Numerical Methods for Fluctuating Hydrodynamics
Aleksandar Donev, CIMS

Joint work with John Bell (Lawrence Berkeley National Labs), Alejandro Garcia (San Jose State University), Berni Alder (Lawrence Livermore National Labs), Eric Vanden-Eijnden and Jonathan Goodman (Courant Institute)

I will describe our recent and ongoing research focused on fluid mechanics in regimes where thermal fluctuations are important. Notable examples include flows at micro and nano scales typical of new microfluidic, nanofluidic and microelectromechanical devices; biological systems such as lipid membranes, Brownian molecular motors, nanopores; as well as processes where the effect of fluctuations is amplified by strong non-equilibrium effects, such as combustion of lean flames, capillary dynamics, hydrodynamic instabilities, and others. Computational issues at play include coarse-graining to bridge the large gap in timescales and length scales, coupling between disparate methods such as molecular dynamics and Navier-Stokes solvers, the inclusion of thermal fluctuations, and others.

I will first review our work on developing coarse-grained stochastic particle models that build upon the Direct Simulation Monte Carlo (DSMC) method, and is also related to the dissipative particle dynamics (DPD) and the multi-particle collision (also called stochastic rotation) dynamics techniques. I will then consider the Landau-Lifshitz Navier-Stokes (LLNS) equations, which incorporate thermal fluctuations into the traditional compressible Navier-Stokes-Fourier system by the addition of white-noise fluxes whose magnitudes are set by a fluctuation-dissipation relation. I will describe the development and analysis of finite-volume methods for solving the equations of fluctuating hydrodynamics and related stochastic partial differential equations. Finally, I will describe a hybrid particle-continuum method that employs bidirectional dynamic coupling between a stochastic particle fluid and a fluctuating continuum. Through several examples I will demonstrate that thermal fluctuations have to be consistently included in the continuum component of hybrid calculations in order not to distort the thermal equilibrium in the particle solver.

I will conclude with a look into the challenges of developing a simulation methodology capable of simulating macroscopic flows of complex fluids with atomistic fidelity.