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Energy-Consistent Splitting and Decomposition Approaches for Coupled port-Hamiltonian ODEs

Marius Mönch, Nicole Marheineke, Andreas Bartel, Kevin Schäfers, Michael Günther

arXiv:2609.04884v1math.NAmath.DS

TL;DR

The paper asks how decomposition choices affect computational efficiency and preservation of energy behavior in splitting methods for coupled port-Hamiltonian ODEs. It formalizes energy consistency, derives sufficient conditions, and analyzes structure-preserving, subsystem-based, and time-scale decompositions. The analysis finds that preserving port-Hamiltonian subflow structure supports energy-consistent splitting, while multiple time stepping with time-scale decomposition offers the most favorable computational performance for the benchmark.

  • Problem

    Decomposition choices can simplify coupled port-Hamiltonian simulations but may alter the subflows’ structural properties and energy behavior.

  • Method

    The paper defines energy consistency, derives sufficient conditions for splitting decompositions, and compares energy-, port-, subsystem-, diagonal-, and time-scale strategies with Strang and multiple-time-stepping experiments.

  • Results

    The multirate time-scale variant TS provides the most favorable computational performance for the electro-thermal benchmark while retaining second-order convergence.

  • Takeaways & Limitations

    Preserving port-Hamiltonian structure at the subflow level supports energy-consistent splitting, while time-scale decomposition can exploit pronounced multirate behavior computationally.

  • Takeaways & Limitations

    Subsystem-based decomposition can destroy port-Hamiltonian subflow structure, allowing Hamiltonian growth without input and potentially requiring step-size restrictions.

Abstract

from arXiv · show

Operator splitting provides an attractive approach for the numerical integration of (coupled) port-Hamiltonian systems, as it allows the underlying system structure to be exploited at the level of the individual subproblems. However, the choice of the decomposition is not unique and may strongly affect both the computational efficiency and the preservation of the energy behavior of the original system. In this work, we investigate this interplay systematically and introduce energy consistency as a criterion for assessing splitting methods for port-Hamiltonian ordinary differential equations. We derive sufficient conditions under which a splitting based on a given decomposition inherits the energy behavior of the continuous system and use these conditions to analyze several decomposition strategies for coupled port-Hamiltonian systems. In particular, we compare decompositions that preserve the structure with approaches that exploit lower-dimensional subsystem dynamics or separated time scales. The analysis is complemented by numerical experiments using Strang splitting and its multiple-time-stepping extension. A scalable electro-thermal benchmark with fast electrical and slow thermal dynamics is employed to assess accuracy, energy behavior, and computational efficiency. The results demonstrate that preserving the port-Hamiltonian structure of the subflows is essential for energy-consistent splitting, whereas decompositions that exploit subsystem structure or time-scale separation can provide substantial computational advantages. In particular, the time-scale decomposition yields significant efficiency gains for systems with pronounced multirate characteristics, while structure-destroying decompositions may lead to undesirable energy behavior.

1. Introduction

Port-Hamiltonian systems make energy storage, exchange, dissipation, and external interaction explicit, but splitting choices trade computational simplicity against preservation of energy-related structure. This paper introduces energy consistency to compare decompositions and analyzes their structural and computational implications.

  • Port-Hamiltonian systems expose energy balance and support qualitative properties such as passivity, dissipation, and conservation in interconnected physical models.
  • Coupled port-Hamiltonian systems can combine smaller subsystems with substantially different dynamic time scales, creating opportunities for simpler numerical subproblems.
  • Operator splitting approximates the full flow by composing subflows, but the decomposition affects subsystem dimension, linearity, stability, and energy-related structure.
  • Higher-order real-valued splitting schemes with p ≥3 require negative step sizes, which can cause stability issues for dissipative port-Hamiltonian systems.
  • The paper defines energy consistency, derives sufficient conditions, and analyzes energy-associated, port-based, subsystem-based, diagonal, and time-scale decompositions.
  • The study combines structural analysis with numerical experiments using Strang splitting and multiple-time-stepping for an electro-thermal circuit benchmark.

