Fourth-Order Energy-Consistent Average Vector Field Discrete Gradient Methods for Dissipative Hamiltonian Systems
Abstract
Discrete gradient methods provide a structure-preserving approach to energy-consistent time integration of dissipative Hamiltonian systems. In their conventional formulation, however, they generally attain at most second-order convergence. Extending the perturbation principle previously developed for Poisson systems to the dissipative setting, we derive fourth-order average vector field discrete gradient schemes. A congruence-based completion preserves the positive semi-definiteness of the dissipation matrix, thereby guaranteeing unconditional energy dissipativity for all positive step sizes. This completion concept provides a systematic route to higher-order energy-consistent schemes. For purely dissipative systems, an alternative exponential completion is derived, together with structure-preserving rational approximations. In particular, we characterize Padé approximants that preserve positive semi-definiteness and can be incorporated into energy-associated splitting schemes. Numerical experiments on a damped physical pendulum and a dissipative Fermi--Pasta--Ulam system demonstrate the importance of the positive semi-definite completion for avoiding nonphysical numerical instabilities and capturing the correct qualitative dynamics. They further show that the proposed methods achieve the desired accuracy and energy consistency at a computational cost competitive with existing Galerkin-type approaches. Moreover, the fourth-order schemes outperform the conventional second-order discrete gradient method in computational efficiency over the considered accuracy range, despite their higher per-step cost.
Related
- Nonlinear differential-algebraic systems with dissipative port-Hamiltonian structure
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BibTeX
@misc{lautwein2026fourthorder,
title = {{Fourth-Order Energy-Consistent Average Vector Field Discrete Gradient Methods for Dissipative Hamiltonian Systems}},
author = {Lucas Lautwein and Nicole Marheineke and Håkon Noren Myhr and Kevin Schäfers},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24692},
eprint = {2609.24692},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24692},
}
RIS
TY - UNPB
TI - Fourth-Order Energy-Consistent Average Vector Field Discrete Gradient Methods for Dissipative Hamiltonian Systems
AU - Lucas Lautwein
AU - Nicole Marheineke
AU - Håkon Noren Myhr
AU - Kevin Schäfers
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24692
UR - https://arxiv.org/abs/2609.24692
AB - Discrete gradient methods provide a structure-preserving approach to energy-consistent time integration of dissipative Hamiltonian systems. In their conventional formulation, however, they generally attain at most second-order convergence. Extending the perturbation principle previously developed for Poisson systems to the dissipative setting, we derive fourth-order average vector field discrete gradient schemes. A congruence-based completion preserves the positive semi-definiteness of the dissipation matrix, thereby guaranteeing unconditional energy dissipativity for all positive step sizes. This completion concept provides a systematic route to higher-order energy-consistent schemes. For purely dissipative systems, an alternative exponential completion is derived, together with structure-preserving rational approximations. In particular, we characterize Padé approximants that preserve positive semi-definiteness and can be incorporated into energy-associated splitting schemes. Numerical experiments on a damped physical pendulum and a dissipative Fermi--Pasta--Ulam system demonstrate the importance of the positive semi-definite completion for avoiding nonphysical numerical instabilities and capturing the correct qualitative dynamics. They further show that the proposed methods achieve the desired accuracy and energy consistency at a computational cost competitive with existing Galerkin-type approaches. Moreover, the fourth-order schemes outperform the conventional second-order discrete gradient method in computational efficiency over the considered accuracy range, despite their higher per-step cost.
ER -