  Semiflexible slender filaments are ubiquitous in nature and cell biology,
including in the cytoskeleton, where reorganization of actin filaments allows
the cell to move and divide. Most methods for simulating semiflexible
inextensible fibers/polymers are based on discrete (bead-link or blob-link)
models, which become prohibitively expensive in the slender limit when
hydrodynamics is accounted for. In this paper, we develop a novel
coarse-grained approach for simulating fluctuating slender filaments with
hydrodynamic interactions. Our approach is tailored to relatively stiff fibers
whose persistence length is comparable to or larger than their length, and is
based on three major contributions. First, we discretize the filament
centerline using a coarse non-uniform Chebyshev grid, on which we formulate a
discrete constrained Gibbs-Boltzmann equilibrium distribution and overdamped
Langevin equation. Second, we define the hydrodynamic mobility at each point on
the filament as an integral of the Rotne-Prager-Yamakawa kernel along the
centerline, and apply a spectrally-accurate quadrature to accurately resolve
the hydrodynamics. Third, we propose a novel midpoint temporal integrator which
can correctly capture the Ito drift terms that arise in the overdamped Langevin
equation. We verify that the equilibrium distribution for the Chebyshev grid is
a good approximation of the blob-link one, and that our temporal integrator
samples the equilibrium distribution for sufficiently small time steps. We also
study the dynamics of relaxation of an initially straight filament, and find
that as few as 12 Chebyshev nodes provides a good approximation to the dynamics
while allowing a time step size two orders of magnitude larger than a resolved
blob-link simulation. We conclude by studying how bending fluctuations aid the
process of bundling in cross-linked networks of semiflexible fibers.
