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首页> 外文期刊>Numerical Heat Transfer, Part B. Fundamentals: An International Journal of Computation and Methodology >Further Development of the Elliptic PDE Formulation of the P N Approximation and its Marshak Boundary Conditions
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Further Development of the Elliptic PDE Formulation of the P N Approximation and its Marshak Boundary Conditions

机译：P N近似的椭圆PDE公式及其Marshak边界条件的进一步发展

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摘要

The expansion of the radiative transfer equation (RTE) into spherical harmonics results in the P N approximation, consisting of (N+1) 2 simultaneous, first-order partial differential equations (PDEs). This system of equations is generally solved subject to a set of so-called Marshak's boundary conditions, although some ambiguity exists in multidimensional media, for which the set provides more than the necessary number of conditions. In recent work Modest has shown that the general 3-D P N approximation can be formulated as a set of N(N+1)/2 second-order, elliptic PDEs, using the original set of Marshak's conditions, and which can be solved with standard PDE solution packages. In this article the Marshak boundary conditions are reexamined in the light of the elliptic formulation, culminating in a self-consistent set of N(N+1)/2 conditions along the boundary of the enclosure. The elliptic set of PDEs is reformulated and reduced considerably by limiting considerations to isotropic scattering. As an example, the 2-D P 3 approximation is extracted, and sample 2-D P 1, P 3, and P 5 computations are compared with Monte Carlo results.
机译：将辐射传递方程（RTE）扩展为球谐函数会得到P N近似值，该近似值由（N + 1）2个同时的一阶偏微分方程（PDE）组成。通常在一组所谓的Marshak边界条件的约束下求解此方程组，尽管多维介质中存在一些歧义，为此，多维介质组提供了更多的必要条件。在最近的工作中，Modest表明，可以使用原始的Marshak条件集，将一般的3-DPN近似公式化为一组N（N + 1）/ 2个二阶椭圆PDE，并且可以使用标准方程求解。 PDE解决方案包。在本文中，根据椭圆公式重新检查了Marshak边界条件，最终沿着封闭空间的边界形成了N（N + 1）/ 2个条件的自洽集合。通过限制对各向同性散射的考虑，可以重新构造椭圆形的PDE并大大减少。例如，提取2-D P 3近似值，并将样本2-D P 1，P 3和P 5计算与蒙特卡洛结果进行比较。

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