A new algorithm for two-dimensional transport for astrophysical simulations. I. General formulation and tests for the one-dimensional spherical case

We derive new equations using the mixed-frame approach tor one- and two-dimensional (axisymmetnc) time-dependent radiation transport and the associated couplings with matter. Our formulation is multigroup and multiangle and includes anisotropic scattering, frequency (energy)-dependent scattering and absorption, complete velocity dependence to order v/c, rotation, and energy redistribution due to inelastic scattering. Hence, the "2D" realization is actually "6 1/2"-dimensional. The effects of radiation viscosity are automatically incorporated. Moreover, we develop accelerated lambda iteration, Krylov subspace (GMRES), discontinuous finite element, and Feautrier numerical methods for solving the equations and present the results of one-dimensional numerical tests of the new formalism. The virtues of the mixed-frame approach include simple velocity dependence with no velocity derivatives, straight characteristics, simple physical interpretation, and clear generalization to higher dimensions. Our treatment can be used for both photon and neutrino transport, but we focus on neutrino transport and applications to core-collapse supernova theory in the discussions and examples.