Temporal Integrators for Fluctuating Hydrodynamics

TL;DR

This paper develops finite-volume spatial discretizations and implicit-explicit predictor-corrector temporal integrators for fluctuating hydrodynamics, ensuring the preservation of fluctuation-dissipation balance and accurate long-time statistical behavior.

Contribution

It introduces new stochastic Runge-Kutta schemes that are weakly second-order accurate and robust for large time steps in fluctuating hydrodynamics simulations.

Findings

Discretizations obey fluctuation-dissipation balance.

Schemes accurately reproduce equilibrium distribution.

Midpoint scheme is highly robust across time steps.

Abstract

Including the effect of thermal fluctuations in traditional computational fluid dynamics requires developing numerical techniques for solving the stochastic partial differential equations of fluctuating hydrodynamics. These Langevin equations possess a special fluctuation-dissipation structure that needs to be preserved by spatio-temporal discretizations in order for the computed solution to reproduce the correct long-time behavior. In particular, numerical solutions should approximate the Gibbs-Boltzmann equilibrium distribution, and ideally this will hold even for large time step sizes. We describe finite-volume spatial discretizations for the fluctuating Burgers and fluctuating incompressible Navier-Stokes equations that obey a discrete fluctuation-dissipation balance principle just like the continuum equations. We develop implicit-explicit predictor-corrector temporal integrators for the resulting stochastic method-of-lines discretization. These stochastic Runge-Kutta schemes treat diffusion implicitly and advection explicitly, are weakly second-order accurate for additive noise for small time steps, and give a good approximation to the equilibrium distribution even for very strong fluctuations. Numerical results demonstrate that a midpoint predictor-corrector scheme is very robust over a broad range of time step sizes.