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arxiv: 2606.06153 · v1 · pith:GFUNI35Knew · submitted 2026-06-04 · 🧮 math.NA · cs.NA

Structure-Preserving Operator Splitting via JR-Decomposition for Circuit Models

Andreas Bartel , Malak Diab This is my paper

Pith reviewed 2026-06-28 00:21 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords JR-decompositionmodified nodal analysisport-Hamiltonian systemsoperator splittingcircuit simulationstructure preservationenergy conservation
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The pith

Enhanced JR-decomposition enables energy-conform splitting for modified nodal analysis circuit models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the standard JR-decomposition cannot be applied directly to modified nodal analysis formulations of circuits. By introducing a relaxed enhanced JR-decomposition tailored specifically to these models, the authors obtain an operator splitting that respects the port-Hamiltonian structure and conserves energy. A numerical example confirms both convergence and the preservation of the underlying structure. A sympathetic reader would care because energy-conform splittings can yield more stable long-term behavior in circuit simulation without artificial dissipation or drift.

Core claim

The enhanced JR-decomposition relaxes the standard JR-decomposition so that it applies to modified nodal analysis systems in the port-Hamiltonian framework, thereby delivering an energy-conform operator splitting.

What carries the argument

The enhanced JR-decomposition, a relaxation of the standard version that is tailored to circuit models and enables structure-preserving splitting.

If this is right

  • The splitting preserves the energy balance of the port-Hamiltonian formulation.
  • Numerical solutions converge while retaining the algebraic and geometric structure of the original system.
  • The method applies to modified nodal analysis models that previously could not use JR-decomposition.
  • Structure preservation holds across the demonstrated numerical test case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same relaxation technique might apply to other index-1 or index-2 differential-algebraic systems that share port-Hamiltonian structure.
  • Long-term integration of large-scale circuits could benefit from reduced artificial energy loss compared with generic splitting schemes.
  • The approach may connect to existing structure-preserving integrators for Hamiltonian systems with algebraic constraints.

Load-bearing premise

Relaxing the standard JR-decomposition remains valid and structure-preserving when applied to modified nodal analysis systems.

What would settle it

A concrete circuit simulation in which the enhanced splitting produces measurable energy drift or violates the port-Hamiltonian structure would falsify the claim.

read the original abstract

We investigate circuit models, namely, modified nodal analysis (MNA) in the port-Hamiltonian framework. Based on this, the JR-decomposition for the numerical treatment would offer an energy conform splitting. However, for circuit models, the application of the standard JR-decomposition is restricted. To enable a JR-decomposition for MNA, we need to relax the decomposition. To this end, we introduce the enhanced JR-decomposition, which is particularly tailored to the application to circuits. We conclude with a numerical example that illustrates the applicability of the proposed approach as well as its convergence and structure-preserving properties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper investigates modified nodal analysis (MNA) circuit models in the port-Hamiltonian framework and proposes an enhanced JR-decomposition, tailored to circuits, to overcome restrictions of the standard JR-decomposition and enable an energy-conform operator splitting. The contribution is illustrated by a numerical example demonstrating applicability, convergence, and structure-preserving properties.

Significance. If the enhanced JR-decomposition rigorously preserves the required algebraic structure (skew-symmetry of J and positive-semidefiniteness of R) for general MNA systems, the method would provide a structure-preserving numerical tool for circuit simulation. The numerical example supplies concrete evidence of applicability and observed preservation, which is a strength, but the absence of explicit derivation, error analysis, or invariant verification in the provided text limits the assessed impact.

major comments (2)
  1. [Abstract] Abstract: the central claim that the enhanced JR-decomposition 'enables a JR-decomposition for MNA' and yields an 'energy conform splitting' requires that the relaxation preserves J^T = -J and R = R^T ≥ 0 (or the dissipation inequality) on the MNA incidence and conductance matrices; no such algebraic verification or explicit construction is supplied, leaving the structure-preserving guarantee dependent on the numerical example alone.
  2. [Numerical example] Numerical example (final section): while the example is cited to illustrate convergence and structure preservation, it does not include explicit checks (e.g., computed norms of J + J^T or eigenvalues of R) or a general proof that the tailored relaxation maintains the port-Hamiltonian invariants for arbitrary MNA systems; this is load-bearing for the claim that the splitting remains energy-conform.
minor comments (1)
  1. [Abstract] The abstract refers to 'the standard JR-decomposition' without a brief recall of its definition or the precise restriction for circuits; adding one sentence would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and constructive comments. We address each major comment below and plan to revise the manuscript to strengthen the algebraic guarantees and the numerical validation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the enhanced JR-decomposition 'enables a JR-decomposition for MNA' and yields an 'energy conform splitting' requires that the relaxation preserves J^T = -J and R = R^T ≥ 0 (or the dissipation inequality) on the MNA incidence and conductance matrices; no such algebraic verification or explicit construction is supplied, leaving the structure-preserving guarantee dependent on the numerical example alone.

    Authors: We agree that an explicit algebraic verification is crucial to support the central claim. The enhanced JR-decomposition is tailored to the MNA structure by relaxing the standard JR conditions in a specific manner that preserves the required properties by construction. However, the current manuscript does not provide the detailed algebraic derivation. In the revised version, we will include an explicit construction showing how the relaxation maintains J skew-symmetric and R positive semidefinite for general MNA incidence and conductance matrices. revision: yes

  2. Referee: [Numerical example] Numerical example (final section): while the example is cited to illustrate convergence and structure preservation, it does not include explicit checks (e.g., computed norms of J + J^T or eigenvalues of R) or a general proof that the tailored relaxation maintains the port-Hamiltonian invariants for arbitrary MNA systems; this is load-bearing for the claim that the splitting remains energy-conform.

    Authors: We concur that adding explicit numerical checks would provide stronger evidence. In the revision, we will augment the numerical example with computations of the norms of J + J^T and the eigenvalues of R to verify the structure preservation in the example. The general proof will be incorporated as part of the new algebraic verification section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; enhanced JR-decomposition is an independent tailored construction

full rationale

The paper states that standard JR-decomposition is restricted for circuit MNA models and introduces an enhanced version 'particularly tailored to the application to circuits' to enable energy-conform splitting in the port-Hamiltonian framework. This is presented as a methodological relaxation followed by a numerical example demonstrating applicability, convergence, and structure-preserving properties. No equations or claims reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the derivation chain consists of an explicit new operator definition whose invariants are asserted to hold under the relaxation and then verified numerically. The contribution is therefore self-contained as an independent construction rather than a renaming or re-derivation of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the port-Hamiltonian framework applies to MNA circuit models and that the standard JR-decomposition is restricted in this setting, requiring the proposed relaxation.

axioms (1)
  • domain assumption The port-Hamiltonian framework is suitable for modified nodal analysis of circuits
    Stated as the basis for the investigation in the abstract.

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discussion (0)

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