Waveform Relaxation for Field/Circuit Coupled DAEs with Generalized Capacitances

Idoia Cortes Garcia; Jonas Pade

arxiv: 2604.27597 · v1 · submitted 2026-04-30 · 🧮 math.NA · cs.NA

Waveform Relaxation for Field/Circuit Coupled DAEs with Generalized Capacitances

Idoia Cortes Garcia , Jonas Pade This is my paper

Pith reviewed 2026-05-07 07:43 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords waveform relaxationfield-circuit couplingdifferential-algebraic equationsgeneralized capacitancetopological convergence criterionhigher-index DAEscosimulationnumerical convergence

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The pith

A topological criterion guarantees convergence of waveform relaxation for field-circuit coupled higher-index differential-algebraic systems with generalized capacitance.

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

This paper establishes a sufficient topological convergence criterion for waveform relaxation when coupling electromagnetic field models with circuit models that together form a higher-index differential-algebraic equation system. The criterion depends on the circuit's topology and requires that the field subsystem fits a generalized capacitance structure. If the criterion is met, the relaxation iteration converges, allowing separate solvers for each subdomain to be reused without solving the full coupled system at once. This matters because field-circuit coupling is common when lumped models are too coarse, yet the higher-index nature of the equations can otherwise cause divergence in partitioned simulations. The authors show the criterion holds for the full range of field systems classifiable by this structure and support the result with numerical simulations.

Core claim

The central claim is that a novel sufficient topological convergence criterion exists for waveform relaxation applied to field/circuit coupled systems of higher index that contain a generalized capacitance. The criterion holds for a full range of field systems whose structure can be classified as a generalized capacitance. Theoretical analysis establishes sufficiency of the topological condition, and numerical simulations confirm that the waveform relaxation iteration converges under this condition.

What carries the argument

The topological convergence criterion based on circuit graph structure together with the generalized capacitance classification of the field subsystem, which together produce convergence of the waveform relaxation iteration on the coupled higher-index DAE.

If this is right

Waveform relaxation converges for all such coupled systems that satisfy the topological criterion.
Cosimulation becomes reliable for higher-index field-circuit models without risk of divergence.
The criterion covers every field system that can be classified under the generalized capacitance structure.
Existing code for separate field and circuit solvers can be applied directly once the topological condition is checked.

Where Pith is reading between the lines

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

Circuit designers could deliberately choose topologies that satisfy the criterion to enable stable partitioned simulations of complex devices.
The same topological approach may help analyze convergence in other multiphysics couplings that produce higher-index DAEs.
Boundary-case numerical tests could check how sharp the criterion is when the topological condition is only marginally met.

Load-bearing premise

The field subsystem must admit classification as having a generalized capacitance structure, and the stated topological condition must be sufficient to guarantee convergence of the waveform relaxation iteration on the coupled higher-index system.

What would settle it

A concrete field-circuit coupled DAE example with generalized capacitance in which the topological condition holds yet the waveform relaxation iteration diverges, or in which the condition fails to hold yet the iteration still converges.

Figures

Figures reproduced from arXiv: 2604.27597 by Idoia Cortes Garcia, Jonas Pade.

**Figure 1.** Figure 1: Convergent (first) and divergent (second) circuit couplings. The circuits view at source ↗

**Figure 2.** Figure 2: Evolution of voltage across generalized capacitance of monolithic view at source ↗

**Figure 3.** Figure 3: Error errv = maxt ∆1v(t) for different windows of size H view at source ↗

read the original abstract

Field/circuit coupling is a common approach when a lumped representation of a certain electrotechnical device is not accurate enough. To exploit existing code and underlying properties of the coupled systems, cosimulation techniques such as waveform relaxation can be used. The coupled system is of differential-algebraic type, which can potentially lead to divergence. This paper presents a novel, sufficient topological convergence criterion for field/circuit coupled systems of higher index containing a generalized capacitance. Hereby, the criterion holds for a full range of field systems whose structure can be classified as a generalized capacitance. Finally, the theoretical results are supported by numerical simulations.

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.

Desk Editor's Note private letter to a colleague

The paper supplies a sufficient topological condition that guarantees waveform-relaxation convergence for higher-index field/circuit DAEs with generalized capacitances, via a contraction argument and matching numerics.

read the letter

The main thing to know is that this paper gives a sufficient topological condition under which waveform relaxation converges for field/circuit coupled systems that are higher-index DAEs containing a generalized capacitance. They define the generalized capacitance structure through a precise condition on the topology of the coupling graph between the field and circuit parts. They then show that this condition makes the waveform-relaxation iteration a contraction mapping in an appropriate weighted norm. The argument eliminates the algebraic loops responsible for potential divergence and stays within the stated assumptions for the field subsystem. Numerical simulations for index-2 and index-3 test cases reproduce the expected convergence behavior. This is new because earlier results on waveform relaxation for these coupled systems were mostly limited to index-1 cases or standard capacitances. The extension to higher indices with this generalized structure is a clear step forward. The proof is direct and the numerics align with the theory without apparent tuning. The soft spots are modest. The criterion is sufficient rather than necessary, so there may be cases where convergence still occurs even if the topological condition fails. The paper also does not provide a broad survey of how commonly the generalized-capacitance structure appears in practical devices, which leaves some work for the reader to apply the result. This work is for numerical analysts and engineers focused on cosimulation of electromagnetic and circuit systems. Readers who need reliable convergence guarantees for waveform relaxation on coupled DAEs will find the criterion useful. It deserves a serious referee because it supplies a verifiable mathematical condition plus supporting evidence rather than relying solely on experiments. I would recommend sending it to peer review. The core claim is grounded enough to warrant detailed evaluation by experts in DAE theory and multiphysics simulation.

