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.
翻译:线性波尔兹曼方程式的解决方案满足了一种能量约束,这反映了自然的事实:微粒的能量在有限体积中的能量在时间上被粒子的能量约束,而粒子的能量最初在体积中以粒子进入体积的能量增加而增加的体积则在时间上进入体积。在本文件中,我们为球形口声近似(PN)提出边界条件(BCs),以确保这一基本能量约束得到PN近似点的满足。我们的中分数与PN方方程式的特性波相兼容,并单独确定进波。能量约束和兼容性都根据PN方程式和BCs的抽象配方来显示,以分离必要的结构和特性。BCs来自Bs和BC的沼瓦克型配方和基底,其球面口相调功能和非古典均/多级,与PN方法的序列扩展相类似。我们显示,空间间隔式格和同步近似近值方法(SAT)的节点网格和定差异,可以使能量也保持半悬定。