### Abstract

The burnup equations can in principle be solved by
computing the exponential of the burnup matrix. However,
due to the difficult numerical characteristics of burnup
matrices, the problem is extremely stiff, and the matrix
exponential solution was long considered infeasible for
an entire burnup system containing over a thousand
nuclides. After discovering that the eigenvalues of
burnup matrices are generally confined to a region near
the negative real axis, the Chebyshev rational
approximation method (CRAM) was introduced as a novel
method to solve the burnup equations. It can be
characterized as the best rational function on the
negative real axis and it has been shown to be capable of
simultaneously solving an entire burnup system both
accurately and efficiently. The main difficulty in using
CRAM for computing the matrix exponential is determining
the coefficients of the rational function for a given
approximation order. Some polynomial CRAM coefficients
have been published in 1984, and based on these
literature values, CRAM approximations up to the order 16
have been thus far applied in burnup calculations. The
topic of this paper is the computation of CRAM
approximations and their application to burnup equations.
A Remez-type method utilizing the equioscillation
property of best approximations is used to construct the
CRAM approximants for approximation orders 1, ..., 50.
Numerical results are presented for a large burnup system
and for a decay system. It is demonstrated that
higher-order CRAM can be used to accurately solve the
burnup equations even with time steps of the order of
millions of years.

Original language | English |
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Title of host publication | Proceedings of the international conference on physics of reactors |

Subtitle of host publication | PHYSOR 2014 |

Number of pages | 11 |

Volume | JAEA-Conf 2014-003 |

DOIs | |

Publication status | Published - 2015 |

MoE publication type | A4 Article in a conference publication |

Event | International Conference on the Physics of Reactors, PHYSOR 2014: The Role of Reactor Physics toward Sustainable Future - Kyoto, Japan Duration: 28 Sep 2014 → 3 Oct 2014 |

### Conference

Conference | International Conference on the Physics of Reactors, PHYSOR 2014 |
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Abbreviated title | PHYSOR2014 |

Country | Japan |

City | Kyoto |

Period | 28/09/14 → 3/10/14 |

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## Cite this

Pusa, M. (2015). Higher-order Chebyshev Rational Approximation Method (CRAM). In

*Proceedings of the international conference on physics of reactors: PHYSOR 2014*(Vol. JAEA-Conf 2014-003). [1119422] https://doi.org/10.11484/jaea-conf-2014-003