  Mixed-frame formulations of radiation-hydrodynamics (RHD), where the
radiation quantities are computed in an inertial frame but matter quantities
are in a comoving frame, are advantageous because they admit algorithms that
conserve energy and momentum to machine precision and combine more naturally
with adaptive mesh techniques, since unlike pure comoving-frame methods they do
not face the problem that radiation quantities must change frame every time a
cell is refined or coarsened. However, implementing multigroup RHD in a
mixed-frame formulation presents challenges due to the complexity of handling
frequency-dependent interactions and the Doppler shift of radiation boundaries.
In this paper, we introduce a novel method for multigroup RHD that integrates a
mixed-frame formulation with a piecewise powerlaw approximation for frequency
dependence within groups. This approach ensures the exact conservation of total
energy and momentum while effectively managing the Lorentz transformation of
group boundaries and evaluation of group-averaged opacities. Our method takes
advantage of the locality of matter-radiation coupling, allowing the source
term for $N_g$ frequency groups to be handled with simple equations with a
sparse Jacobian matrix of size $N_g + 1$, which can be inverted with $O(N_g)$
complexity. This results in a computational complexity that scales linearly
with $N_g$ and requires no more communication than a pure hydrodynamics update,
making it highly efficient for massively parallel and GPU-based systems. We
implement our method in the GPU-accelerated RHD code QUOKKA and demonstrate
that it passes a wide range of numerical tests. We demonstrate that the
piecewise powerlaw method shows significant advantages over traditional opacity
averaging methods for handling rapidly variable opacities with modest frequency
resolution.
