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非齐次燃耗方程数值解法

付元光 邓力 李刚

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非齐次燃耗方程数值解法

付元光, 邓力, 李刚

Preliminary study on numerical solver of inhomogeneous burnup equations

Fu Yuan-Guang, Deng Li, Li Gang
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  • 非齐次燃耗方程常用于描述具有显著核素迁移效应的核能系统中核素含量随时间的变化规律.国内外许多燃耗计算程序无法求解方程非齐次项含时的情况.本文在方程非齐次项能够被有限阶关于时间的泰勒展开逼近这一前提下,研究了非齐次项含时情况下方程的两种解法.首先通过Laplace变换推导出了方程基于线性子链方法的解析解形式,然后使用Carathéodory-Fejér方法计算出了方程矩阵级数解的近最佳Chebyshev有理逼近式.将两种方法在燃耗计算程序JBURN中实现,并进行了数值计算,绝大部分计算结果符合很好,部分结果在较长有效数字内仍能保持一致,验证了方法的正确性和精度.同时为求解具有其他非齐次项形式的燃耗方程提供了一种思路.
    The inhomogeneous burnup equation is often used for describing the time evolution of nuclides' depletion in nuclear systems which have a significant nuclide migration effect. However, lots of burnup calculations codes only deal with the homogeneous cases instead of the inhomogeneous ones, among them there are a few codes that can work only when the inhomogeneous term of the equation is constant. Based on the condition that the inhomogeneous term can be approximated by finite-order Taylor expansion, two methods are introduced to solve the inhomogeneous burnup equation whose inhomogeneous term is time dependent. For the first method, the transmutation trajectory analysis method is used to decompose the connections between nuclides into linear chains, for one chain the analytical solution is derived strictly by using the Laplace transform. For the second method, a solution of the inhomogeneous equation in the form of summation of infinite matrix series is first derived, and then the sum function of the series is found. Furthermore, the different-order nearly-best rational approximation function of the sum function is found by using Carathéodory-Fejér method. The error between the sum function and the rational function fluctuates in a certain range without exceeding a limit value, while the maximum error decreases exponentially with the order of rational function increasing. By adopting the nearly-best rational approximation, the summation of infinite matrix series converts into a finite expansion of matrix fraction, which is much easier to deal with. These two methods are implemented in the burnup calculation code JBURN and numerical tests are done through using two examples. The first example is a small-scale matrix example and the result shows that the results from the two methods agree well in at least 6 decimal precision together with the results from the reference solution. The second example is a large-scale problem based on real nuclides' reaction database, and the result shows that less than 1% among all nuclides have a deviation larger than 10% between two methods, while about 8% nuclides have a deviation larger than 0.01% and the remaining ones have a deviation smaller than 0.01%. These results validate the correctness and accuracy for each of the two methods. Finally, this paper provides a possible implementation process for solving inhomogeneous burnup equations which have other time-dependent forms of inhomogeneous term.
      通信作者: 邓力, deng_li@iapcm.ac.cn
    • 基金项目: 能源局重大专项子项(批准号:2015ZX06002008)、国防科工局国防基础科研计划(批准号:C1520110002)、国家磁约束核聚变能源研究专项(批准号:2015GB108002)和国家自然科学基金-广东联合基金(第二期)超级计算科学应用专项(批准号:U1501501)资助的课题.
      Corresponding author: Deng Li, deng_li@iapcm.ac.cn
    • Funds: Project supported by the Sub-item of Special Projects of the National Energy Bureau, China (Grant No. 2015ZX06002008), the National Defense Basic Scientific Research Program of China (Grant No. C1520110002), the National Magnetic Constrained Nuclear Fusion Energy Research Project, China (Grant No. 2015GB108002), and the National Science Foundation of China-Guangdong United Fund (second phase) Projects of Application of Supercomputing Science (Grant No. U1501501).
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    Li C J, Zhu X J, Gu C Q 2011 Appl. Math. 2 247

    [10]

    Cleve M, Charles V L 2003 SIAM Rev. 45 3

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    England T R 1961 CINDER–A One-Point Depletion and Fission Product Program (Los Alamos:Los Alamos National Laboratory) p1

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    He B Q, Wang X H 2007 Higher Mathematics (Beijing:Science Press) p298 (in Chinese)[何柏庆, 王晓华 2007 高等数学(北京:科学出版社) 第298页]

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    Isotalo A E, Aarnio P A 2011 Ann. Nucl. Energy 38 261

    [14]

