论文标题
动力学传输方程的渐近保存和统一无条件稳定的有限差异方案
Asymptotic preserving and uniformly unconditionally stable finite difference schemes for kinetic transport equations
论文作者
论文摘要
在本文中，针对扩散缩放中的动力学传输方程开发了无条件稳定的一阶和二阶有限差方案。我们首先从分布函数的形式解决方案中得出了宏观密度的近似演化方程，然后通过以下传输部分的以下特性，具有向后有限的差异差半拉格朗日方法，而扩散部分则被隐含地分配。宏观密度可用后，即使在完全隐式的时间离散化中，分布函数也可以有效地求解，因为所有离散速度都被解耦，从而从每个离散速度下从空间离散化中产生了低维线性系统。考虑到空间和及时的一阶和二阶离散化。所得方案可以证明是在扩散极限内是渐近保存（AP）。通过基于相应扩增矩阵的特征值的傅立叶分析来验证均匀的无条件稳定性。数值实验（包括高维问题）已证明了空间和时间上的相应准确性顺序，均匀的稳定性，AP特性以及我们所提出的方法的良好性能。
In this paper, uniformly unconditionally stable first and second order finite difference schemes are developed for kinetic transport equations in the diffusive scaling. We first derive an approximate evolution equation for the macroscopic density, from the formal solution of the distribution function, which is then discretized by following characteristics for the transport part with a backward finite difference semi-Lagrangian approach, while the diffusive part is discretized implicitly. After the macroscopic density is available, the distribution function can be efficiently solved even with a fully implicit time discretization, since all discrete velocities are decoupled, resulting in a low-dimensional linear system from spatial discretizations at each discrete velocity. Both first and second order discretizations in space and in time are considered. The resulting schemes can be shown to be asymptotic preserving (AP) in the diffusive limit. Uniformly unconditional stabilities are verified from a Fourier analysis based on eigenvalues of corresponding amplification matrices. Numerical experiments, including high dimensional problems, have demonstrated the corresponding orders of accuracy both in space and in time, uniform stability, AP property, and good performances of our proposed approach.
