  We formulate low Mach number fluctuating hydrodynamic equations appropriate
for modeling diffusive mixing in isothermal mixtures of fluids with different
density and transport coefficients. These equations eliminate the fluctuations
in pressure associated with the propagation of sound waves by replacing the
equation of state with a local thermodynamic constraint. We demonstrate that
the low Mach number model preserves the spatio-temporal spectrum of the slower
diffusive fluctuations. We develop a strictly conservative finite-volume
spatial discretization of the low Mach number fluctuating equations in both two
and three dimensions and construct several explicit Runge-Kutta temporal
integrators that strictly maintain the equation of state constraint. The
resulting spatio-temporal discretization is second-order accurate
deterministically and maintains fluctuation-dissipation balance in the
linearized stochastic equations. We apply our algorithms to model the
development of giant concentration fluctuations in the presence of
concentration gradients, and investigate the validity of common simplifications
such as neglecting the spatial non-homogeneity of density and transport
properties. We perform simulations of diffusive mixing of two fluids of
different densities in two dimensions and compare the results of low Mach
number continuum simulations to hard-disk molecular dynamics simulations.
Excellent agreement is observed between the particle and continuum simulations
of giant fluctuations during time-dependent diffusive mixing.
