On finite difference approximation of a matrix-vector product in the Jacobian-free Newton-Krylov method

被引：32

作者：

An, Heng-Bin ^{[1
]}

Wen, Ju ^{[2
]}

Feng, Tao ^{[3
,4
]}

机构：

[1] Inst Appl Phys & Computat Math, Key Lab Computat Phys, Beijing 100094, Peoples R China

[2] Ningxia Univ, Grad Sch, Yinchuan 750021, Peoples R China

[3] Univ Sci & Technol China, Sch Math Sci, Hefei 230052, Peoples R China

[4] China Acad Engn Phys, Grad Sch, Beijing 100083, Peoples R China

来源：

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS | 2011年 / 236卷 / 06期

基金：

国家高技术研究发展计划(863计划); 中国国家自然科学基金;

关键词：

Nonlinear equations; Jacobian-free Newton-Krylov method; Matrix-vector product; Finite difference step; NONEQUILIBRIUM RADIATION DIFFUSION; NONLINEAR-SYSTEMS; LINEAR-SYSTEMS; EQUATIONS; GMRES;

D O I：

10.1016/j.cam.2011.09.003

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

The Jacobian-free Newton-Krylov (JFNK) method is a special kind of Newton-Krylov algorithm, in which the matrix-vector product is approximated by a finite difference scheme. Consequently, it is not necessary to form and store the Jacobian matrix. This can greatly improve the efficiency and enlarge the application area of the Newton-Krylov method. The finite difference scheme has a strong influence on the accuracy and robustness of the JFNK method. In this paper, several methods for approximating the Jacobian-vector product, including the finite difference scheme and the finite difference step size, are analyzed and compared. Numerical results are given to verify the effectiveness of different finite difference methods. (C) 2011 Elsevier B.V. All rights reserved.

引用

页码：1399 / 1409

页数：11

共 14 条

[1] A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations [J].

An, Heng-Bin ;

Bai, Zhong-Zhi .

APPLIED NUMERICAL MATHEMATICS, 2007, 57 (03) :235-252

[2] On choosing a nonlinear initial iterate for solving the 2-D 3-T heat conduction equations [J].

An, Heng-Bin ;

Mo, Ze-Yao ;

Xu, Xiao-Wen ;

Liu, Xu .

JOURNAL OF COMPUTATIONAL PHYSICS, 2009, 228 (09) :3268-3287

[3] On using approximate finite differences in matrix-free Newton-Krylov methods [J].

Brown, Peter N. ;

Walker, Homer F. ;

Wasyk, Rebecca ;

Woodward, Carol S. .

SIAM JOURNAL ON NUMERICAL ANALYSIS, 2008, 46 (04) :1892-1911

[4] CONVERGENCE THEORY OF NONLINEAR NEWTON-KRYLOV ALGORITHMS [J].

BROWN, PN ;

SAAD, Y .

SIAM JOURNAL ON OPTIMIZATION, 1994, 4 (02) :297-330

[5] HYBRID KRYLOV METHODS FOR NONLINEAR-SYSTEMS OF EQUATIONS [J].

BROWN, PN ;

SAAD, Y .

SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING, 1990, 11 (03) :450-481

[6]

COLLIER AM, 2006, UCRLSM208116 CTR APP

[7]

Kelley C.T., 1987, Iterative Methods for Linear and Nonlinear Equations

[8] Jacobian-free Newton-Krylov methods: a survey of approaches and applications [J].

Knoll, DA ;

Keyes, DE .

JOURNAL OF COMPUTATIONAL PHYSICS, 2004, 193 (02) :357-397

[9] Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion [J].

Mousseau, VA ;

Knoll, DA ;

Rider, WJ .

JOURNAL OF COMPUTATIONAL PHYSICS, 2000, 160 (02) :743-765

[10] New physics-based preconditioning of implicit methods for non-equilibrium radiation diffusion [J].

Mousseau, VA ;

Knoll, DA .

JOURNAL OF COMPUTATIONAL PHYSICS, 2003, 190 (01) :42-51

← 1 2 →