Performance Assessment of Energy-preserving, Adaptive Time-step Variational Integrators

TL;DR

This paper evaluates energy-preserving, adaptive time-step variational integrators, demonstrating their numerical stability and effectiveness in conserving energy and momentum in nonintegrable systems through theoretical analysis and practical examples.

Contribution

It introduces and assesses an energy-preserving, adaptive time-step variational integrator that maintains multiple invariants and analyzes its numerical stability and performance.

Findings

The integrator conserves energy and momentum effectively.

Numerical stability is confirmed via backward error analysis.

Performance demonstrated on Kepler's problem and simple mechanical systems.

Abstract

A fixed time-step variational integrator cannot preserve momentum, energy, and symplectic form simultaneously for nonintegrable systems. This barrier can be overcome by treating time as a discrete dynamic variable and deriving adaptive time-step variational integrators that conserve the energy in addition to being symplectic and momentum-preserving. Their utility, however, is still an open question due to the numerical difficulties associated with solving the discrete governing equations. In this work, we investigate the numerical performance of energy-preserving, adaptive time-step variational integrators. First, we compare the time adaptation and energy performance of the energy-preserving adaptive algorithm with the adaptive variational integrator for Kepler's two-body problem. Second, we apply tools from Lagrangian backward error analysis to investigate numerical stability of the energy-preserving adaptive algorithm. Finally, we consider a simple mechanical system example to illustrate the backward stability of this energy-preserving, adaptive time-step variational integrator.