Operator splitting for coupled linear port-Hamiltonian systems

Authors

Jan Lorenz, Tom Zwerschke, Michael Günther, Kevin Schäfers

Abstract

Operator splitting methods tailored to coupled linear port-Hamiltonian systems are developed. We present algorithms that are able to exploit scalar coupling, as well as multirate potential of these coupled systems. The obtained algorithms preserve the dissipative structure of the overall system and are convergent of second order. Numerical results for coupled mass–spring–damper chains illustrate the computational efficiency of the splitting methods compared to a straight-forward application of the implicit midpoint rule to the overall system.

Keywords

Port-Hamiltonian systems; Operator splitting; Multiple time stepping

Citation

• Journal: Applied Mathematics Letters
• Year: 2025
• Volume: 160
• Issue:
• Pages: 109309
• Publisher: Elsevier BV
• DOI: 10.1016/j.aml.2024.109309

BibTeX

@article{Lorenz_2025,
  title={{Operator splitting for coupled linear port-Hamiltonian systems}},
  volume={160},
  ISSN={0893-9659},
  DOI={10.1016/j.aml.2024.109309},
  journal={Applied Mathematics Letters},
  publisher={Elsevier BV},
  author={Lorenz, Jan and Zwerschke, Tom and Günther, Michael and Schäfers, Kevin},
  year={2025},
  pages={109309}
}

Download the bib file

References

• Van Der Schaft, Port-Hamiltonian systems: an introductory survey. (2006)
• Gonzalez, O. Time integration and discrete Hamiltonian systems. Journal of Nonlinear Science vol. 6 449–467 (1996) – 10.1007/bf02440162
• Bartel, A., Günther, M., Jacob, B. & Reis, T. Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems. Numerische Mathematik vol. 155 1–34 (2023) – 10.1007/s00211-023-01369-5
• McLachlan, R. I. & Quispel, G. R. W. Splitting methods. Acta Numerica vol. 11 341–434 (2002) – 10.1017/s0962492902000053
• Frommer, (2023)
• Suzuki, M. General theory of fractal path integrals with applications to many-body theories and statistical physics. Journal of Mathematical Physics vol. 32 400–407 (1991) – 10.1063/1.529425
• Strang, G. On the Construction and Comparison of Difference Schemes. SIAM Journal on Numerical Analysis vol. 5 506–517 (1968) – 10.1137/0705041
• Macchelli, A. Control Design for a Class of Discrete-Time Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 68 8224–8231 (2023) – 10.1109/tac.2023.3292180
• Arnold, M. & Günther, M. Bit Numerical Mathematics vol. 41 1–25 (2001) – 10.1023/a:1021909032551
• Hairer, Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. (2006)
• Gugercin, S., Polyuga, R. V., Beattie, C. & van der Schaft, A. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica vol. 48 1963–1974 (2012) – 10.1016/j.automatica.2012.05.052
• Mönch, (2024)
• Omelyan, I. P., Mryglod, I. M. & Folk, R. Symplectic analytically integrable decomposition algorithms: classification, derivation, and application to molecular dynamics, quantum and celestial mechanics simulations. Computer Physics Communications vol. 151 272–314 (2003) – 10.1016/s0010-4655(02)00754-3
• Schäfers, (2024)
• Günther, (2023)
• Schäfers, (2023)