A previously developed hybrid particle-continuum method [J. B. Bell, A. Garcia, and S. A. Williams, Multiscale Model. Simul., 6 (2008), pp. 1256–1280] is generalized to dense fluids and two- and three-dimensional flows. The scheme couples an explicit fluctuating compressible Navier–Stokes solver with the isotropic direct simulation Monte Carlo (DSMC) particle method [A. Donev, A. L. Garcia, and B. J. Alder, J. Stat. Mech. Theory Exp., 2009 (2009), article P11008]. To achieve bidirectional dynamic coupling between the particle (microscale) and continuum (macroscale) regions, the continuum solver provides state-based boundary conditions to the particle subdomain, while the particle solver provides flux-based boundary conditions for the continuum subdomain. This type of coupling ensures both state and flux continuity across the particle-continuum interface analogous to coupling approaches for deterministic parabolic partial differential equations; here, when fluctuations are included, a small (< 1%) mismatch is expected and observed in the mean density and temperature across the interface. By calculating the dynamic structure factor for both a “bulk” (periodic) and a finite system, it is verified that the hybrid algorithm accurately captures the propagation of spontaneous thermal fluctuations across the particle-continuum interface. The equilibrium diffusive (Brownian) motion of a large spherical bead suspended in a particle fluid is examined, demonstrating that the hybrid method correctly reproduces the velocity autocorrelation function of the bead but only if thermal fluctuations are included in the continuum solver. Finally, the hybrid is applied to the well-known adiabatic piston problem, and it is found that the hybrid correctly reproduces the slow nonequilibrium relaxation of the piston toward thermodynamic equilibrium but, again, only if the continuum solver includes stochastic (white-noise) flux terms. These examples clearly demonstrate the need to include fluctuations in continuum solvers employed in hybrid multiscale methods.
Alejandro Garcia, A. Donev, J. B. Bell, and B. Alder. "A hybrid particle-continuum method for hydrodynamics of complex fluids" SIAM Multiscale Modeling and Simulation (2010): 871-911. doi:http://dx.doi.org/10.1137/090774501