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.

无雅可比的 Newton-Krylov (JFNK) 方法是求解计算物理学中产生的非线性代数方程的流行方法。关键问题是雅可比向量积的计算，通常通过有限差分方法完成。然而，这些方法同时存在截断误差和舍入误差，精度在很大程度上取决于差分步长的复杂选择。在一些极端情况下，即使选择了最佳的差异步长，精度可能仍然无法满足内部 Krylov 迭代的要求。在本文中，我们将复杂步长有限差分 (CSFD) 方法扩展到 JFNK 方法。提供了一些加速该方法的技巧。提供了多个示例来揭示 JFNK 与 CSFD 的性能，并比较了逼近雅可比向量乘积的不同方法。用一种相对简单的实现方式证明了CSFD 方法非常适合JFNK 方法，从而导致极其准确和稳定的数值性能。与传统的有限差分方法形成鲜明对比的是，它使我们摆脱了对差分步长的干扰选择，并且可以完全依赖该方法而没有任何准确性问题。