The accuracy check in numerical integration of dynamical systems

A routine method for checking the accuracy of the numerical solution in the case of dynamical systems has been based on the use of known integrals of dynamical systems. It is suspected that the uncritical use of such integrals as accuracy diagnostic tools could, at least occasionally, lead to incorrect results. The present investigation is concerned with questions related to the reliability of the considered accuracy tests. Attention is given to the observation that the use of the total energy integral as an accuracy check in a three-body numerical experiment is quite unreliable. A suggestion made by Szebehely and Bettis (1970) regarding the use of an invariant integral relation as an accuracy check is also considered. It is found that numerical solutions have an 'inclination to keep each integral constant' so that they have much higher apparent accuracy than the coordinates and velocities. A revised technique is suggested and tested. This technique is found to provide a good check in stable regions.