几种常用燃耗方程组求解方法的性能对比分析
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摘要
为求解点堆燃耗方程组,通过自主编程燃耗计算模块,分别采用Chebyshev有理近似方法(Chebyshev Rational Approximation Method,CRAM)、Padé近似方法、Krylov Subspace方法以及线性子链法(Transmutation Trajectory Analysis,TTA)对不同规模的点堆燃耗系统进行燃耗计算,并与高精度下16阶系数的CRAM的结果进行对比分析。结果表明,CRAM能够高效地获得精确的计算结果,而且其系数阶数越高,精度越高,步长稳定性越高,因此目前燃耗计算多采用这种方法;Padé近似方法在双精度下对于中小规模燃耗系统能够快速获得精确结果,受步长影响小,而对于大规模燃耗系统需要采用高精度计算,且受步长影响也小;Krylov Subspace方法在双精度下可准确求解中小规模燃耗矩阵,但不适用于大规模燃耗求解;TTA适用于求解各规模燃耗方程,但由于线性链的简化,相对参考值精度较低,求解速度相对较慢。本文所获结果对于不同燃耗计算系统的开发及应用具有较好的借鉴意义。
[Background] It is very important to trace fuel concentration changing with high precision and efficiency in the current core simulation calculations, where an essential part is to solve depletion equations precisely. [Purpose] This study aims to solve point-core depletion equations in different sizes with various methods and find out the characteristics of these methods by a self-programming burnup module. [Methods] Four methods including Chebyshev rational approximation method(CRAM), Padé approximation method, Krylov Subspace method and transmutation trajectory analysis(TTA) were used to realize burnup calculations and comparative analysis with the results by CRAM with 16 orders in high precision for different-scale burnup systems. [Results] The results suggested that CRAM could provide very accurate solutions in a very short computing time, and the higher the coefficient order, the higher the precision and step stability. Padé approximation method could get accurate results affected marginally by step length with double precision in small and medium systems quickly, but high precision computation should be adopted for large systems while little affected by step length. Krylov Subspace method had accurate solutions in small and medium systems with double floats. However, it can't be applied to large cases. TTA was applicable to burnup equations at any scale with a weak accuracy and a slower speed because of linear chain simplification. [Conclusion] The four methods can be adapted to the depletion calculations of the cores in different cases, and the responding results can be treated as reference for the development and application of different burnup calculation systems.
[Background] It is very important to trace fuel concentration changing with high precision and efficiency in the current core simulation calculations, where an essential part is to solve depletion equations precisely. [Purpose] This study aims to solve point-core depletion equations in different sizes with various methods and find out the characteristics of these methods by a self-programming burnup module. [Methods] Four methods including Chebyshev rational approximation method(CRAM), Padé approximation method, Krylov Subspace method and transmutation trajectory analysis(TTA) were used to realize burnup calculations and comparative analysis with the results by CRAM with 16 orders in high precision for different-scale burnup systems. [Results] The results suggested that CRAM could provide very accurate solutions in a very short computing time, and the higher the coefficient order, the higher the precision and step stability. Padé approximation method could get accurate results affected marginally by step length with double precision in small and medium systems quickly, but high precision computation should be adopted for large systems while little affected by step length. Krylov Subspace method had accurate solutions in small and medium systems with double floats. However, it can't be applied to large cases. TTA was applicable to burnup equations at any scale with a weak accuracy and a slower speed because of linear chain simplification. [Conclusion] The four methods can be adapted to the depletion calculations of the cores in different cases, and the responding results can be treated as reference for the development and application of different burnup calculation systems.
引文
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4范文玎,孙光耀,张彬航,等.基于切比雪夫有理逼近方法的蒙特卡罗燃耗计算研究与验证[J].核技术,2016,39(4):040501.DOI:10.11889/j.0253-3219.2016.hjs.39.040501.FAN Wending,SUN Guangyao,ZHANG Binhang,et al.Research and verification of Monte Carlo burnup calculations based on Chebyshev rational approximation method[J].Nuclear Techniques,2016,39(4):040501.DOI:10.11889/j.0253-3219.2016.hjs.39.040501.
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6 Cody W J,Meinardus G,Varga R S.Chebyshev rational approximations to exp(-x)in[0,+∞)and applications to heat-conduction problems[J].Journal of Approximation Theory,1969,2(1):50-65.DOI:10.1016/0021-9045(69)90030-6.
7 Schmelzer T.Carathéodory-Fejér approximation,MATLAB central[CP/OL].2008.http://www.mathworks.com/matlabcentral.
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9 Pusa M.Correction to partial fraction decomposition coefficients for Chebyshev rational approximation on the negative real axis[J/OL].ar Xiv:1206.2880[math.NA],2012.
10张竞宇,马亚栋,陈义学,等.CRAM在放射性核素存量计算中的应用[J].核技术,2017,40(8):080502.DOI:10.11889/j.0253-3219.2017.hjs.40.080502.ZHANG Jingyu,MA Yadong,CHEN Yixue,et al.Research and verification of Monte Carlo burnup calculations based on Chebyshev rational approximation method[J].Nuclear Techniques,2017,40(8):080502.DOI:10.11889/j.0253-3219.2017.hjs.40.080502.
