  In the current study a meshfree Lagrangian particle method for the
Landau-Lifshitz Navier-Stokes (LLNS) equations is developed. The LLNS equations
incorporate thermal fluctuation into macroscopic hydrodynamics by the addition
of white noise fluxes whose magnitudes are set by a fluctuation-dissipation
theorem. The study focuses on capturing the correct variance and correlations
computed at equilibrium flows, which are compared with available theoretical
values. Moreover, a numerical test for the random walk of standing shock wave
has been considered for capturing the shock location.
