  We present a formulation and numerical algorithm to extend the scheme for
grey radiation magneto-hydrodynamics (MHD) developed by Jiang (2021) to include
the frequency dependence via the multi-group approach. The entire frequency
space can be divided into arbitrary number of groups in the lab frame, and we
follow the time dependent evolution of frequency integrated specific
intensities along discrete rays inside each group. Spatial transport of photons
is done in the lab frame while all the coupling terms are solved in the fluid
rest frame. Lorentz transformation is used to connect different frames.
Radiation transport equation is solved fully implicitly in time while the MHD
equations are evolved explicitly so that time step is not limited by the speed
of light. A finite volume approach is used for transport in both spatial and
frequency spaces to conserve radiation energy density and momentum. The
algorithm includes photon absorption, electron scattering as well as Compton
scattering, which is calculated by solving the Kompaneets equation. The
algorithm is accurate for a wide range of optical depth conditions and can
handle both radiation pressure and gas pressure dominated flows. It works for
both Cartesian and curvilinear coordinate systems with adaptive mesh
refinement. We provide a variety of test problems including radiating sphere,
shadow test, absorption of a moving gas, Bondi type flows as well as a
collection of test problems for thermal and bulk Compton scattering. We also
discuss examples where frequency dependence can make a big difference compared
with the grey approach.
