  Continuum kinetic theories provide an important tool for the analysis and
simulation of particle suspensions. When those particles are anisotropic, the
addition of a particle orientation vector to the kinetic description yields a
$2d-1$ dimensional theory which becomes intractable to simulate, especially in
three dimensions or near states where the particles are highly aligned.
Coarse-grained theories that track only moments of the particle distribution
functions provide a more efficient simulation framework, but require closure
assumptions. For the particular case where the particles are apolar, the
Bingham closure has been found to agree well with the underlying kinetic
theory; yet the closure is non-trivial to compute, requiring the solution of an
often nearly-singular nonlinear equation at every spatial discretization point
at every timestep. In this paper, we present a robust, accurate, and efficient
numerical scheme for evaluating the Bingham closure, with a controllable
error/efficiency tradeoff. To demonstrate the utility of the method, we carry
out high-resolution simulations of a coarse-grained continuum model for a
suspension of active particles in parameter regimes inaccessible to kinetic
theories. Analysis of these simulations reveals that inaccurately computing the
closure can act to effectively limit spatial resolution in the coarse-grained
fields. Pushing these simulations to the high spatial resolutions enabled by
our method reveals a coupling between vorticity and topological defects in the
suspension director field, as well as signatures of energy transfer between
scales in this active fluid model.