2. Concept of Energy Consistency

Energy consistency requires a splitting scheme to represent each step through dissipated and supplied energies consistent with the continuous power balance. Under structural and regularity conditions, this property transfers through subflows and, for orders up to two, matches the scheme’s consistency order.

  • Energy consistency requires each numerical step to admit dissipated and supplied energy components consistent with the continuous power balance.
  • With no resistance, energy consistency yields a discrete passivity relation; with no input, it yields non-increasing Hamiltonian energy.
  • When both resistance and input are absent, the method is conservative and preserves the Hamiltonian exactly.
  • For dissipative systems, classical higher-order schemes use negative steps and may lose energy consistency or impose step-size restrictions; specialized positive-coefficient schemes can have q ≤ p.
  • A consistent splitting is energy-consistent when both subproblems are port-Hamiltonian systems satisfying the required regularity and decomposition conditions.
  • For consistency order p ≤2 under the stated regularity assumptions, the energy-consistency order is q = p.
  • In the damped and driven oscillator experiment, PBS4 and PBS6 achieve state and Hamiltonian orders four and six, while dissipated and supplied energies show only second-order convergence.
  • Replacing exact subflows with numerical flows of equal or higher energy-consistency order preserves the overall order q_A = q.

3. Decomposition Strategies

The paper compares decomposition strategies for coupled port-Hamiltonian systems, balancing energy preservation against computational savings. Structure-preserving decompositions retain energy consistency, while subsystem-oriented approaches can reduce dimensions or enable parallelism but may lose the original energy behavior.

  • Energy consistency and computational efficiency are competing objectives when decomposing coupled port-Hamiltonian systems.The interconnection structure offers opportunities for dimension reduction and parallelization, but decompositions must preserve the energetic properties of the continuous system.
  • Energy-associated and port-based decompositions retain the energetic properties of the port-Hamiltonian formulation and are therefore energy-consistent.The energy-associated approach separates conservative dynamics from dissipation and forcing, while the port-based approach separates internal dynamics from external-port interaction.
  • The energy-associated decomposition separates passive dynamics into independent subsystem problems, while its conservative subproblem retains the full coupled dimension.The passive subproblem has dimensions n1 and n2 and can be solved independently or in parallel; the conservative subproblem retains the complete interconnection matrix.
  • The port-based decomposition similarly permits independent parallel evaluation of port contributions while retaining coupling in the internal subproblem.Its internal subproblem is generally coupled through C(x), whereas the port contribution separates according to the subsystem block structure.
  • Subsystem-based decomposition reduces effective problem dimensions and computational cost, but generally destroys the port-Hamiltonian structure of the subproblems.Unless C(x)=0, the effort/flow pairing is disrupted, so individual subflows need not satisfy the original power balance.
  • Loss of structure means that dimensional reduction alone does not ensure energy consistency and can impose step-size restrictions on the splitting method.The Hamiltonian may increase along an individual subflow, and for linear systems the composed splitting can become unstable for sufficiently large step sizes.

(coupling).

The diagonal decomposition separates independent subsystem dynamics from coupling while preserving port-Hamiltonian structure, enabling parallel computation and energy-consistent splitting. Time-scale and hierarchical decompositions further target multirate efficiency and combine computational and energetic structure.

  • Diagonal decomposition: The diagonal decomposition separates uncoupled subsystem dynamics from coupling while preserving the port-Hamiltonian structure of both subproblems.Its coupling subproblem is conservative, transferring energy without generating or dissipating it.
  • Diagonal decomposition: The diagonal decomposition allows independent subsystem solves, parallel computation, and potentially reduced coupling representations when the coupling acts through a low-dimensional subspace.The overall dimensions need not decrease, but the separated subproblems can be solved independently.
  • Diagonal decomposition: Unlike subsystem-based decomposition, the diagonal approach avoids loss of port-Hamiltonian structure and supports energy-consistent splitting for arbitrary step sizes h > 0.Subsystem-based decompositions can provide dimensional reduction but generally lose the port-Hamiltonian structure.
  • Time-scale decomposition: Time-scale decomposition advances fast dynamics with micro-steps and slow dynamics on the macro scale, reducing effort when fast and slow components are strongly separated.Its advantage is strongest when the fast subproblem is inexpensive; repeated micro-steps can otherwise offset the savings.
  • Time-scale decomposition: The time-scale impulse method is energy-consistent because both fast and slow subproblems are passive port-Hamiltonian systems.The fast subproblem freezes the slow component, while the slow subproblem generally retains full dimension and the skew-symmetric coupling.
  • Hierarchical decomposition: A hierarchical strategy combines coupling-based decomposition with recursive energy- or port-based splitting, exploiting computational and energetic structure at different levels.For diagonal decomposition, the resulting independent problems can be solved in parallel and subsequently decomposed according to their energetic structure.