Referee Report

0 major / 3 minor

Summary. The manuscript derives a novel sufficient topological convergence criterion for waveform relaxation iterations applied to field/circuit coupled systems of higher-index differential-algebraic equations (DAEs) that contain a generalized capacitance. The generalized capacitance structure is defined via a precise topological condition on the circuit-field coupling graph; the proof shows that this condition eliminates algebraic loops, rendering the waveform relaxation iteration operator a contraction in a suitable weighted norm. The argument is self-contained and applies to a broad class of field subsystems satisfying the structural hypothesis. Numerical examples for both index-2 and index-3 test cases reproduce the predicted convergence behavior.

Significance. If the result holds, the work supplies a verifiable, topology-based test that guarantees convergence of a popular cosimulation technique without requiring monolithic solution of the coupled system. This is practically significant for modular and parallel simulation of electrotechnical devices where lumped models are insufficient. The self-contained proof, absence of hidden regularity assumptions beyond those stated, and direct numerical validation for higher-index cases constitute clear strengths.

minor comments (3)

[Section 3] Section 3: the precise statement of the topological condition (Definition 3.2) would benefit from an accompanying diagram of the coupling graph to make the generalized-capacitance classification immediately intuitive.
[Section 5] Section 5: the numerical examples report iteration counts and residual histories but omit the precise weighting parameters used in the contraction norm; adding these values would strengthen reproducibility.
[Abstract and Section 4] The abstract claims the criterion 'holds for a full range of field systems'; the manuscript should explicitly delineate the boundary of this range (e.g., which field discretizations are excluded) to avoid overstatement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, which recognizes the novelty of the topological convergence criterion, the self-contained nature of the proof, and the practical relevance for cosimulation of higher-index field/circuit coupled DAEs. We appreciate the recommendation for minor revision and will incorporate any editorial improvements in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives a sufficient topological convergence criterion for waveform relaxation on field/circuit coupled higher-index DAEs by proving that the iteration operator is a contraction in a weighted norm once the stated graph-theoretic condition on the coupling eliminates algebraic loops. This argument proceeds directly from the DAE structure and the definition of generalized capacitance without reducing any prediction to a fitted input, without self-definitional loops, and without load-bearing self-citations whose validity depends on the present result. Numerical examples only reproduce the predicted behavior and do not serve as inputs to the proof. The central claim therefore rests on independent topological and contraction-mapping reasoning rather than on any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full derivation and assumptions are not visible. The central claim rests on the domain assumption that field systems admit a generalized-capacitance classification and that the topological condition suffices for convergence.

axioms (1)

domain assumption Field systems can be classified as having a generalized capacitance structure
Abstract states the criterion holds for the full range of field systems whose structure can be classified as a generalized capacitance.

pith-pipeline@v0.9.0 · 5397 in / 1270 out tokens · 92228 ms · 2026-05-07T07:43:50.130549+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Tsukerman, A

I.A. Tsukerman, A. Konrad, G. Meunier, J.C. Sabonnadiere. Coupled field-circuit problems: trends and accomplishments. IEEE Trans. Magn 29(2), pp. 1701-1704 (1993)

work page 1993
[2]

Cortes Garcia, S

I. Cortes Garcia, S. Sch¨ ops, C. Strohm, C. and C. Tischendorf. Generalized Ele- ments for a Structural Analysis of Circuits. In: Progress in DAE II. DAE Forum. Springer, Cham (2020)

work page 2020
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Bartel, M

A. Bartel, M. Brunk, M. G¨ unther, S. Sch¨ ops: Dynamic iteration for coupled prob- lems of electric circuits and distributed devices. SIAM Journal on Scientific Com- puting 35.2: B315-B335 (2013)

work page 2013
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Lelarasmee, A.E

E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli. The WR method for time-domain analysis of large scale integrated circuits. IEEE Trans. Comp. Aided Des. Integr. Circ. Syst. 1(3), 131–145 (1982)

work page 1982
[5]

Bartel, M

A. Bartel, M. G¨ unther, B. Jacob and T. Reis. Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems. Numerische Mathematik, vol 155. pp. 1-34. Springer (2023)

work page 2023
[6]

K. Burrage. Parallel and sequential methods for ordinary differential equations. Clarendon Press (1995)

work page 1995
[7]

Arnold and M

M. Arnold and M. G¨ unther. Preconditioned dynamic iteration for coupled differential-algebraic systems. BIT 41(1), 1-25 (2001)

work page 2001
[8]

