  Continuing on our previous work [ArXiv:1212.2644], we develop semi-implicit
numerical methods for solving low Mach number fluctuating hydrodynamic
equations appropriate for modeling diffusive mixing in isothermal mixtures of
fluids with different densities and transport coefficients. We treat viscous
dissipation implicitly using a recently-developed variable-coefficient Stokes
solver [ArXiv:1308.4605]. This allows us to increase the time step size
significantly compared to the earlier explicit temporal integrator. For
viscous-dominated flows, such as flows at small scales, we develop a scheme for
integrating the overdamped limit of the low Mach equations, in which inertia
vanishes and the fluid motion can be described by a steady Stokes equation. We
also describe how to incorporate advanced higher-order Godunov advection
schemes in the numerical method, allowing for the treatment of fluids with high
Schmidt number including the vanishing mass diffusion coefficient limit. We
incorporate thermal fluctuations in the description in both the inertial and
overdamped regimes. We apply our algorithms to model the development of giant
concentration fluctuations during the diffusive mixing of water and glycerol,
and compare numerical results with experimental measurements. We find good
agreement between the two, and observe propagative (non-diffusive) modes at
small wavenumbers (large spatial scales), not reported in published
experimental measurements of concentration fluctuations in fluid mixtures. Our
work forms the foundation for developing low Mach number fluctuating
hydrodynamics methods for miscible multi-species mixtures of chemically
reacting fluids.
