Polynomial Chaos Functions and Neutron Diffusion

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.