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

Authors:
Heng-Bin An;Ju Wen;Tao Feng
Affiliations:
Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100094, China;Graduate School of Ningxia University, Yinchuan 750021, China;School of Mathematical Sciences, University of Science and Technology of China, Hefei 230052, China and Graduate School of China Academy Engineering Physics, Beijing 100083, China
Venue:
Journal of Computational and Applied Mathematics
Year:
2011

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Abstract

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.