Splitting techniques for DAEs with port-Hamiltonian applications
Authors
Andreas Bartel, Malak Diab, Andreas Frommer, Michael Günther, Nicole Marheineke
Abstract
In the simulation of differential-algebraic equations (DAEs), it is essential to employ numerical schemes that take into account the inherent structure and maintain explicit or hidden algebraic constraints. This paper focuses on operator splitting techniques for coupled systems and aims at preserving the structure in the port-Hamiltonian framework. The study explores two decomposition strategies: one considering the underlying coupled subsystem structure and the other addressing energy-associated properties such as conservation and dissipation. We show that for coupled index-1 DAEs with and without private index-2 variables, the splitting schemes on top of a dimension-reducing decomposition achieve the same convergence rate as in the case of ordinary differential equations. Additionally, we discuss an energy-associated decomposition for linear time-invariant port-Hamiltonian index-1 DAEs and introduce generalized Cayley transforms to uphold energy conservation. The effectiveness of both strategies is evaluated using port-Hamiltonian benchmark examples from electric circuits.
Keywords
DAEs; Port-Hamiltonian systems; Cayley transform; Strang splitting
Citation
- Journal: Applied Numerical Mathematics
- Year: 2025
- Volume: 214
- Issue:
- Pages: 28–53
- Publisher: Elsevier BV
- DOI: 10.1016/j.apnum.2025.03.004
BibTeX
@article{Bartel_2025,
title={{Splitting techniques for DAEs with port-Hamiltonian applications}},
volume={214},
ISSN={0168-9274},
DOI={10.1016/j.apnum.2025.03.004},
journal={Applied Numerical Mathematics},
publisher={Elsevier BV},
author={Bartel, Andreas and Diab, Malak and Frommer, Andreas and Günther, Michael and Marheineke, Nicole},
year={2025},
pages={28--53}
}
References
- Brenan, (1995)
- Kunkel, (2006)
- Riaza, (2008)
- Beattie, Linear port-Hamiltonian descriptor systems. Math. Control Signals Syst. (2018)
- van der Schaft, Port-Hamiltonian differential-algebraic systems. (2013)
- van der Schaft, A. & Maschke, B. Generalized port-Hamiltonian DAE systems. Systems & Control Letters vol. 121 31–37 (2018) – 10.1016/j.sysconle.2018.09.008
- Kotyczka, P. & Lefèvre, L. Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters vol. 133 104530 (2019) – 10.1016/j.sysconle.2019.104530
- Mehl, C., Mehrmann, V. & Wojtylak, M. Linear Algebra Properties of Dissipative Hamiltonian Descriptor Systems. SIAM Journal on Matrix Analysis and Applications vol. 39 1489–1519 (2018) – 10.1137/18m1164275
- Blanes, S., Casas, F. & Murua, A. Splitting methods with complex coefficients. SeMA Journal vol. 50 47–60 (2010) – 10.1007/bf03322541
- McLachlan, R. I. & Quispel, G. R. W. Splitting methods. Acta Numerica vol. 11 341–434 (2002) – 10.1017/s0962492902000053
- Blanes,
- Strang, G. On the Construction and Comparison of Difference Schemes. SIAM Journal on Numerical Analysis vol. 5 506–517 (1968) – 10.1137/0705041
- Altmann, R. & Ostermann, A. Splitting methods for constrained diffusion–reaction systems. Computers & Mathematics with Applications vol. 74 962–976 (2017) – 10.1016/j.camwa.2017.02.044
- Flohr, (2019)
- Diab, Splitting Methods for Linear Coupled Field-Circuit DAEs. (2024)
- Diab, (2023)
- Frommer, Operator Splitting for Port-Hamiltonian Systems. (2025)
- Bartel, Operator Splitting for Semi-Explicit Differential-Algebraic Equations and Port-Hamiltonian DAEs. (2025)
- Hairer, (2006)
- 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
- van der Schaft, (2014)
- Rüth, Time stepping algorithms for partitioned multi-scale multi-physics in precice. (2018)
- Mönch, M. & Marheineke, N. Commutator-based operator splitting for linear port-Hamiltonian systems. Applied Numerical Mathematics vol. 210 25–38 (2025) – 10.1016/j.apnum.2024.12.007
- Maier, (2024)
- Jansen, (2015)
- Mehrmann, Structure-preserving discretization for port-Hamiltonian descriptor systems. (2019)
- 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
- Hairer, (1989)
- Lopez, L. & Politi, T. Applications of the Cayley approach in the numerical solution of matrix differential systems on quadratic groups. Applied Numerical Mathematics vol. 36 35–55 (2001) – 10.1016/s0168-9274(99)00049-5
- Blanes, S. & Casas, F. Splitting methods for non-autonomous separable dynamical systems. Journal of Physics A: Mathematical and General vol. 39 5405–5423 (2006) – 10.1088/0305-4470/39/19/s05