  In this paper, we propose to combine the fifth order Hermite weighted
essentially non-oscillatory (HWENO) scheme and fast sweeping method (FSM) for
the solution of the steady-state $S_{N}$ transport equation in the finite
volume framework. It is well-known that the $S_{N}$ transport equation
asymptotically converges to a macroscopic diffusion equation in the limit of
optically thick systems with small absorption and sources. Numerical methods
which can preserve the asymptotic limit are referred to as asymptotic
preserving methods. In the one-dimensional case, we provide the analysis to
demonstrate the asymptotic preserving property of the high order finite volume
HWENO method, by showing that its cell-edge and cell-average fluxes possess the
thick diffusion limit. Numerical results in both one- and two- dimensions are
presented to validate its asymptotic preserving property. A hybrid strategy to
compute the nonlinear weights in the HWENO reconstruction is introduced to save
computational cost. Extensive one- and two-dimensional numerical experiments
are performed to verify the accuracy, asymptotic preserving property and
positivity of the proposed HWENO FSM.
