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Jacobian-free Multigrid Preconditioner for Discontinuous Galerkin Methods applied to Numerical Weather Prediction

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Pith's one-line read This paper argues that a Jacobian-free multigrid preconditioner built on a low-order finite-volume subgrid makes implicit, high-order discontinuous Galerkin time stepping competitive for two-dimensional atmospheric flows.

desk verdict Solid empirical extension of a known FV-multigrid preconditioner to 2D viscous atmospheric flows; the benchmark data support the main claim, but the conclusions overreach and the missing artifacts weaken reproducibility. read the letter →

arxiv 2505.24504 v1 pith:OGM6DWMQ submitted 2025-05-30 math.NA cs.NAphysics.comp-ph

Philipp Birken , Andreas Dedner , Robert Klöfkorn This is my paper
classification math.NAcs.NAphysics.comp-ph MSC 65M6065M5565F0876M1086A10
keywords discontinuousGalerkinfinitevolumesubgridmultigridpreconditionerJacobian-freeNewton-Krylovimplicittimeintegrationnumericalweatherpredictionwell-balancedschemepseudo-timesmoother
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that implicit time stepping for high-order discontinuous Galerkin (DG) discretizations of atmospheric flow can be made competitive with explicit time stepping by preconditioning the Newton--Krylov solver with a multigrid method defined on a low-order finite-volume subgrid. In three 2D test cases, the preconditioner sharply reduces the number of GMRES iterations and lets the implicit method take time steps hundreds to thousands of times larger than the explicit stability limit. The practical stake is that DG methods are attractive for numerical weather prediction but have lacked efficient, matrix-free, parallel solvers for the large algebraic systems produced by implicit time integration. The paper's solver is Jacobian-free and uses little extra memory, and on the finest grids tested its wall-clock time is roughly half that of the explicit reference scheme.

What carries the argument

The load-bearing object is the composite preconditioner $Q^{-1}=T^{-1}q^{-1}T$, where $T$ interpolates the high-order DG solution at the cell centers of a finite-volume subgrid and $q^{-1}$ is a geometric multigrid cycle for the first-order finite-volume discretization built on that subgrid. The multigrid restriction is volume averaging of child-quad averages, prolongation is injection of the agglomerate average, and the smoother is a one-stage explicit Euler pseudo-time iteration on the finite-volume levels, with optional smoothing on the DG data before transfer. This machinery matters because it approximates the inverse of the high-order DG Jacobian using only operators that are local, parallel, and matrix-free; the outer GMRES iteration then sees a better-conditioned system without the memory cost of forming the full Jacobian.

What would settle it

Run the same preconditioner on a problem whose solution error is dominated by components that oscillate inside each DG cell, or refine the grid while holding the time step fixed and record the multigrid convergence factor: if the preconditioned GMRES iteration count grows proportionally to the number of levels, or if the V-cycle convergence factor approaches one, the finite-volume subgrid is failing to represent the error modes that the solver must remove. A more direct calculation is to compare the energy of representative high-order error modes before and after transfer through $T^{-1}T$; modes whose energy is significantly reduced by the transfer are not being smoothed, and their presence in the residual would falsify the assumption.

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Extended reading notes

Core claim

The central claim is that the linear systems from an SDIRK2 time discretization of a DG space discretization of the 2D Euler or viscous flow equations with gravity can be solved efficiently by preconditioned Jacobian-free Newton-GMRES, provided the preconditioner is an agglomeration multigrid V-cycle applied to a first-order finite-volume discretization on a subgrid with the same number of degrees of freedom. Transfer between the DG polynomial space and the finite-volume cell averages is done by interpolation, with an optional mass-conservative correction, and the multigrid smoother is an explicit Runge-Kutta pseudo-time iteration, so no Jacobian matrix is ever formed. The stated result is that the preconditioner increases the stability of the solver while substantially reducing the number of iterations, making the implicit method competitive; on the fine-grid runs, the implicit method finished in about half the wall-clock time of the explicit SSP time stepping while using time steps around 500 times larger.

Load-bearing premise

The load-bearing premise is that a coarse cell-averaged version of the same equations, connected to the high-order DG discretization by simple interpolation, retains the error modes that dominate the high-order Jacobian, a property the paper relies on from earlier empirical comparisons rather than proving here.