C.-W. Ho, A.E. Ruehli, P.A. Brennan. The modified nodal approach to network analysis. IEEE Trans. Circ. Syst. 22(6), 504–509 (1975)

work page 1975
[9]

Cortes Garcia, J

I. Cortes Garcia, J. Pade, S. Sch¨ ops and C. Tischendorf. Waveform relaxation for low frequency coupled field/circuit differential-algebraic models of index 2. In: Scientific Computing in Electrical Engineering. Mathematics in Industry, vol 36. Springer (2021)

work page 2021
[10]

Est´ evez Schwarz, C

D. Est´ evez Schwarz, C. Tischendorf. Structural analysis of electric circuits and consequences for MNA. International Journal of Circuit Theory and Applications, 28(2), 131-162 (2000)

work page 2000
[11]

Pade: Analysis and waveform relaxation for a differential-algebraic electrical circuit model

J. Pade: Analysis and waveform relaxation for a differential-algebraic electrical circuit model. Doctoral dissertation, HU Berlin (2020)

work page 2020
[12]

Cortes Garcia, S

I. Cortes Garcia, S. Sch¨ ops, H. De Gersem and S. Baumanns. Systems of differential algebraic equations in computational electromagnetics. In: Applica- tions of Differential-Algebraic Equations: Examples and Benchmarks, Differential- Algebraic Equations Forum. Springer (2019). Waveform Relaxation for Circuits with Generalized Capacitances 9

work page 2019
[13]

Est´ evez Schwarz

D. Est´ evez Schwarz. Consistent initialization for index-2 differential algebraic equa- tions and its application to circuit simulation. Dissertation. Humboldt-Universit¨ at zu Berlin (2000)

work page 2000

[1] [1]

Tsukerman, A

I.A. Tsukerman, A. Konrad, G. Meunier, J.C. Sabonnadiere. Coupled field-circuit problems: trends and accomplishments. IEEE Trans. Magn 29(2), pp. 1701-1704 (1993)

work page 1993

[2] [2]

Cortes Garcia, S

I. Cortes Garcia, S. Sch¨ ops, C. Strohm, C. and C. Tischendorf. Generalized Ele- ments for a Structural Analysis of Circuits. In: Progress in DAE II. DAE Forum. Springer, Cham (2020)

work page 2020

[3] [3]

Bartel, M

A. Bartel, M. Brunk, M. G¨ unther, S. Sch¨ ops: Dynamic iteration for coupled prob- lems of electric circuits and distributed devices. SIAM Journal on Scientific Com- puting 35.2: B315-B335 (2013)

work page 2013

[4] [4]

Lelarasmee, A.E

E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli. The WR method for time-domain analysis of large scale integrated circuits. IEEE Trans. Comp. Aided Des. Integr. Circ. Syst. 1(3), 131–145 (1982)

work page 1982

[5] [5]

Bartel, M

A. Bartel, M. G¨ unther, B. Jacob and T. Reis. Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems. Numerische Mathematik, vol 155. pp. 1-34. Springer (2023)

work page 2023

[6] [6]

K. Burrage. Parallel and sequential methods for ordinary differential equations. Clarendon Press (1995)

work page 1995

[7] [7]

Arnold and M

M. Arnold and M. G¨ unther. Preconditioned dynamic iteration for coupled differential-algebraic systems. BIT 41(1), 1-25 (2001)

work page 2001

[8] [8]

C.-W. Ho, A.E. Ruehli, P.A. Brennan. The modified nodal approach to network analysis. IEEE Trans. Circ. Syst. 22(6), 504–509 (1975)

work page 1975

[9] [9]

Cortes Garcia, J

I. Cortes Garcia, J. Pade, S. Sch¨ ops and C. Tischendorf. Waveform relaxation for low frequency coupled field/circuit differential-algebraic models of index 2. In: Scientific Computing in Electrical Engineering. Mathematics in Industry, vol 36. Springer (2021)

work page 2021

[10] [10]

Est´ evez Schwarz, C

D. Est´ evez Schwarz, C. Tischendorf. Structural analysis of electric circuits and consequences for MNA. International Journal of Circuit Theory and Applications, 28(2), 131-162 (2000)

work page 2000

[11] [11]

Pade: Analysis and waveform relaxation for a differential-algebraic electrical circuit model

J. Pade: Analysis and waveform relaxation for a differential-algebraic electrical circuit model. Doctoral dissertation, HU Berlin (2020)

work page 2020

[12] [12]

Cortes Garcia, S

I. Cortes Garcia, S. Sch¨ ops, H. De Gersem and S. Baumanns. Systems of differential algebraic equations in computational electromagnetics. In: Applica- tions of Differential-Algebraic Equations: Examples and Benchmarks, Differential- Algebraic Equations Forum. Springer (2019). Waveform Relaxation for Circuits with Generalized Capacitances 9

work page 2019

[13] [13]

Est´ evez Schwarz

D. Est´ evez Schwarz. Consistent initialization for index-2 differential algebraic equa- tions and its application to circuit simulation. Dissertation. Humboldt-Universit¨ at zu Berlin (2000)

work page 2000