  Neutrino-matter interactions play an important role in core-collapse
supernova (CCSN) explosions as they contribute to both lepton number and/or
four-momentum exchange between neutrinos and matter, and thus act as the agent
for neutrino-driven explosions. Due to the multiscale nature of neutrino
transport in CCSN simulations, an implicit treatment of neutrino-matter
interactions is desired, which requires solutions of coupled nonlinear systems
in each step of the time integration scheme. In this paper we design and
compare nonlinear iterative solvers for implicit systems with energy coupling
neutrino-matter interactions commonly used in CCSN simulations. Specifically,
we consider electron neutrinos and antineutrinos, which interact with static
matter configurations through the Bruenn~85 opacity set. The implicit systems
arise from the discretization of a non-relativistic two-moment model for
neutrino transport, which employs the discontinuous Galerkin (DG) method for
phase-space discretization and an implicit-explicit (IMEX) time integration
scheme. In the context of this DG-IMEX scheme, we propose two approaches to
formulate the nonlinear systems -- a coupled approach and a nested approach.
For each approach, the resulting systems are solved with Anderson-accelerated
fixed-point iteration and Newton's method. The performance of these four
iterative solvers has been compared on relaxation problems with various degree
of collisionality, as well as proto-neutron star deleptonization problems with
several matter profiles adopted from spherically symmetric CCSN simulations.
Numerical results suggest that the nested Anderson-accelerated fixed-point
solver is more efficient than other tested solvers for solving implicit
nonlinear systems with energy coupling neutrino-matter interactions.