Editorial extensions

If this is right

  • Unpreconditioned Jacobian-free GMRES is prohibitively slow or fails to converge within the allocated time on the inertia-gravity and rising-bubble tests; the multigrid preconditioner is what makes those runs finish.
  • Implicit time steps of roughly 500 to 2000 times the explicit stability limit reproduce the reference solution closely in the rising-bubble and density-current cases, with larger steps degrading accuracy.
  • On the finer grids tested, the preconditioned implicit method used roughly half the wall-clock time of the explicit SSP reference, for example 7915 seconds versus 14000 seconds for the rising bubble at 12.5 meter spacing.
  • GMRES iteration counts grow sublinearly with increasing time step and by less than a factor of two when the grid is refined, although they are not fully grid independent.
  • The most efficient configuration was a V-cycle with one pre- and post-smoothing step on the DG data and all finite-volume levels; W-cycles improve iteration counts but roughly double the run time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the preconditioner acts on the residual operator rather than on any specific flux or geometry, the same transfer-plus-subgrid-multigrid construction should carry over to three-dimensional DG-SEM and simplicial DG discretizations, with the node layout and subgrid subdivision adjusted accordingly.
  • A sharper test of the transfer assumption would be to measure, for a fixed error mode, the ratio of its energy after $T^{-1}T$ to its original energy; modes that the finite-volume subgrid annihilates are exactly the ones the smoother must handle, and their loss would predict when iteration reductions fail to generalize.
  • In flows where global mass conservation is a hard constraint, such as moist or tracer transport, the optional mass-conservative correction in the transfer may become essential even though it made no difference in the dry test cases reported here.
  • The observed growth of iterations under mesh refinement indicates the scheme is not yet an optimal-order solver; a natural next step would be to identify whether the limiting factor is the pseudo-time smoother or the coarse-grid correction of the finite-volume subgrid.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a Jacobian-free multigrid preconditioner for implicit time integration of high-order discontinuous Galerkin (DG) discretizations of compressible atmospheric flow equations. The preconditioner transfers the DG solution to a low-order finite volume subgrid with the same number of degrees of freedom, applies a geometric multigrid V- or W-cycle with explicit Runge-Kutta pseudo-time smoothers, and transfers back with a mass-conservative mapping. The method is embedded in a Jacobian-free Newton-GMRES framework with SDIRK2 time stepping and is tested on three two-dimensional benchmark problems: non-hydrostatic inertia gravity, rising warm air bubble, and density current. The paper reports GMRES iteration counts and wall-clock times, showing significant reductions relative to unpreconditioned GMRES and, in some high-resolution runs, CPU times competitive with an explicit strong-stability-preserving Runge-Kutta method.

Significance. If the results hold, the paper offers a practical path toward making implicit high-order DG methods affordable for numerical weather prediction, a setting where stiffness from gravity and low Mach numbers makes explicit time stepping unattractive. The strengths are the use of standard benchmark test cases, direct comparison of iteration counts and wall-clock times, the mass-conservative transfer construction, and the availability of the implementation within the open DUNE-FEM framework. The main weaknesses are that several strong claims—parallel scalability, low memory footprint, and near-optimal arithmetic intensity—are not supported by measurements, the h-dependence of iteration counts is only demonstrated over one refinement step per test case, and solution accuracy is only assessed visually rather than through quantitative error norms.

major comments (4)
  1. [Section 2.4.4 and Section 3] The pseudo-CFL number used for the explicit Euler smoother is never reported. The text states that the pseudo time step is 'computed based on a pseudo CFL number' and that the parameters are 'chosen based on [18]', but no value or tuning procedure is given. Since the iteration counts in Figures 5, 7, and 10 depend directly on this smoother parameter, the experiments cannot be reproduced without it. Please provide the pseudo-CFL value(s) used, and clarify how the choice of one-stage explicit Euler relates to the optimized multi-stage Runge-Kutta smoothers in [18].
  2. [Section 3.2, Fig. 7(c); Section 3.3, Fig. 10(b); Section 4] The central claim that the preconditioned implicit solver becomes competitive at high resolution is supported by only two resolutions in each test case, and the iteration counts grow under refinement. The paper explicitly acknowledges that the rising-bubble iteration count grows by less than a factor of two when the grid width is halved from 25 m to 12.5 m and that the density-current count grows by less than 50% from 320 m to 160 m. This is insufficient to establish that the preconditioner will remain effective at the much finer resolutions typical of NWP. Please either add experiments at one or two additional refinement levels, or revise the conclusions to state the demonstrated resolution regime.
  3. [Section 4 (Summary and Conclusions)] The statements that the preconditioner 'achieves close to optimal arithmetic intensity for large problems on parallel machines' and 'uses little extra memory' are not supported by any measurement in the manuscript. There are no FLOP-to-byte ratios, roofline data, memory footprint measurements, or parallel scaling results. The abstract also promises a 'fast, highly parallel, low-memory iterative solver.' These claims should either be demonstrated with appropriate experiments or removed/tempered, as they currently go beyond what the paper shows.
  4. [Section 3 and Section 4] Solution accuracy is only assessed visually. Figures 3, 4, and 9 show line plots, and the text uses qualitative language such as 'broadly follow' and 'some deviation.' No L1, L2, or L-infinity error norms relative to the explicit reference solution are reported. Since the conclusion states that 'the implicit high order solver produces accurate results even for large CFL numbers,' quantitative accuracy metrics are needed to support this claim.
minor comments (5)
  1. [Abstract and Section 2.4.2] There are typos: 'precondtioner' in the abstract and 'straighforward' in Section 2.4.2. Please correct them.
  2. [Section 3.2] In the sentence 'at ∆x=25m we the DG solution is computed' the verb is missing, and later the configuration 'mg0011111V' appears with an extra '1'; it should be 'mg001111V'.
  3. [Section 3.1] The domain notation '[0, 300 000]×[0, 10 000]km^2' is dimensionally inconsistent; presumably the lengths are in meters.
  4. [Section 3.3] The value 'cp = 1, 004J/(kg K)' uses a comma as a decimal separator inconsistently with the rest of the text; consider writing 1004 J/(kg K).
  5. [Section 2.4.1] The phrase 'defined such that the functions are close in some sense' is vague. Please specify the relevant norm or cite a precise approximation result from [16, 30] for the interpolation-based transfer operator.

