Conservative single-step time-integration schemes with higher-order accuracy for multi-particle dynamics with local two-point potentials

A general framework for algorithms that conserve linear and angular momenta for problems of multi-particle mechanics is presented. Conditions for energy conservation are derived, and different manners in which this may be achieved are discussed. A detailed examination of the relative equilibrium states is carried out, and conditions under which algorithms preserve these states are given; in particular, algorithms can be designed to capture the exact solutions of relative equilibrium problems, although these algorithms are unlikely to be energy-conserving. Following on from the approach proposed by Argyris et al. [J.H. Argyris, P.C. Dunne, T. Angelopoulos, Dynamic response by large step integration, Earthquake Engrg. Struct. Dynam. 2 (1973) 185-203], the local accuracy characteristics of algorithms are investigated thoroughly, and it is shown that there is no limit to the order of accuracy that can be achieved by algorithms in this framework, even for problems with time-dependent forces. No extra stages of calculation or additional degrees of freedom are required to be present, although the sparsity of the resulting system of equations is compromised. A few examples of new algorithms are given, and their properties verified on some model problems.