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arxiv: 2606.27300 · v1 · pith:B2VZSROGnew · submitted 2026-06-25 · 🧮 math.NA · cs.NA

Automated Galerkin time stepping in Irksome

Boris D. Andrews , Pablo Brubeck , Patrick E. Farrell , Robert C. Kirby , Scott P. MacLachlan This is my paper

Pith reviewed 2026-06-26 03:11 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Galerkin time steppingdiscontinuous Galerkincontinuous Petrov-Galerkinvariational problemstime discretizationstructure preservationfinite element methodsRunge-Kutta methods
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The pith

Automation now supports discontinuous and continuous Galerkin time stepping for semidiscrete variational problems.

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

The paper seeks to remove the need for bespoke code by automating both discontinuous Galerkin and continuous Petrov-Galerkin time discretizations inside an existing framework for variational problems. This matters because Galerkin-in-time schemes are known to preserve more structure than Runge-Kutta methods, yet their adoption has been limited by implementation effort. The work shows that auxiliary variables, flexible temporal quadrature, and monolithic solvers can be handled automatically, so users can switch formulations with only small code edits. Numerical tests on representative PDE systems confirm that accuracy, solver behavior, and structure preservation remain intact under the automation.

Core claim

Automation has been added for discontinuous Galerkin and continuous Petrov-Galerkin time stepping of semidiscrete variational problems; the implementation handles auxiliary variables, flexible temporal quadrature, and monolithic algebraic solvers while allowing users to switch between Runge-Kutta and Galerkin-in-time formulations with minimal changes to their code.

What carries the argument

The automated extension of spatial discretization and algebraic solver infrastructure to incorporate temporal degrees of freedom and quadrature rules for Galerkin-in-time schemes.

If this is right

  • Users can switch from Runge-Kutta to Galerkin-in-time formulations by changing only a small number of lines.
  • Both discontinuous and continuous Petrov-Galerkin variants become available without separate implementations.
  • Monolithic solvers remain applicable to the enlarged space-time systems.
  • Structure-preservation properties demonstrated for the underlying Galerkin methods carry over to the automated versions.
  • Accuracy and solver performance can be checked on standard PDE examples without bespoke coding.

Where Pith is reading between the lines

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

  • The same automation layer could be reused to test hybrid schemes that combine Runge-Kutta stages with Galerkin time elements inside one variational form.
  • Extension to problems with time-dependent coefficients or moving domains would require only quadrature-rule updates rather than new solver infrastructure.
  • Wider availability of these schemes may allow direct numerical comparison of structure preservation across families of methods on identical spatial meshes.

Load-bearing premise

The existing infrastructure for spatial discretization and algebraic solvers can be extended to handle the added temporal degrees of freedom and quadrature rules without introducing new bottlenecks or losing structure preservation.

What would settle it

A concrete Hamiltonian system whose exact energy is known to be conserved under a continuous Petrov-Galerkin scheme but is observed to drift when the automated implementation is applied.

Figures

Figures reproduced from arXiv: 2606.27300 by Boris D. Andrews, Pablo Brubeck, Patrick E. Farrell, Robert C. Kirby, Scott P. MacLachlan.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 1.1
Figure 1.1. Figure 1.1: Firedrake/Irksome listing for the Navier–Stokes lid-driven cavity problem. whatever choices are made in the spatial discretization, leading to Vh. To discretize the space of functions from In to Vh, we denote by Vh,s(In) the space of all polynomial maps from In into Vh, (2.1) Vh,s(In) := Ps(In; Vh). Basis expansion & quadrature. Equipping Ps(In) with a basis {ϕj} s j=0, we can write elements of Vh,s(In) … view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Comparing conservation for the implicit Gauss–Legendre and CPG schemes [PITH_FULL_IMAGE:figures/full_fig_p015_5_1.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Monotonic energy decay for the Allen–Cahn equation discretized with CPG [PITH_FULL_IMAGE:figures/full_fig_p016_5_2.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Computational domain for Navier–Stokes flow past a cylinder. [PITH_FULL_IMAGE:figures/full_fig_p017_5_3.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Schematic showing the degrees of freedom for pressure (left) and velocity [PITH_FULL_IMAGE:figures/full_fig_p018_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Typical Vanka patch for Alfeld–Sorokina pair. Pressure degrees of freedom [PITH_FULL_IMAGE:figures/full_fig_p018_5_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Drag and lift versus time for Navier–Stokes flow past a cylinder using [PITH_FULL_IMAGE:figures/full_fig_p019_5_6.png] view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Relative L 2 ([0, 8]) error in lift and drag versus the time step size for DG and CPG time stepping schemes. “CPG” refers to CPG with the pressure treated as a discontinuous-in-time auxiliary variable; “stabilized CPG” additionally under￾integrates the (div uh, qh) term using s-stage right-Radau quadrature. 10−2 10−1 10−7 10−6 10−5 10−4 10−3 ∆t Divergence DG(1) DG(2) CPG(1) CPG(2) stabilized CPG(1) stabi… view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: Total run-time versus time step size for DG and stabilized CPG schemes. this gap may be because our reference solution is itself a fine DG simulation, so DG-discretized schemes may benefit from a small inherited bias. The bulk of the gap, however, is more likely due to DG being better suited than CPG to this problem. We also report on the performance of our Newton/Krylov/multigrid schemes for the DG and … view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 5.10
Figure 5.10. Figure 5.10: Solver performance across time steps for DG and stabilized CPG schemes. [PITH_FULL_IMAGE:figures/full_fig_p021_5_10.png] view at source ↗
Figure 5.11
Figure 5.11. Figure 5.11: Generation of entropy for the Navier–Stokes–Fourier model. [PITH_FULL_IMAGE:figures/full_fig_p023_5_11.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
read the original abstract

