engleski

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

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. The main idea investigated is that algorithmic parameters, which are normally taken as constants, may be made dependent on the actual configuration in a non-linear way 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 iterative solution process. The family of algorithms presented includes several well-known implicit algorithms with constant integration parameters 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 general or specific 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. Additionally, the Keplerian motion is specifically analysed to show how the anomalous perihelion drift present in some of the known algorithms may be successfully removed.

conserving time integration; non-linear configuration-dependent integration parameters; relative equilibria; Laplace-Runge-Lenz vector; Kepler's problem

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