On the design of energy-decaying momentum-conserving integrator for nonlinear dynamics using energy splitting and perturbation techniques

TL;DR

This paper introduces energy-splitting based integrators for nonlinear Hamiltonian systems that are energy-decaying and momentum-conserving, avoiding numerical instability issues of traditional quotient-based methods.

Contribution

It develops a novel class of structure-preserving integrators using energy splitting and perturbation techniques, eliminating the need for quotient formulas and ensuring stability.

Findings

The proposed integrators are stable and preserve invariants.

They demonstrate effective energy decay and momentum conservation in numerical tests.

The methods serve as alternatives to classical integrators when instability occurs.

Abstract

This work proposes a suite of numerical techniques to facilitate the design of structure-preserving integrators for nonlinear dynamics. The celebrated LaBudde-Greenspan integrator and various energy-momentum schemes adopt a difference quotient formula in their algorithmic force definitions, which suffers from numerical instability as the denominator gets close to zero. There is a need to develop structure-preserving integrators without invoking the quotient formula. In this work, the potential energy of a Hamiltonian system is split into two parts, and specially developed quadrature rules are applied separately to them. The resulting integrators can be regarded as classical ones perturbed with first- or second-order terms, and the energy split guarantees the dissipative nature in the numerical residual. In the meantime, the conservation of invariants is respected in the design. A complete analysis of the proposed integrators is given, with representative numerical examples provided to demonstrate their performance. They can be used either independently as energy-decaying and momentum-conserving schemes for nonlinear problems or as an alternate option with a conserving integrator, such as the LaBudde-Greenspan integrator, when the numerical instability in the difference quotient is detected.