In our previous paper (SIAM J. Math. Anal. 28 (1997), 381-388), we showed that the qualitative properties of a Morse-Smale gradient-like flow are preserved by its discretization mapping obtained via numerical methods. In this paper, we extend the result to flows which satisfy Axiom A and the strong transversality condition. We prove that if p ≥ 2, Φt is a Cp + 1 flow on a compact manifold satisfying Axiom A and the strong transversality condition, and Nh is a numerical method of step size h and order p, then for all sufficiently small h, there are a homeomorphism Hh and a continuous real-valued function τh on M such that Hh, ○ Φh+hτh(x)(x) = Nh ○ Hh(x) and Hh is O(hp)-close to the identity map on M.