JFNK方法在SN输运计算中的应用

doi:10.19596/j.cnki.1001-246x.8282

摘要/Abstract

摘要：

JFNK(Jacobian-free Newton-Krylov)方法是一种求解非线性方程的高效迭代算法。传统输运计算中的负通量修正与k-特征值迭代使得原本线性的输运计算转变为非线性问题数值求解。为提高非线性输运问题的计算效率，将这两类非线性问题离散成残差形式的非线性方程组，并采用JFNK方法对其进行迭代求解。分析不同约束条件对JFNK方法性能的影响，并将其与NK(Newton-Krylov)方法进行对比。针对JFNK方法的内迭代过程，分析两类子空间方法(GMRES(m)与LGMRES)对整体计算效率的影响。数值结果表明：①相比于传统的幂迭代方法，JFNK方法具有更高的计算效率；②Jacobian矩阵向量积的差分近似对结果没有影响，且基于物理的约束条件比标准的数学约束更加高效；③LGMRES可以充分利用子空间的信息，从而使得JFNK方法整体表现更加高效。

关键词: 负通量修正, k-特征值问题, 非线性输运, 离散纵标, JFNK方法

Abstract:

JFNK method is an efficient method for nonlinear problems. We consider two nonlinearities of neutron transport equation, the negative flux correction and the k-eigenvalue problem. The nonlinear problems are transformed into nonlinear residual equation form. And then we use JFNK method to solve them. We analyze impact of different constraints on the performance of JFNK, and compare JFNK method with NK method. LGMRES is used instead of restarting GMRES(m) method. Numerical results show that: ① Compared with SI method, JFNK has higher computational efficiency, even in the case of high scattering ratio; ② Difference approximation of Jacobian matrix-vector multiplication has no effect on the result, and the physics-based constraints are more efficient than standard mathematical constraints; ③ In addition, as an alternative to GMRES(m), LGMRES makes JFNK more efficient.

Key words: negative-flux fixup, k-eigenvalue, nonlinear transport, discrete ordinates, JFNK

中图分类号:

TL323

朱凯博, 徐龙飞, 潘流俊, 沈华韵. JFNK方法在S_N输运计算中的应用[J]. 计算物理, 2021, 38(4): 381-392.

Kaibo ZHU, Longfei XU, Liujun PAN, Huayun SHEN. Application of JFNK Method in S_N Transport Calculations[J]. Chinese Journal of Computational Physics, 2021, 38(4): 381-392.

图/表 16

参考文献 16

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区域	Σ_t	Σ_S	Q
(I)	1.0	1.0	1.0
(II)	0.001	0.0	0.0
(III)	100.0	0.0	0.0
(IV)	10.0	9.999	0.0

区域	Σ_t	Σ_S	Q
(I)	1.0	1.0	1.0
(II)	0.001	0.0	0.0
(III)	100.0	0.0	0.0
(IV)	10.0	9.999	0.0

方法	误差	迭代次数	扫描次数	时间/s
SI	9.30×10^-6	160	160	10.395
GMRES	5.39×10^-6	14	18	1.207
JFNK	6.28×10^-6	(17)	22	1.404

方法	误差	迭代次数	扫描次数	时间/s
SI	9.30×10^-6	160	160	10.395
GMRES	5.39×10^-6	14	18	1.207
JFNK	6.28×10^-6	(17)	22	1.404

方法	误差	迭代次数	扫描次数	时间/s
SI	9.60×10^-6	158	158	12.744
JFNK(η₁)	6.43×10^-7	(6，8，15，16)	55	4.665
JFNK(η₂)	6.23×10^-8	(16，19，18)	61	5.094