Solving Burnup Equations by Numerical Inversion of the Laplace Transform Using Padé Rational Approximation

Abstract

Burnup calculations play a very important role in nuclear reactor design and analysis, and solving burnup equations is an essential topic in burnup calculations. In the last decade, several high-accuracy methods, mainly including the Chebyshev rational approximation method (CRAM), the quadrature-based rational approximation method, the Laguerre polynomial approximation method, and the mini-max polynomial approximation method, have been proposed to solve the burnup equations. Although these methods have been demonstrated to be quite successful in the burnup calculations, limitations still exist in some cases, one of which is that the accuracy becomes compromised when treating the time-dependent polynomial external feed rate. In this work, a new method called the Padé rational approximation method (PRAM) is proposed. Without directly approximating the matrix exponential, this new method is derived by using the Padé rational function to approximate the scalar exponential function in the formula of the inverse Laplace transform of burnup equations. Several test cases are carried out to verify the proposed new method. The high accuracy of the PRAM is validated by comparing the numerical results with the high-precision reference solutions. Against CRAM, PRAM is significantly superior in handling the burnup equations with time-dependent polynomial external feed rates and is much more efficient in improving the accuracy by using substeps, which demonstrates that PRAM is the attractive method for burnup calculations.

Keywords:

• Burnup calculations
• numerical inverse Laplace transform
• Padé rational approximation
• external feed
• substep strategy

Acknowledgments

This work is supported by the “Strategic Priority Research Program” of the Chinese Academy of Science (number XDA02001005) and the Shanghai Leading Talents Project which is funded by Science and Technology Commission of Shanghai Municipality (STCSM).

Notes

^a In this section, error analysis does not take into account the external feed rate; the accuracy of PRAM when introducing the external feed rate will be discussed in Sec. II.B.3.