High Order Asymptotic Preserving Hermite WENO Fast Sweeping Method for the Steady-State \(S_{N}\) Transport Equations

In this paper, we propose to combine the fifth-order Hermite weighted essentially non-oscillatory (HWENO) scheme and the 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 diffusion 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. A hybrid strategy to compute the nonlinear weights in the HWENO reconstruction is introduced to save computational costs. Extensive one- and two-dimensional numerical experiments are performed to verify the accuracy, asymptotic preserving property and positivity of the proposed HWENO FSM. The proposed HWENO method can also be combined with the Diffusion Synthetic Acceleration algorithm to improve computational efficiency.