the stochastic drift terms that arise because of the kinetic multiplicative noise in the limiting dynamics. We illustrate our integrators on two applications involving the development of giant nonequilibrium concentration fluctuations in diffusively-mixing fluids. We first study the development of giant fluctuations in recent experiments performed in microgravity using an overdamped integrator. We then include the effects of gravity, and find that we also need to include the effects of fluid inertia, which affects the dynamics of the concentration fluctuations greatly at small wavenumbers.