Study on the convergence criterion of linear iteration in inexact Newton methods

FENG Xuanyan; YAN Zhenguo; ZHU Huajun; MA Yankai; FENG Xinlong

doi:10.7638/kqdlxxb-2023.0001

English Version Journal Login

您当前的位置：

首页 >

文章列表页 >

Study on the convergence criterion of linear iteration in inexact Newton methods

更新时间：2026-05-28

- Study on the convergence criterion of linear iteration in inexact Newton methods
- Vol. 41, Issue 12, Pages: 28-36(2023)
- 作者机构：
  
  1. 新疆大学数学与系统科学学院,乌鲁木齐,830000
  2. 空天飞行空气动力科学与技术全国重点实验室,绵阳,621000
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2023.0001
  CLC： O242.2;V211.3
- Published：2023
- 稿件说明：
移动端阅览
冯选燕, 燕振国, 朱华君, 等. 非精确Newton方法中线性迭代收敛判据研究[J]. 空气动力学学报, 2023, 41(12): 28−36.

FENG Xuanyan, YAN Zhenguo, ZHU Huajun, et al. Study on the convergence criterion of linear iteration in inexact Newton methods[J]. 2023, 41(12): 28-36.
冯选燕, 燕振国, 朱华君, 等. 非精确Newton方法中线性迭代收敛判据研究[J]. 空气动力学学报, 2023, 41(12): 28−36. DOI： 10.7638/kqdlxxb-2023.0001

FENG Xuanyan, YAN Zhenguo, ZHU Huajun, et al. Study on the convergence criterion of linear iteration in inexact Newton methods[J]. 2023, 41(12): 28-36. DOI： 10.7638/kqdlxxb-2023.0001.

摘要

在计算流体力学中，采用隐式时间推进方法时通常需要采用Newton类迭代方法求解大型非线性离散系统。每步非线性迭代需求解由非线性系统Jacobian矩阵组成的大型线性方程组，其中线性方程组求解误差会对非线性系统的收敛性产生显著影响，然而对存在Jacobian矩阵误差情况下的线性迭代收敛判据缺乏深入的研究。本文针对上述问题，首先给出了存在Jacobian矩阵误差和线性迭代误差情况下Newton迭代式的形式，并通过数值测试验证了Jacobian矩阵误差对迭代产生较大影响的可能性；其次对常见的两种不同类型的线性迭代收敛判据进行了数值测试，重点研究了存在Jacobian矩阵误差情况下容易产生的过度求解问题；最后，结合上述两类判据的特点发展了一种新的线性迭代收敛判据，结果表明：新提出的迭代收敛判据能够有效缓解过度求解问题，从而提高计算效率。

Abstract

Newton-like iteration methods are usually used for implicit time stepping to solve large scale nonlinear systems. Each of the nonlinear iterative step requires solving large linear equations composed of a Jacobian matrix of the nonlinear system

and the error of solving such linear equations can have a significant impact on the convergence of the nonlinear system. However

the convergence criterion of the linear iteration is less studied when the Jacobian matrix error exists in the Newton method. Aiming at the above problem

this work first presents the Newton iterative formula with both the Jacobian matrix error and the linear iteration error

and verifies the significant influence of the Jacobian matrix error on the iteration through numerical tests. Then

two different types of linear iterative convergence criteria are tested numerically

and the produced over-solving problem is investigated at the presence of Jacobian matrix error. Finally

a new convergence criterion of the linear iteration is developed based on two of the previous convergence criteria. Numerical tests suggest that the proposed method is effective to relieve over-solving problems

thus improves the computational efficiency.

关键词

Keywords

references

CAI X, KEYES D E, VENKATAKRISHNAN V. Newton-Krylov-Schwarz: An Implicit Solver for CFD[R]. Institute for Computer Applications in Science and Engineering (ICASE), 1995.

DENG X G, LIU X, MAO M L, et al. Advances in high-order accurate weighted compact nonlinear schemes[J]. Advances in Mechanics, 2007, 37(3): 417-427 (in Chinese).

LEI G D, LI W A, REN Y X. A high-order unstructured-grid WENO FVM for compressible flow computation[J]. Chinese Journal of Computational Physics, 2011, 28(5): 633-640 (in Chinese).

LUO H, ABSILLIS G, NOURGALIEV R. A moving discontinuous Galerkin finite element method with interface condition enforcement for compressible flows[J]. Journal of Computational Physics, 2021, 445: 110618. DOI: 10.1016/j.jcp.2021.110618

YAN Z G. Investigation of implicit time integration methods with HDCS schemes[D]. Mianyang, China: China Aerodynamica Research and Development Center, 2013 (in Chinese).