As the study of temporal and spatial discretization schemes continues to advance, recent work has focused on the use of Galerkin-in-time discretization schemes that enable broader structure-preservation than is known for Runge-Kutta integrators. While the promise of such discretizations is immense, their realization has, until now, generally relied on bespoke implementations that have limited their wider use. In this work, we present automation in Irksome for both discontinuous Galerkin and continuous Petrov-Galerkin time stepping of semidiscrete variational problems. The implementation supports auxiliary variables, flexible temporal quadrature, and monolithic algebraic solvers, and it enables switching between Runge-Kutta and Galerkin-in-time formulations with minimal changes to user code. Numerical examples illustrate accuracy, solver performance, and structure preservation across representative PDE systems.

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

0 major / 2 minor

Summary. The paper presents automation of discontinuous Galerkin (DG) and continuous Petrov-Galerkin (CPG) time stepping schemes within the Irksome library for semidiscrete variational problems. The implementation supports auxiliary variables, flexible temporal quadrature, and monolithic algebraic solvers, and allows switching between Runge-Kutta and Galerkin-in-time formulations with minimal changes to user code. Numerical examples across representative PDE systems demonstrate accuracy, solver performance, and structure preservation.

Significance. If the implementation claims hold, the work is significant because it removes a major practical barrier to adopting Galerkin-in-time methods, which offer broader structure preservation than standard Runge-Kutta integrators. By embedding the automation inside an existing finite-element framework with documented minimal code changes and working examples, the contribution directly enables wider experimentation and use of these methods in the community.

minor comments (2)
  1. [Abstract] The abstract states that examples 'illustrate accuracy, solver performance, and structure preservation' but does not name the specific PDE systems or spatial discretizations used; adding one sentence with this information would improve clarity for readers scanning the contribution.
  2. Section describing the user interface changes would benefit from an explicit side-by-side code comparison (original Runge-Kutta vs. new Galerkin-in-time) rather than a prose description alone, to make the 'minimal changes' claim immediately verifiable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and their recommendation to accept. The referee's assessment correctly identifies the core contribution: automation of DG and CPG Galerkin-in-time schemes inside Irksome that supports auxiliary variables, flexible quadrature, monolithic solvers, and minimal user-code changes when switching from Runge-Kutta methods.

Circularity Check

0 steps flagged

No significant circularity: implementation paper with no derivation chain

full rationale

This is a software implementation and automation paper describing extensions to the Irksome library for DG and CPG time discretizations. No mathematical derivations, predictions, fitted parameters, or first-principles results are presented that could reduce to their own inputs by construction. The central claim is the existence and usability of the implemented automation, supported by numerical examples and code-change demonstrations rather than any self-referential equations or load-bearing self-citations. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The central claim rests on the unexamined assumption that the Irksome code base already contains the necessary hooks for temporal variational formulations.

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    Software used in ‘Automated Galerkin time stepping in Irksome’, jun 2026, https://doi.org/10. 5281/zenodo.20784213, https://doi.org/10.5281/zenodo.20784213

    work page doi:10.5281/zenodo.20784213 2026