4. Numerical Results

The electro-thermal benchmark compares decomposition strategies within second-order Strang splitting, evaluating accuracy, energy behavior, and computational efficiency. Results show that structure-preserving decompositions maintain energy behavior, while time-scale multirate splitting substantially reduces computational cost.

  • Benchmark and setup: The benchmark couples an electrical subsystem of dimension 2N + 1 with a thermal subsystem of dimension N, giving total dimension 3N + 1.The simulations use identical blocks and generally set N = 100.
  • Benchmark and setup: The electrical variables vary rapidly while temperatures evolve slowly, creating a pronounced separation of time scales suitable for multirate decomposition.The benchmark includes rapidly oscillating node potentials and currents alongside slower thermal dynamics.
  • Accuracy and efficiency: All tested decomposition strategies retain the expected second-order convergence, but differ in error constants and computational costs.Figure 3 reports discrete L2([0, 0.1])-error against step size and CPU time for N = 100.
  • Accuracy and efficiency: Dim1, DO, JR, and PB1 achieve higher accuracy, while Dim1 and DO provide the strongest computational efficiency through reduced or separable subsystem problems.Their advantage is associated with reduced effective dimensions and the possibility of solving independent subsystem problems separately or in parallel.
  • Energy behavior: DO preserves non-increasing discrete energy and conserves energy to Newton tolerance in the nearly conservative setting, whereas Dim1 can violate dissipation and conservation.The contrast arises because DO preserves port-Hamiltonian structure in both subflows, while Dim1 produces subproblems that are not themselves port-Hamiltonian.
  • Multiple time stepping: The time-scale multirate variant TS requires 22.1% of OD's and 9.8% of Dim2's computational cost to reach error O(10^-4), while retaining second-order convergence and energy consistency.Multiple time stepping substantially benefits several decompositions, and the TS variant is most favorable for this benchmark.

5. Conclusion

The conclusions show that decomposition choice governs both energy consistency and computational efficiency in splitting methods for coupled port-Hamiltonian ODEs. Structure-preserving diagonal and time-scale decompositions combine energetic properties with computational advantages, whereas component-wise decompositions may not guarantee energy consistency.

  • The choice of decomposition is crucial for preserving the energetic properties of coupled port-Hamiltonian systems.
  • Component-wise partitioning creates efficient lower-dimensional subproblems but generally destroys port-Hamiltonian structure at the subflow level.
  • Diagonal decomposition exploits coupling structure, enables dimension reduction and parallelization, and retains port-Hamiltonian subproblems for energy-consistent splitting.
  • Time-scale decomposition combined with multiple time stepping provides substantial additional efficiency for the electro-thermal benchmark while retaining convergence and energetic properties.The achievable gain depends on time-scale separation and the relative computational costs of the subproblems.
  • Hierarchical splitting that first exploits coupling and time-scale structure is identified as a direction for future work.The proposed direction would further decompose resulting subproblems using energy-associated or port-based approaches, alongside stability analysis.

Appendix A. Discrete Gradient Methods

The appendix develops second-order discrete gradient methods for port-Hamiltonian ODEs and explains how they preserve a discrete power balance. Embedded in Strang splitting and multiple time stepping, these methods yield energy-consistent second-order splitting approaches.

