title: Multiscale temporal integrators for fluctuating hydrodynamics.
abstract: 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.
authors.affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA.
authors.forename: Steven
authors.initials: S
authors.lastname: Delong
authors.affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA.
authors.forename: Yifei
authors.initials: Y
authors.lastname: Sun
authors.affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA and Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3250, USA.
authors.forename: Boyce E
authors.initials: BE
authors.lastname: Griffith
authors.affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA.
authors.forename: Eric
authors.initials: E
authors.lastname: Vanden-Eijnden
authors.affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA.
authors.forename: Aleksandar
authors.initials: A
authors.lastname: Donev