PULLIAM T H. Development of implicit methods in CFD NASA Ames Research Center 1970s-1980s[J]. Computers Fluids, 2011, 41(1): 65-71. DOI: 10.1016/j.compfluid.2010.09.016

YAN C, QU F, ZHAO Y T, et al. Review of development and challenges for physical modeling and numerical scheme of CFD in aeronautics and astronautics[J]. Acta Aerodynamica Sinica, 2020, 38(5): 829-857 (in Chinese).

SHANNO D F, KETTLER P C. Optimal conditioning of quasi-Newton methods[J]. Mathematics of Computation, 1970, 24(111): 657-664.

CHISHOLM T T , ZINGG D W . A Jacobian-free Newton–Krylov algorithm for compressible turbulent fluid flows[J]. Journal of Computational Physics, 2009, 228(9): 3490-3507.

LE D V, WHITE J, PERAIRE J, et al. An implicit immersed boundary method for three-dimensional fluid-membrane interactions[J]. Journal of Computational Physics, 2009, 228(22): 8427-8445. DOI: 10.1016/j.jcp.2009.08.018

PERSSON P-O, PERAIRE J. Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the navier-stokes equations[J]. SIAM Journal on Scientific Computing, 2008, 30(6): 2709-2733. DOI: 10.1137/070692108

ZHAO X Y, SUN D F, TOH K C. A Newton-CG augmented Lagrangian method for semidefinite programming[J]. SIAM Journal on Optimization, 2010, 20(4): 1737-1765. DOI: 10.1137/080718206

DEMBO R S, ELSENSTAT S T, STEIHAUG T. Inexact Newton Methods[J]. SIAM Journal on Numerical Analysis, 1982, 19(2): 400-408.

VANDEN K J, ORKWIS P D. Comparison of numerical and analytical Jacobians[J]. AIAA Journal, 1996, 34(6): 1125-1129.

EZERTAS A , EYI S. Performances of numerical and analytical Jacobians in flow and sensitivity analysis[C]// 19th AIAA Computational Fluid Dynamics, San Antonio, Texas, 2009.

YANG X Q, CHENG J , LUO H, et al. Robust implicit direct discontinuous Galerkin method for simulating the compressible turbulent flows[J]. AIAA Journal , 2019, 57(3): 1113-1132.

EISENSTAT S C, WALKER H F. Choosing the forcing terms in an inexact Newton method[J]. SIAM Journal on Scientific Computing, 1996, 17(1): 16-32. DOI: 10.1137/0917003

AN H B, MO Z Y, LIU X P. A choice of forcing terms in inexact Newton method[J]. Journal of Computational and Applied Mathematics, 2007, 200(1): 47-60. DOI: 10.1016/j.cam.2005.12.030

KNOLL D A, MCHUGH P R, KEYES D E. Newton-Krylov methods for low-Mach-number compressible combustion[J]. AIAA Journal, 1996, 34(5): 961-967.

HAYAMI K, YIN J F, ITO T. GMRES methods for least squares problems[J]. SIAM Journal on Matrix Analysis and Applications, 2010, 31(5): 2400-2430. DOI: 10.1137/070696313

YAN Z G, LIU H Y, MAO M L, et al. Convergence property investigation of GMRES method based on high-order dissipative compact scheme[J]. Acta Aeronautica et Astronautica Sinica, 2014, 35(5): 1181-1192 (in Chinese).

PARK H, NOURGALIEV R R, MARTINEAU R C, et al. On physics-based preconditioning of the Navier-Stokes equations[J]. Journal of Computational Physics, 2009, 228(24): 9131-9146. DOI: 10.1016/j.jcp.2009.09.015

ZHAO Z, ZHANG L P, HE L, et al. PHengLEI: a large scale parallel CFD framework for arbitrary grids[J]. Chinese Journal of Computers, 2019, 42(11): 2368-2383 (in Chinese).

Views

下载量

CSCD

Alert me when the article has been cited

提交

Tools

Publicity Resources

Gust loads alleviation of flying wing aircraft based on jet flow control

A parallel fast iteration method for wall distance implemented in PHengLEI

Aerodynamic and wind energy harvesting integrative design for UAVs

Engine cooling fan layout study of a medium size pickup truck

Development and application of a credibility assessment platform for CFD software

Related Author

Xia Yuhang

Guo Pengyu

Liao Xueqi

Zhang Weiyuan

Shi Zhiwei

Mei Yang

Ren Yuxin

Zhou Hanyu

Related Institution

College of Aerospace Engineering, Nanjing University of Aeronautics And Astronautics, Nanjing

School of Aerospace Engineering, Tsinghua University, Beijing

School of Aeronautics, Northwestern Polytechnical University, Xi'an

National Key Laboratory of Aircraft Configuration Design, Xi'an

Vehicle Technology section, Model Development Department, Technology Center, Zhengzhou Nissan Automobile CO. , LTD, Zhengzhou

AI问答

⁰