论文标题

PN方程的稳定边界条件和离散化

Stable Boundary Conditions and Discretization for PN Equations

论文作者

Bünger, Jonas, Sarna, Neeraj, Torrilhon, Manuel

论文摘要

线性玻尔兹曼方程的解决方案满足了一个能量结合,这反映了一个自然事实:有限体积中粒子的能量是通过最初占据颗粒的能量来限制的,最初占据了粒子的能量,这些粒子的体积占据了体积的增加,而粒子通过粒子传输到体积中的能量随着时间的推移而进入体积。在本文中,我们介绍了球形谐波(PN)近似的边界条件(BCS),这确保PN近似满足了这种基本能量结合。我们的BC与PN方程的特征波兼容,并独特地确定传入波。基于PN方程和BC的抽象公式,可以显示能量结合和兼容性,以隔离必要的结构和特性。 BC源自BC的Marshak类型公式,并在球形谐波函数的非古典偶数/奇数分类和稳定步骤上得出,这与PN方法中的串联扩展的截断相似。我们表明,零件(SBP)在空间中交错的网格上的有限差异以及同时近似项(SAT)的方法允许在半混凝土水平上保持绑定的能量。

A solution to the linear Boltzmann equation satisfies an energy bound, which reflects a natural fact: The energy of particles in a finite volume is bounded in time by the energy of particles initially occupying the volume augmented by the energy transported into the volume by particles entering the volume over time. In this paper, we present boundary conditions (BCs) for the spherical harmonic (PN) approximation, which ensure that this fundamental energy bound is satisfied by the PN approximation. Our BCs are compatible with the characteristic waves of PN equations and determine the incoming waves uniquely. Both, energy bound and compatibility, are shown based on abstract formulations of PN equations and BCs to isolate the necessary structures and properties. The BCs are derived from a Marshak type formulation of BC and base on a non-classical even/odd-classification of spherical harmonic functions and a stabilization step, which is similar to the truncation of the series expansion in the PN method. We show that summation by parts (SBP) finite differences on staggered grids in space and the method of simultaneous approximation terms (SAT) allows to maintain the energy bound also on the semi-discrete level.

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