  Simulations of decaying magnetohydrodynamic (MHD) turbulence are performed
with a fluid and a kinetic code. The initial condition is an ensemble of
long-wavelength, counter-propagating, shear-Alfv\'{e}n waves, which interact
and rapidly generate strong MHD turbulence. The total energy is conserved and
the rate of turbulent energy decay is very similar in both codes, although the
fluid code has numerical dissipation whereas the kinetic code has kinetic
dissipation. The inertial range power spectrum index is similar in both the
codes. The fluid code shows a perpendicular wavenumber spectral slope of
$k_{\perp}^{-1.3}$. The kinetic code shows a spectral slope of
$k_{\perp}^{-1.5}$ for smaller simulation domain, and $k_{\perp}^{-1.3}$ for
larger domain. We estimate that collisionless damping mechanisms in the kinetic
code can account for the dissipation of the observed nonlinear energy cascade.
Current sheets are geometrically characterized. Their lengths and widths are in
good agreement between the two codes. The length scales linearly with the
driving scale of the turbulence. In the fluid code, their thickness is
determined by the grid resolution as there is no explicit diffusivity. In the
kinetic code, their thickness is very close to the skin-depth, irrespective of
the grid resolution. This work shows that kinetic codes can reproduce the MHD
inertial range dynamics at large scales, while at the same time capturing
important kinetic physics at small scales.
