Implicit one-step dynamic algorithms with configuration-dependent parameters: application to central force fields

Abstract

This work presents a family of implicit one-step algorithms for non-linear dynamics possessing the ability to conserve some of the integrals of the underlying Hamiltonian motion. Algorithmic parameters, normally taken as constants, are here made dependent on the actual configuration and thus become particularly suitable for capturing some of the additional properties of a non-linear motion, while at the same time not increasing the computational burden in the Newton–Raphson solution process. The family of algorithms presented includes several well-known implicit algorithms as special cases. The idea is presented on a two-degree-of-freedom problem of a point mass moving in a central force field. The concept is illustrated on two model examples taken to represent some of the issues in non-linear elastodynamics and celestial mechanics: a stiff mechanical problem and a problem possessing an additional integral of motion. It is shown that in this way overall motion of a stiff problem may be described much better even using very large time steps while, for the Keplerian motion, the anomalous perihelion drift may be successfully removed.