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

Conserving and controllable numerically dissipative implicit algorithm designs in the single-field form via a new Total Energy framework for a system of N particles are shown in this paper which provides new avenues with improved physical interpretation, and can also explain analogous past efforts. Unlike the traditional approach in the Newtonian framework with vector formalism, the scalar formalism via the Total Energy framework naturally provides an improved physical insight in both continuous and discrete time systems with several attractive computational features. We show various algorithm designs via two distinct approaches, namely, the mean value theorem and the classical and normalized time weighted residual methodologies, and draw comparisons among the various time-stepping algorithms and features. Although the parent general framework is in the single-field form, it covers most past efforts in the two-field form in the sense of equivalence to conserving algorithms. A simple numerical simulation is also shown to demonstrate the pros/cons for the conserving properties of the various algorithms.