Stabilization of constraints and integrals of motion in dynamical systems

Conventional numerical methods, when applied to the ordinary differential equations of motion of classical mechanics, conserve the total energy and angular momentum only to the order of the truncation error. Since these constants of the motion play a central role in mechanics, it is a great advantage to be able to conserve them exactly. A new numerical method is developed, which is a generalization to arbitrary order of the "discrete mechanics" described in earlier work, and which conserves the energy and angular momentum to all orders. This new method can be applied much like a "corrector" as a modification to conventional numerical approximations, such as those obtained via Taylor series, Runge-Kutta, or predictor-corrector formulae. The theory is extended to a system of particles in Part II of this work.

https://link.springer.com/content/pdf/10.1007/BF01396562.pdf

Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion

The work of Aarseth and Zare (1974) is extended to provide aglobal regularisation of the classical gravitational three-body problem: by transformation of the variables in a way that does not depend on the particular configuration, we obtain equations of motion which are regular with respect to collisions between any pair of particles. The only cases excepted are those in which collisions between more than one pair occur simultaneously and those in which at least one of the masses vanishes. However, by means of the same principles the restricted problem is regularised globally if collisions between the two primaries are excluded. Results of numerical tests are summarised, and the theory is generalised to provide global regularisations, first, for perturbed three-body motion and, second, for theN-body problem. A way of increasing the number of degrees of freedom of a dynamical system is central to the method, and is the subject of an Appendix.

A global regularisation of the gravitational N -body problem

Extending Nacozy's idea of manifold correction by using the concept of the integral invariant relation, we propose a new approach to numerically integrate quasi-Keplerian orbits. The method integrates the time evolution of the Kepler energy and the usual equation of motion simultaneously. Then it directly adjusts the integrated position and velocity by a spatial scale transformation in order to satisfy the Kepler energy relation rigorously at every integration step. The scale factor is determined by solving an associated cubic equation precisely with the help of Newton's method. In treating multiple bodies, the Kepler energies are integrated for each body and the scale factors are adjusted separately. The implementation of the new method is simple, the added cost of computation is low, and its applicability is wide. Numerical experiments show that the scaling reduces the integration error drastically. In the case of pure Keplerian orbits, the truncation error grows linearly with respect to time, and the round-off error grows more slowly than that. When perturbations exist, a component that grows with the second or a higher power of time appears in the truncation error, but its magnitude is reduced significantly as compared with the case without scaling. The rate of decrease varies roughly as the 5/4 to 5/2 power of the strength of the perturbing acceleration, where the power index depends on the type of perturbation. The method seems to suppress the accumulation of round-off errors in the perturbed cases, although the details remain to be investigated. The new approach provides a fast and high-precision device with which to simulate the orbital motions of major and minor planets, natural and artificial satellites, comets, and space vehicles at a negligible increase in computational cost.

/pdf/efficient-orbit-integration-by-scaling-for-kepler-energy-1ed3z5pf34.pdf

Efficient orbit integration by scaling for Kepler energy consistency

We develop a brand-new method of manifold correction that extends Nacozy's approach to correcting individual Kepler energies that vary with time in n-body problems of planetary dynamics. Besides discussing this method, we also closely study other existing manifold correction techniques, including the two scaling methods of Fukushima and Liu & Liao. We give a detailed discussion about them analytically and numerically. Our numerical experiments show that all three methods are effective and work almost equivalently in raising the precision of the numerical integration when the adopted basic integrators can give sufficient precision to chosen integrals or quasi-integrals. For simplicity of application, Liu & Liao's approach is worth recommending. However, this approach cannot improve the orbital eccentricity at all. Because of this, we prefer to use this brand-new extension, as well as the scaling method of Fukushima.

https://iopscience.iop.org/article/10.1086/516827/pdf

COMPARISON AMONG CORRECTION METHODS OF INDIVIDUAL KEPLER ENERGIES IN n-BODY SIMULATIONS

A stabilization of the classical equations of two-body motion is offered. It is characterized by the use of the regularizing independent variable (eccentric anomaly) and by the addition of a control-term to the differential equations. This method is related to the KS-theory (Stiefel, 1970) which performed for the first time a stabilization of the Kepler motion. But in contrast to the KS-theory our method does not transform the coordinates of the particle. As far as the theory of stability and the numerical experiments are concerned we restrict ourselves to thepure Kepler motion. But, of course, the stabilizing devices will also improve the accuracy of the computation of perturbed orbits. We list, therefore, also the equations of the perturbed motion.

Numerical stabilization of the differential equations of Keplerian motion

Numerical integration of unstable differential equations should be avoided since a numerical error during thenth step produces erroneous initial values for the next step and thus deteriorates the subsequent integration in an unstable manner. A method is offered to stabilize the equations of motion corresponding to a given HamiltonianH by transformingH into a new HamiltonianH
* which is equivalent to the Hamiltonian of a harmonic oscillator. In contrast to other methods of stabilization the realm of canonical mechanics is thus not abandoned. Perturbations are discussed and as examples the Keplerian motion and the motion of a gyroscope are presented.

Stabilization by Manipulation of the Hamiltonian

In the publication Baumgarte and Stiefel (1974a) a stabilization of the Keplerian motion was offered by making use of a manipulation of the Hamiltonian. By this stabilization technique the given HamiltonianH(pi,qi) is replaced by a new HamiltonianH*, which leads to Lyapunov-stable differential equations of motion.

https://link.springer.com/content/pdf/10.1007/BF01228537.pdf

Stabilized Kepler motion connected with analytic step adaptation

The paper introduces a new time element to be used with a general time transformation for satellite equations of motion. The purpose of this time element is to reduce the growth of the numerical errors with respect to the time integration. It is characteristic for the new time element τ, that the relation between the timet and the element τ does not depend on the independent variables.

J. Baumgarte

Papers

Numerical stabilization of the differential equations of Keplerian motion

Stabilization by Manipulation of the Hamiltonian

Stabilized Kepler motion connected with analytic step adaptation

A new time element for a general time transformation