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

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.