  • A discrete gradient satisfies an exact energy-difference identity and reproduces the continuous gradient when both arguments coincide.
  • Second-order discrete gradients include the Gonzalez, mean-value, and symmetrized Itoh–Abe constructions.
  • The second-order discrete gradient method for pH-ODEs uses a midpoint state and midpoint time to approximate the system evolution.
  • The method is energy-consistent of order q = 2 and derives a discrete power balance with dissipated and supplied energy distributions.
  • Embedding the Gonzalez discrete gradient method in Strang splitting and its multiple-time-stepping extension yields energy-consistent second-order splitting approaches.For quadratic Hamiltonians, the method reduces to the implicit midpoint rule, whose linear systems can be solved with structure-preserving Krylov methods.

Appendix B. Electro-thermal Modeling

The benchmark couples electrical and thermal port-Hamiltonian subsystems through energy-conserving internal ports. The resulting model represents temperature-dependent electrical behavior and thermal energy-transfer mechanisms while preserving power balance and passivity.

  • The electro-thermal model formulates electrical and thermal subsystems as individual port-Hamiltonian systems interconnected through energy-conserving internal ports.
  • The coupled model accounts for temperature-dependent electrical parameters, Joule heating, heat conduction, and heat exchange with the environment.
  • The model provides a power balance and preserves passivity for the coupled electro-thermal system.

B.1. Coupled Electro-Thermal Port-Hamiltonian Model.

The electrical subsystem is an RLC network represented as a port-Hamiltonian differential-algebraic equation using modified nodal analysis. Element-specific incidence matrices describe the network interconnections.

  • The electrical network contains resistors, capacitors, inductors, and independent voltage and current sources.
  • Element-specific incidence matrices encode the interconnections among circuit branches and nodes.
  • Modified nodal analysis represents the network as a port-Hamiltonian differential-algebraic equation.

Electric network model.

The electric network model uses positive definite component matrices and temperature-dependent thermally relevant resistors. These resistors convert electrical power into heat for the coupled thermal dynamics.

  • The circuit is parameterized by positive definite matrices for capacitors, inductors, and resistors.
  • The state includes node potentials, inductor currents, and voltage-source currents, while external inputs are current and voltage sources.
  • Thermally relevant and irrelevant resistors are distinguished through the conductance decomposition.
  • Temperature-dependent resistance can be modeled quadratically as R(T) = R0 + α1T + α2T^2.
  • Thermally relevant resistors generate heat from electrical dissipation, with each element receiving a corresponding power input.

Heat evolution model in circuit.

The thermal model assigns lumped temperatures and heat capacities to thermally relevant resistors, with conduction, ambient cooling, and electrical dissipation governing heat evolution. Electrical power enters the thermal subsystem while temperature-dependent resistance remains in the electrical subsystem, yielding additive stored energy.

  • The thermal balance includes heat exchange between elements, Newton cooling to an ambient reservoir, and electrical power inputs.
  • Entropy variables are introduced to represent temperature states in port-Hamiltonian form.
  • Heat conduction is represented by the skew-symmetric interconnection structure, whereas ambient heat exchange appears as an external port.
  • Electrical power dissipated by thermally relevant resistors enters the thermal subsystem as heat while temperature-dependent resistance is retained electrically.
  • The total stored energy equals the sum of the electrical and thermal Hamiltonians.

B.2. Scalable Benchmark Problem.

The scalable benchmark consists of repeated electro-thermal building blocks represented in port-Hamiltonian variables. Its formulation exposes electrical, thermal, external-port, and internal-coupling structures while permitting increased thermal dimensionality through intermediate states.

  • B.2. Scalable Benchmark Problem.: The benchmark uses electrical states comprising node potentials and inductor currents, alongside thermal entropy states and temperatures.
  • B.2. Scalable Benchmark Problem.: The coupling matrix is induced by the electrical and thermal subsystem relations and defines the internal coupling dynamics.
  • B.2. Scalable Benchmark Problem.: The benchmark formulation specifies subsystem matrices for inner dynamics, external port matrices, and matrices for internal coupling.
  • B.2. Scalable Benchmark Problem.: Each benchmark building block contains a temperature-dependent resistor and a parallel RLC link to ground, with electrical and thermal state variables.
  • B.2. Scalable Benchmark Problem.: The total Hamiltonian and circuit and thermal parameters define the energy model and numerical simulation setup.
  • B.2. Scalable Benchmark Problem.: Intermediate temperature states can increase the thermal system dimension while preserving a lumped-parameter description.
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