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

Ju Liu

doi:10.1016/j.jcp.2023.112177

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arXiv · 2212.09099

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

Ju Liu

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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.

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Ju Liu. 2022-12-18. On the design of energy-decaying momentum-conserving integrator for nonlinear dynamics using energy splitting and perturbation techniques. https://doi.org/10.1016/j.jcp.2023.112177

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