11 Moler C,Loan C F.Nineteen dubious ways to compute the exponential of a matrix,twenty-five years later[J].SIAM Review,2003,45(1):3-49.DOI:10.1137/S00361445024180.
12 Higham N J.The scaling and squaring method for the matrix exponential revisited[J].SIAM Review,2009,51(4):747-764.DOI:10.1137/090768539.
13 Sidje R B.Expokit:a software package for computing matrix exponentials[J].ACM Transactions on Mathematical Software,1998,24(1):130-156.DOI:10.1145/285861.285868.
14 Yamamoto A,Tatsumi M,Sugimura N.Numerical solution of stiff burnup equation with short half lived nuclides by the Krylov Subspace method[J].Journal of Nuclear Science and Technology,2007,44(2):147-154.DOI:10.1080/18811248.2007.9711268.
15 James R B,Donald J R.Sparse matrix computations[M].New York:Academic Press,1976:3-22.DOI:10.1016/B978-0-12-141050-6.50006-4.
16 Pusa M,Lepp?nen J.Solving linear systems with sparse gaussian elimination in the Chebyshev rational approximation method[J].Nuclear Science and Engineering,2013,175(3):250-258.DOI:10.13182/nse12-52.
2 Pusa M,Lepp?nen J.Computing the matrix exponential in burnup calculations[J].Nuclear Science and Engineering,2010,164(Pt23):140-150.DOI:10.13182/NSE09-14.
3 Isotalo A E,Aarnio P A.Comparison of depletion algorithms for large systems of nuclides[J].Annals of Nuclear Energy,2011,38(2):261-268.DOI:10.1016/j.anucene.2010.10.019.
4范文玎,孙光耀,张彬航,等.基于切比雪夫有理逼近方法的蒙特卡罗燃耗计算研究与验证[J].核技术,2016,39(4):040501.DOI:10.11889/j.0253-3219.2016.hjs.39.040501.FAN Wending,SUN Guangyao,ZHANG Binhang,et al.Research and verification of Monte Carlo burnup calculations based on Chebyshev rational approximation method[J].Nuclear Techniques,2016,39(4):040501.DOI:10.11889/j.0253-3219.2016.hjs.39.040501.
5吴明宇,王事喜,杨勇,等.回溯算法在燃耗计算中的应用[J].原子能科学技术,2013,47(7):1127-1132.DOI:10.7538/yzk.2013.47.07.1127.WU Mingyu,WANG Shixi,YANG Yong,et al.Application of backtracking algorithm to depletion calculations[J].Atomic Energy Science and Technology,2013,47(7):1127-1132.DOI:10.7538/yzk.2013.47.07.1127.
6 Cody W J,Meinardus G,Varga R S.Chebyshev rational approximations to exp(-x)in[0,+∞)and applications to heat-conduction problems[J].Journal of Approximation Theory,1969,2(1):50-65.DOI:10.1016/0021-9045(69)90030-6.
7 Schmelzer T.Carathéodory-Fejér approximation,MATLAB central[CP/OL].2008.http://www.mathworks.com/matlabcentral.
8 Carpenter A J,Ruttan A,Varga R S.Extended numerical computations on the“1/9”conjecture in rational approximation theory[J].Lecture Notes in Mathematics,1984,1105(4):383-411.DOI:10.1007/BFb0072427.
9 Pusa M.Correction to partial fraction decomposition coefficients for Chebyshev rational approximation on the negative real axis[J/OL].ar Xiv:1206.2880[math.NA],2012.
10张竞宇,马亚栋,陈义学,等.CRAM在放射性核素存量计算中的应用[J].核技术,2017,40(8):080502.DOI:10.11889/j.0253-3219.2017.hjs.40.080502.ZHANG Jingyu,MA Yadong,CHEN Yixue,et al.Research and verification of Monte Carlo burnup calculations based on Chebyshev rational approximation method[J].Nuclear Techniques,2017,40(8):080502.DOI:10.11889/j.0253-3219.2017.hjs.40.080502.
11 Moler C,Loan C F.Nineteen dubious ways to compute the exponential of a matrix,twenty-five years later[J].SIAM Review,2003,45(1):3-49.DOI:10.1137/S00361445024180.
12 Higham N J.The scaling and squaring method for the matrix exponential revisited[J].SIAM Review,2009,51(4):747-764.DOI:10.1137/090768539.
13 Sidje R B.Expokit:a software package for computing matrix exponentials[J].ACM Transactions on Mathematical Software,1998,24(1):130-156.DOI:10.1145/285861.285868.
14 Yamamoto A,Tatsumi M,Sugimura N.Numerical solution of stiff burnup equation with short half lived nuclides by the Krylov Subspace method[J].Journal of Nuclear Science and Technology,2007,44(2):147-154.DOI:10.1080/18811248.2007.9711268.
15 James R B,Donald J R.Sparse matrix computations[M].New York:Academic Press,1976:3-22.DOI:10.1016/B978-0-12-141050-6.50006-4.
16 Pusa M,Lepp?nen J.Solving linear systems with sparse gaussian elimination in the Chebyshev rational approximation method[J].Nuclear Science and Engineering,2013,175(3):250-258.DOI:10.13182/nse12-52.