Comparison of Two Kinds of Scaling Methods in Two-Body Problems

Manifold correction methods are not only useful in controlling the growth of the energy error, but also easy to operate in application. By comparison, we found that the velocity scaling method of Ma et al. is a one-step method, the linear transformation idea discovered by Fukushima is a two-step algorithm. From numerical examples, it is easy to see the two methods have almost the same effectiveness in preserving energy integral. Both can improve the orbital semi-major axis and the mean anomaly similarly. However, since the velocity scaling factor of the latter is obtained from energy integral and Laplace integral, it is better at improving the accuracy of the eccentricity and the argument of perihelion by several orders.