  Hydrodynamic fluctuations in simple fluids under shear flow are demonstrated
to be spatially correlated, in contrast to the fluctuations at equilibrium,
using mesoscopic hydrodynamic simulations. The simulation results for the
equal-time hydrodynamic correlations in a multiparticle collision dynamics
(MPC) fluid in shear flow are compared with the explicit expressions obtained
from fluctuating hydrodynamic calculations. For large wave vectors $k$, the
nonequlibrium contributions to transverse and longitudinal velocity
correlations decay as $k^{-4}$ for wave vectors along the flow direction, and
as $k^{-2}$ for the off-flow directions. For small wave vectors, a cross-over
to a slower decay occurs, indicating long-range correlations in real space. The
coupling between the transverse velocity components, which vanishes at
equilibrium, also exhibits a $k^{-2}$ dependence on the wave vector. In
addition, we observe a quadratic dependency on the shear rate of the
non-equilibrium contribution to pressure.
