In this paper, we present two-level nonlinear iteration methods for the transport equation in 1D slab geometry approximated by means of the linear discontinuous finite element method (LDFEM). We develop transport schemes based on the quasidiffusion (QD) method in which the low-order QD (LOQD) equations are discretized by the linear continuous finite element method (LCFEM). This requires a mapping of the LCFEM low-order solution to the LDFEM high-order solution to define the scattering term. Several mappings are proposed and analyzed. Another proposed transport discretization scheme is based on the step characteristics for the transport equation and LCFEM for the LOQD equations. We also develop new nonlinear synthetic acceleration (NSA) methods based on the LCFEM discretization of the QD equation. To gain iterative stability, the NSA algorithms are combined with the nonlinear Krylov acceleration method. We present numerical results that demonstrate performance and basic properties of the proposed discretization schemes and iterative solution methods.

在本文中，我们提出了通过线性不连续有限元方法（LDFEM）近似的一维平板几何中的输运方程的两级非线性迭代方法。我们开发了基于拟扩散（QD）方法的运输方案，其中通过线性连续有限元方法（LCFEM）离散低阶QD（LOQD）方程。这要求将LCFEM低阶解映射到LDFEM高阶解以定义散射项。提出并分析了几种映射。另一种建议的输运离散化方案是基于输运方程的步长特征和LOQD方程的LCFEM。我们还基于QD方程的LCFEM离散化开发了新的非线性合成加速度（NSA）方法。为了获得迭代稳定性，NSA算法与非线性Krylov加速方法相结合。我们提供的数值结果证明了所提出的离散化方案和迭代求解方法的性能和基本特性。