Extension of complex step finite difference method to Jacobian-free Newton–Krylov​ method

Abstract

Jacobian-free Newton–Krylov (JFNK) method is a popular approach to solve nonlinear algebraic equations arising from computational physics. The key issue is the calculation of Jacobian-vector product, commonly done through finite difference methods. However, these approaches suffer from both truncation error and round-off error, and the accuracy heavily depends on a sophisticated choice of the difference step size. In some extreme cases, even with the best choice of the difference step size, the accuracy may still not meet the requirement for the inner Krylov iteration. In this paper, we extend the complex step finite difference (CSFD) method to the JFNK method. Some tips are presented for accelerating the method. Multiple​ examples are presented to reveal the performance of the JFNK with the CSFD, and different methods for approximating the Jacobian-vector product are compared. It is demonstrated with a relatively easy way of implementation that the CSFD method is well-suited for the JFNK method, leading to extremely accurate and stable numerical performance. In strong contrast to traditional finite difference approaches, it frees us from the disturbing choice for the difference step size, and one can fully rely on the method without any accuracy concerns.