A general framework for conservative single-step time-integration schemes with higher-order accuracy for a central-force system

Abstract

A general framework for algorithms that conserve angular momentum for single-body central-force problems is presented. It is shown that any family of momentum-conserving algorithms can have at most three free parameters, one of which may be used to ensure energy conservation (and hence will be configuration-dependent). Further restrictions can be made that enable the algorithms to recover the orbits of relative equilibria of the underlying physical problem. In addition, the algorithms can be made time-reversible, whilst still leaving two parameters unspecified. The order of accuracy of a general momentum-conserving family is analysed, and it is shown that energy–momentum algorithms that preserve the underlying physical relative equilibria can have unlimited accuracy if the two remaining parameters are appropriately chosen functions of the configuration and the time-step: this does not require any additional degrees of freedom, extra stages of calculation or information from past solutions. Numerical examples are given that show the performance of some representative higher-order schemes when applied to stiff and non-stiff problems, and the issue of Newton–Raphson convergence is discussed.