An efficient implementation of the Chebyshev Rational Approximation Method (CRAM) for solving the burnup equations

Maria Pusa, Jaakko Leppänen

Research output: Chapter in Book/Report/Conference proceedingConference article in proceedingsScientificpeer-review

6 Citations (Scopus)

Abstract

The Chebyshev Rational Approximation Method (CRAM) has been recently introduced by the authors for solving the burnup equations with excellent results. This method has been shown to be capable of simultaneously solving an entire burnup system with thousands of nuclides both accurately and efficiently. The method was prompted by an analysis of the spectral properties of burnup matrices and it can be characterized as the best rational approximation on the negative real axis. The coefficients of the rational approximation are fixed and have been reported for various approximation orders. In addition to these coefficients, implementing the method only requires a linear solver. This paper describes an efficient method for solving the linear systems associated with the CRAM approximation. The introduced direct method is based on sparse Gaussian elimination where the sparsity pattern of the resulting upper triangular matrix is determined before the numerical elimination phase. The stability of the proposed Gaussian elimination method is discussed based on considering the numerical properties of burnup matrices. Suitable algorithms are presented for computing the symbolic factorization and numerical elimination in order to facilitate the implementation of CRAM and its adoption into routine use. The accuracy and efficiency of the described technique are demonstrated by computing the CRAM approximations for a large test case with over 1600 nuclides
Original languageEnglish
Title of host publicationProceedings
Subtitle of host publicationInternational Conference on the Physics of Reactors 2012: Advances in Reactor Physics, PHYSOR 2012
PublisherAmerican Nuclear Society ANS
Pages952-963
Volume2
ISBN (Print)978-1-6227-6389-4
Publication statusPublished - 2012
MoE publication typeA4 Article in a conference publication
EventInternational Conference on the Physics of Reactors, PHYSOR 2012: Advances in Reactor Physics - Knoxville, United States
Duration: 15 Apr 201220 Apr 2012

Conference

ConferenceInternational Conference on the Physics of Reactors, PHYSOR 2012
CountryUnited States
CityKnoxville
Period15/04/1220/04/12

Fingerprint

Isotopes
Factorization
Linear systems

Keywords

  • Burnup equations
  • Chebyshev Rational Approximation Method
  • CRAM
  • serpent

Cite this

Pusa, M., & Leppänen, J. (2012). An efficient implementation of the Chebyshev Rational Approximation Method (CRAM) for solving the burnup equations. In Proceedings: International Conference on the Physics of Reactors 2012: Advances in Reactor Physics, PHYSOR 2012 (Vol. 2, pp. 952-963). American Nuclear Society ANS.
Pusa, Maria ; Leppänen, Jaakko. / An efficient implementation of the Chebyshev Rational Approximation Method (CRAM) for solving the burnup equations. Proceedings: International Conference on the Physics of Reactors 2012: Advances in Reactor Physics, PHYSOR 2012. Vol. 2 American Nuclear Society ANS, 2012. pp. 952-963
@inproceedings{ae2a0702e72342f486b0a2fcb8083213,
title = "An efficient implementation of the Chebyshev Rational Approximation Method (CRAM) for solving the burnup equations",
abstract = "The Chebyshev Rational Approximation Method (CRAM) has been recently introduced by the authors for solving the burnup equations with excellent results. This method has been shown to be capable of simultaneously solving an entire burnup system with thousands of nuclides both accurately and efficiently. The method was prompted by an analysis of the spectral properties of burnup matrices and it can be characterized as the best rational approximation on the negative real axis. The coefficients of the rational approximation are fixed and have been reported for various approximation orders. In addition to these coefficients, implementing the method only requires a linear solver. This paper describes an efficient method for solving the linear systems associated with the CRAM approximation. The introduced direct method is based on sparse Gaussian elimination where the sparsity pattern of the resulting upper triangular matrix is determined before the numerical elimination phase. The stability of the proposed Gaussian elimination method is discussed based on considering the numerical properties of burnup matrices. Suitable algorithms are presented for computing the symbolic factorization and numerical elimination in order to facilitate the implementation of CRAM and its adoption into routine use. The accuracy and efficiency of the described technique are demonstrated by computing the CRAM approximations for a large test case with over 1600 nuclides",
keywords = "Burnup equations, Chebyshev Rational Approximation Method, CRAM, serpent",
author = "Maria Pusa and Jaakko Lepp{\"a}nen",
year = "2012",
language = "English",
isbn = "978-1-6227-6389-4",
volume = "2",
pages = "952--963",
booktitle = "Proceedings",
publisher = "American Nuclear Society ANS",
address = "United States",