    Trefethen L N 2011 Approximation Theory and Approximation Practice (London:Oxford University) pp61, 168, 171

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    Zadeh L A, Desoer C A 1963 Linear System Theory (New York:McGraw-Hill Book Company) p593

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    [17]

    Pachon R, Trefethen L N 2009 BIT Numer. Math. 49 721

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    Trefethen L N, Gutknecht M H 1983 SIAM J. Numer. Anal. 20 420

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    Deun J V, Trefethen L N 2011 Numer. Math. 51 1039

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    Schmelzer T, Trefethen L N 2007 Electron. Trans. Numer. Anal. 28 1

  • [1]

    Bateman 1910 Cambridge Philos. Soc. Proc. 15 423

    [2]

    Cetnar J 2006 Ann. Nucl. Energy 38 261

    [3]

    Huang K, Wu H C 2016 Ann. Nucl. Energy 87 637

    [4]

    Allen G C 1980 A User's Manual for the ORIGEN2 Computer Code (Tennessee:Oak Ridge National Laboratory) p179

    [5]

    Yamamoto A, Tatsumi M, Sugimura N 2007 J. Nucl. Sci. Technol. 44 147

    [6]

    Pusa M, Leppanen J 2010 Nucl. Sci. Eng. 164 140

    [7]

    Pusa M 2011 Nucl. Sci. Eng. 169 155

    [8]

    Cody W J, Meinardus G 1969 J. Approx. Theory 2 50

    [9]

    Li C J, Zhu X J, Gu C Q 2011 Appl. Math. 2 247

    [10]

    Cleve M, Charles V L 2003 SIAM Rev. 45 3

    [11]

    England T R 1961 CINDER–A One-Point Depletion and Fission Product Program (Los Alamos:Los Alamos National Laboratory) p1

    [12]

    He B Q, Wang X H 2007 Higher Mathematics (Beijing:Science Press) p298 (in Chinese)[何柏庆, 王晓华 2007 高等数学(北京:科学出版社) 第298页]

    [13]

    Isotalo A E, Aarnio P A 2011 Ann. Nucl. Energy 38 261

    [14]

    Trefethen L N 2011 Approximation Theory and Approximation Practice (London:Oxford University) pp61, 168, 171

    [15]

    Zadeh L A, Desoer C A 1963 Linear System Theory (New York:McGraw-Hill Book Company) p593

    [16]

    Amann H 1990 Orinary Differential Equations:An Introduction to Nonlinear Analysis (New York:Walter de Gruyter) p105

    [17]

    Pachon R, Trefethen L N 2009 BIT Numer. Math. 49 721

    [18]

    Trefethen L N, Gutknecht M H 1983 SIAM J. Numer. Anal. 20 420

    [19]

    Deun J V, Trefethen L N 2011 Numer. Math. 51 1039

    [20]

    Schmelzer T, Trefethen L N 2007 Electron. Trans. Numer. Anal. 28 1

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出版历程
  • 收稿日期:  2017-12-14
  • 修回日期:  2018-04-13
  • 刊出日期:  2018-09-05

非齐次燃耗方程数值解法

  • 1. 北京应用物理与计算数学研究所, 北京 100088;
  • 2. 中物院高性能数值模拟软件中心, 北京 100088
  • 通信作者: 邓力, deng_li@iapcm.ac.cn
    基金项目: 能源局重大专项子项(批准号:2015ZX06002008)、国防科工局国防基础科研计划(批准号:C1520110002)、国家磁约束核聚变能源研究专项(批准号:2015GB108002)和国家自然科学基金-广东联合基金(第二期)超级计算科学应用专项(批准号:U1501501)资助的课题.

摘要: 非齐次燃耗方程常用于描述具有显著核素迁移效应的核能系统中核素含量随时间的变化规律.国内外许多燃耗计算程序无法求解方程非齐次项含时的情况.本文在方程非齐次项能够被有限阶关于时间的泰勒展开逼近这一前提下,研究了非齐次项含时情况下方程的两种解法.首先通过Laplace变换推导出了方程基于线性子链方法的解析解形式,然后使用Carathéodory-Fejér方法计算出了方程矩阵级数解的近最佳Chebyshev有理逼近式.将两种方法在燃耗计算程序JBURN中实现,并进行了数值计算,绝大部分计算结果符合很好,部分结果在较长有效数字内仍能保持一致,验证了方法的正确性和精度.同时为求解具有其他非齐次项形式的燃耗方程提供了一种思路.

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