Time and static eigenvalues of the stochastic transport equation by the methods of polynomial chaos

• Published in: Progress in nuclear energy (New series) 2013-08, Vol.67, p.33-55
• Main Authors: Ayres, D., Williams, M.M.R., Eaton, M.D.
• Format: Article
• Language: English
• Subjects: Approximation; Chaos theory; Criticality; Delayed neutrons; Eigenvalues; Nonlinearity; Polycarbonates; Polynomial chaos; Polynomials; Stochasticity; Time eigenvalues; Uncertainty
• ISSN: 0149-1970
• DOI: 10.1016/j.pnucene.2013.03.018

The concepts of static and dynamic eigenvalue problems are discussed and the practical differences between them are noted. Special emphasis is given to the use of these concepts in defining uncertainties in eigenvalues due to uncertainties in cross-sections. We have developed a practical method for calculating the stochastic properties of uncertainties in the time constant, reactivity and multiplication factor. Values have been found for the mean and variance in terms of cross-section uncertainties using both the conventional non-linear polynomial chaos (PC) method and a newly developed linear method. Indeed this is the main purpose of the paper and we compare and contrast the respective advantages and disadvantages of these two approaches. In general, it is found that the conventional non-linear PC methods require considerably more time to evaluate time eigenvalues than the linear methods; in some cases by a factor of more than 100, according to the number of random variables used. An approximate technique based upon simultaneous diagonalisation of matrices is also shown to yield accurate results for eigenvalues and to be a useful approximate tool for uncertainty analysis. •The effect o uncertainty in the nuclear cross sections is studied.•A new method of solving stochastic eigenvalue problems is shown to be very efficient.•Extensive numerical work is presented.•The new and traditional PCE methods are compared.