A velocity scaling method with least-squares correction of several constraints

Abstract

A new scale transformation to the integrated velocity vector is designed to monitor the accumulation of numerical errors in several integrals of motion. The scale factor is derived from the least-squares correction that minimizes the sum of the squares of the errors of these integrals. In order to preserve an invariant, we employ the velocity scaling method for rigorously satisfying the constraint. When adjusting many constants, the new scheme like other existing methods is valid to typically reduce the integration errors below those of an uncorrected integrator. Via integral invariant relations, the new method is also able to treat slowly-varying quantities, such as the Keplerian energy and the Laplace vector, for a perturbed Keplerian problem or each of multiple bodies in the solar system dynamics. Consequently it does nearly agree with the rigorous dual scaling method in the sense of drastically improving the integration accuracy. As one of its advantages, the implementation of the new method is significantly easier than that of other methods. In particular, the method can be simply applied to a complicated dynamical system with some constraints.