  We develop a new relativistic radiation hydrodynamics code based on the
Monte-Carlo algorithm. In this code, we implement a new scheme to achieve the
second-order accuracy in time in the limit of a large packet number for solving
the interaction between matter and radiation. This higher-order time
integration scheme is implemented in the manner to guarantee the
energy-momentum conservation to the precision of the geodesic integrator. The
spatial dependence of radiative processes, such as the packet propagation,
emission, absorption, and scattering, are also taken into account up to the
second-order accuracy. We validate our code by solving various test-problems
following the previous studies; one-zone thermalization, dynamical diffusion,
radiation dragging, radiation mediated shock-tube, shock-tube in the optically
thick limit, and Eddington limit problems. We show that our code reproduces
physically appropriate results with reasonable accuracy and also demonstrate
that the second-order accuracy in time and space is indeed achieved with our
implementation for one-zone and one-dimensional problems.
