Polynomial Chaos Functions and Neutron Diffusion
Polynomial Chaos Functions and Neutron Diffusion
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多项式混沌函数和中子扩散
DOI：
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发表时间：
2007
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通讯作者：
M.M.R. Williams
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作者：
M.M.R. Williams
摘要利用维纳多项式混沌函数求解一类随机微分方程。它示出了各种多项式可根据潜在的随机元素的概率分布。利用勒让德混沌多项式，我们在P1近似下解决了辐射通过随机材料平板的透射问题。对于一个特殊的情况，它是可能的，以获得这个问题的精确解，因此，可以检查的混沌展开的收敛速度。结果以表格和图形的形式示出，其中比较随机平均值与确定性平均值和显着差异被发现。此外，我们计算通量和电流通过平板的方差，从而给出与平均值相关的不确定性的措施。多项式混沌的方法提供了一种替代的程序，通常使用的封闭，或特殊的统计，空间随机性的研究方法，并有可能处理非常复杂的系统，虽然完整的计算影响尚未确定。在附录中，我们展示了如何玻尔兹曼方程，空间随机截面，可以减少到一个耦合的确定性方程组。
Abstract The polynomial chaos functions of Wiener are used to solve a stochastic differential equation. It is shown that a variety of polynomials are available according to the probability distribution of the underlying random element. Using the Legendre chaos polynomials, we have solved the problem of radiation transmission through a slab of random material properties in the P1 approximation. For a special case, it is possible to obtain an exact solution to this problem, and hence the rate of convergence of the chaos expansion can be examined. Results are shown in tabular form and graphically, which compare the stochastic average with the deterministic average and significant differences are found. In addition we calculate the variance in the flux and current across the slab, thereby giving a measure of the uncertainty associated with the average. The method of polynomial chaos offers an alternative procedure to the normally used closure, or special statistics, methods for the study of spatial randomness and has the potential to deal with very complex systems, although the full computational implications have yet to be determined. In the Appendix, we show how the Boltzmann equation, with spatially random cross sections, can be reduced to a coupled set of deterministic equations.