Multiscale temporal integrators for fluctuating hydrodynamics

Following on our previous work [S. Delong, B. E. Griffith, E. Vanden-Eijnden, and A. Donev, Phys. Rev. E 87, 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 time scales. 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 wave numbers.

Figure 1

(Left) Time evolution of the static structure factor $S(k;t)$ in the GRADFLEX experiment [6]. The steady-state spectrum, studied in more detail in Sec. 5 in [35], is also shown and compared to the simple quasiperiodic theory (65). At smaller wave numbers the steady-state fluctuations are damped due to confinement effects. (Right) Static structure factor $S\left(k\right)$ of concentration fluctuations in a binary liquid mixture subjected to a steady temperature gradient [34]. In order to account for spatial discretization artifacts, on the $x$ axes we show the modified wave number $sin\left(k\Delta x/2\right)/\left(\Delta x/2\right)$ [35]. The theory (65) is shown for comparison.

Figure 2

Time correlation functions $S(k,t)$ of concentration fluctuations for concentration fluctuations in a binary liquid mixture subjected to a temperature gradient [34], for the first few wave numbers (listed in the legend in ${\text{cm}}^{-1})$. Solid lines are obtained using the inertial equations, while dashed lines of the same color are obtained using the overdamped approximation. For the smallest two wave numbers clear oscillations are observed, indicating the appearance of propagative modes. These are not captured correctly in the overdamped approximation and are only qualitatively described by the simple theory (69), shown with dash-dotted lines.