  We present a numerical formulation for the solution of non-isothermal,
compressible, Navier-Stokes equations with thermal fluctuations to describe
mesoscale transport phenomena in multispecies fluid mixtures. The novelty of
our numerical method is the use of staggered grid momenta along with a finite
volume discretization of the thermodynamic variables to solve the resulting
stochastic partial differential equations. The key advantages of the numerical
scheme are significantly simplified and compact discretization of the diffusive
and stochastic momentum fluxes, and an unambiguous prescription of boundary
conditions involving pressure. The staggered grid scheme more accurately
reproduces the equilibrium static structure factor of hydrodynamic fluctuations
in gas mixtures compared to a collocated scheme described previously in
Balakrishnan \emph{et al.} [Phys. Rev. E 89, 013017 (2014)]. The numerical
method is tested for ideal noble gases mixtures under various nonequilibrium
conditions, such as applied thermal and concentration gradients, to assess the
role of cross-diffusion effects, such as Soret and Dufour, on the long-ranged
correlations of hydrodynamic fluctuations, which are also more accurately
reproduced compared to the collocated scheme. We numerically study giant
nonequilibrium fluctuations driven by concentration gradients, and
fluctuation-driven Rayleigh-Taylor instability in gas mixtures. Wherever
applicable, excellent agreement is observed with theory and measurements from
the direct simulation Monte Carlo (DSMC) method.
