非精确Newton方法中线性迭代收敛判据研究

冯选燕; 燕振国; 朱华君; 马燕凯; 冯新龙

doi:10.7638/kqdlxxb-2023.0001

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非精确Newton方法中线性迭代收敛判据研究

研究论文 | 更新时间：2026-05-28

- 非精确Newton方法中线性迭代收敛判据研究
- Study on the convergence criterion of linear iteration in inexact Newton methods
- 空气动力学学报 2023年41卷第12期页码：28-36
- 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：
  
  冯选燕（1995—），重庆人，硕士研究生，研究方向：计算数学. E-mail：fengxuanyan@stu.xju.edu.cn
  燕振国*，副研究员，研究方向：高阶精度计算方法及其应用. E-mail：yanzhg@mail.ustc.edu.cn
  朱华君*，副研究员，研究方向：高阶精度计算方法及其应用. E-mail：hjzhu@skla.cardc.cn
- 基金信息：
  
  国家自然科学基金（11902344，12172375）;国家重大项目（GJXM92579）;新疆维吾尔自治区自然科学基金重点项目（2022D01D32）
- DOI：10.7638/kqdlxxb-2023.0001
  中图分类号： O242.2;V211.3
- 纸质出版：2023
- 稿件说明：
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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.

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Keywords

references

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