Conserving/dissipative algorithms and designs for a system of N particles: Total energy framework and single-field form

Conserving and dissipative implicit algorithm designs in the single-field form via the total energy framework for N-body systems are summarized in this paper which provides new avenues with improved physical interpretation, and can also explain analogous past efforts. We show various algorithm designs via two distinct approaches based on the algorithmic time level consistency theorem, namely, the discrete total energy conservation method, and the classical and normalized time weighted residual methodologies; and therein draw comparisons among the various time-stepping algorithms and features. A simple numerical simulation is shown to demonstrate the pros/cons for the conserving properties of various algorithms.