}

Pusa, M & Leppänen, J 2012, An efficient implementation of the Chebyshev Rational Approximation Method (CRAM) for solving the burnup equations. in Proceedings: International Conference on the Physics of Reactors 2012: Advances in Reactor Physics, PHYSOR 2012. vol. 2, American Nuclear Society ANS, pp. 952-963, International Conference on the Physics of Reactors, PHYSOR 2012, Knoxville, United States, 15/04/12.

An efficient implementation of the Chebyshev Rational Approximation Method (CRAM) for solving the burnup equations. / Pusa, Maria; Leppänen, Jaakko.

Proceedings: International Conference on the Physics of Reactors 2012: Advances in Reactor Physics, PHYSOR 2012. Vol. 2 American Nuclear Society ANS, 2012. p. 952-963.

Research output: Chapter in Book/Report/Conference proceedingConference article in proceedingsScientificpeer-review

TY - GEN

T1 - An efficient implementation of the Chebyshev Rational Approximation Method (CRAM) for solving the burnup equations

AU - Pusa, Maria

AU - Leppänen, Jaakko

PY - 2012

Y1 - 2012

N2 - The Chebyshev Rational Approximation Method (CRAM) has been recently introduced by the authors for solving the burnup equations with excellent results. This method has been shown to be capable of simultaneously solving an entire burnup system with thousands of nuclides both accurately and efficiently. The method was prompted by an analysis of the spectral properties of burnup matrices and it can be characterized as the best rational approximation on the negative real axis. The coefficients of the rational approximation are fixed and have been reported for various approximation orders. In addition to these coefficients, implementing the method only requires a linear solver. This paper describes an efficient method for solving the linear systems associated with the CRAM approximation. The introduced direct method is based on sparse Gaussian elimination where the sparsity pattern of the resulting upper triangular matrix is determined before the numerical elimination phase. The stability of the proposed Gaussian elimination method is discussed based on considering the numerical properties of burnup matrices. Suitable algorithms are presented for computing the symbolic factorization and numerical elimination in order to facilitate the implementation of CRAM and its adoption into routine use. The accuracy and efficiency of the described technique are demonstrated by computing the CRAM approximations for a large test case with over 1600 nuclides

AB - The Chebyshev Rational Approximation Method (CRAM) has been recently introduced by the authors for solving the burnup equations with excellent results. This method has been shown to be capable of simultaneously solving an entire burnup system with thousands of nuclides both accurately and efficiently. The method was prompted by an analysis of the spectral properties of burnup matrices and it can be characterized as the best rational approximation on the negative real axis. The coefficients of the rational approximation are fixed and have been reported for various approximation orders. In addition to these coefficients, implementing the method only requires a linear solver. This paper describes an efficient method for solving the linear systems associated with the CRAM approximation. The introduced direct method is based on sparse Gaussian elimination where the sparsity pattern of the resulting upper triangular matrix is determined before the numerical elimination phase. The stability of the proposed Gaussian elimination method is discussed based on considering the numerical properties of burnup matrices. Suitable algorithms are presented for computing the symbolic factorization and numerical elimination in order to facilitate the implementation of CRAM and its adoption into routine use. The accuracy and efficiency of the described technique are demonstrated by computing the CRAM approximations for a large test case with over 1600 nuclides

KW - Burnup equations

KW - Chebyshev Rational Approximation Method

KW - CRAM

KW - serpent

UR - http://www.proceedings.com/16103.html

M3 - Conference article in proceedings

SN - 978-1-6227-6389-4

VL - 2

SP - 952

EP - 963

BT - Proceedings

PB - American Nuclear Society ANS

ER -

Pusa M, Leppänen J. An efficient implementation of the Chebyshev Rational Approximation Method (CRAM) for solving the burnup equations. In Proceedings: International Conference on the Physics of Reactors 2012: Advances in Reactor Physics, PHYSOR 2012. Vol. 2. American Nuclear Society ANS. 2012. p. 952-963