半空间单能中子输运方程的解
Solution of the Monoenergetic Neutron Transport Equation in a Half Space
摘要：
自从1921年Milne半空间问题提出以来，中子输运方程的解析解一直吸引着数学家和物理学家的注意。存在许多数值解，但可以理解的是，只有少数解析解，其中最突出的是1960年Case [2]引入的奇异本征函数展开（SEE）。对于半空间，该方法虽然从半值域完备性得到了一种优美的解析形式，但需要对复杂的积分进行数值计算。此外，人们发现封闭形式的解析表达式，只有无限介质和半空间的情况下。人们只能迭代地求出平板中的通量。也就是说，在一般情况下，必须花费相当大的数值努力，从SEE获得高度精确的基准。因此，研究人员设计了替代方法，例如CN [3]，FN [4]和基于SEE的格林函数方法（GFM）[5]。这些方法以SEE为核心，围绕解析形式构建数值方法。特别是FN方法在生成高度精确的基准方面最为成功。到目前为止，还没有一种方法产生精确的数值解仅仅基于基本的离散化。在这里，我们显示了一个源的均匀介质的真空边界上的问题，精确的数值解是可能的，通过拉格朗日插值在一组离散的方向。
英文摘要：
The analytical solution of neutron transport equation has fascinated mathematicians and physicists alike since the Milne half-space problem was introduce in 1921 [1]. Numerous numerical solutions exist, but understandably, there are only a few analytical solutions, with the prominent one being the singular eigenfunction expansion (SEE) introduced by Case [2] in 1960. For the half-space, the method, though yielding, an elegant analytical form resulting from half-range completeness, requires numerical evaluation of complicated integrals. In addition, one finds closed form analytical expressions only for the infinite medium and half-space cases. One can find the flux in a slab only iteratively. That is to say, in general one must expend a considerable numerical effort to get highly precise benchmarks from SEE. As a result, investigators have devised alternative methods, such as the CN [3], FN [4] and Greens Function Method (GFM) [5] based on the SEE have been devised. These methods take the SEE at their core and construct a numerical method around the analytical form. The FN method in particular has been most successful in generating highly precise benchmarks. No method yielding a precise numerical solution has yet been based solely on a fundamental discretization until now. Here, we show for the albedo problem with a source on the vacuum boundary of a homogeneous medium, a precise numerical solution is possible via Lagrange interpolation over a discrete set of directions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