Circularity Check

0 steps flagged · score 2.0 of 10

No equation-level circularity; the central claim rests on measured benchmark performance, with only minor self-citations for design choices.

full rationale

The paper's derivation chain is: DG discretization (Sec. 2.1), SDIRK time integration (Sec. 2.2), Jacobian-free Newton-GMRES (Sec. 2.3), and a preconditioner Q^{-1}=T^{-1}q^{-1}T built from a low-order FV subgrid discretization with transfer operator T (Sec. 2.4, Eq. 16). The claimed outcome, fewer GMRES iterations and competitive wall-clock times, is obtained by direct measurement on standard atmospheric test cases (inertia gravity, rising bubble, density current) against an explicit reference, not by algebraic prediction from the construction. No fitted parameter is renamed as a prediction: the pseudo-CFL smoother settings are taken from [18], an independent optimization, and the transfer-operator choice is justified by prior comparisons [16,30] but is itself tested in this paper's experiments. The only self-referential elements are citations to the authors' earlier work for the subcell FV multigrid concept and transfer operators; these support implementation choices and do not constitute the empirical conclusion. No equation of the paper reduces a claimed prediction to an input by construction, so no circular step is exhibited.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method rests on the heuristic that a low-order FV operator on the same DoFs approximates the DG Jacobian for preconditioning, plus standard DG-SEM assumptions and a well-balanced perturbation formulation. No new physical entities are introduced; the free parameters are numerical settings, some selected post hoc.

free parameters (4)
  • pseudo-CFL number for smoother time step = not stated (taken from [18])
    Determines the explicit RK pseudo-time step in the smoother; affects convergence but not the solution accuracy.
  • multigrid pre/post smoothing counts (a-f) = e.g., mg111111V, mg001111V
    Configuration is selected based on iteration counts and CPU time in the test cases; this is post-hoc parameter selection.
  • Newton tolerance TOL = 1e-3
    Chosen fixed for tests; convergence behavior depends on it.
  • Eisenstat-Walker forcing parameters = gamma=0.1, alpha=1
    Chosen by standard practice; affects linear solver tolerance.
assumptions (5)
  • standard math DG with Gauss-Legendre nodes yields diagonal mass matrices.
    Section 2.1, from [21-23].
  • domain assumption The pseudo-time ODE (20) is stable if the original spatial discretization is stable.
    Section 2.4.4, not proven.
  • domain assumption Interpolation-based transfer functions perform as well as L2 projection for this preconditioner.
    Section 2.4.1, based on [16,30], not proven in this paper.
  • ad hoc to paper The low-order FV subgrid system is an effective preconditioner for the high-order DG Jacobian.
    This is the central heuristic, validated only empirically in 2D test cases.
  • domain assumption The environmental atmosphere is known a priori for well-balancing.
    Section 3, eq. (22).

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Cite this review

Pith. "Pith review of Jacobian-free Multigrid Preconditioner for Discontinuous Galerkin Methods applied to Numerical Weather Prediction." pith.science (2026). https://pith.science/paper/OGM6DWMQ

@misc{pith2026250524504,
  author       = {Pith},
  title        = {Pith review of: Jacobian-free Multigrid Preconditioner for Discontinuous Galerkin Methods applied to Numerical Weather Prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGM6DWMQ}},
  note         = {Machine review of arXiv:2505.24504}
}
abstract

Discontinuous Galerkin (DG) methods are promising high order discretizations for unsteady compressible flows. Here, we focus on Numerical Weather Prediction (NWP). These flows are characterized by a fine resolution in $z$-direction and low Mach numbers, making the system stiff. Thus, implicit time integration is required and for this a fast, highly parallel, low-memory iterative solver for the resulting algebraic systems. As a basic framework, we use inexact Jacobian-Free Newton-GMRES with a preconditioner. For low order finite volume discretizations, multigrid methods have been successfully applied to steady and unsteady fluid flows. However, for high order DG methods, such solvers are currently lacking. %The lack of efficient solvers suitable for contemporary computer architectures inhibits wider adoption of DG methods. This motivates our research to construct a Jacobian-free precondtioner for high order DG discretizations. The preconditioner is based on a multigrid method constructed for a low order finite volume discretization defined on a subgrid of the DG mesh. We design a computationally efficient and mass conservative mapping between the grids. As smoothers, explicit Runge-Kutta pseudo time iterations are used, which can be implemented in parallel in a Jacobian-free low-memory manner. We consider DG Methods for the Euler equations and for viscous flow equations in 2D, both with gravity, in a well balanced formulation. Numerical experiments in the software framework DUNE-FEM on atmospheric flow problems show the benefit of this approach.

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    Dedner, A., Kl¨ ofkorn, R., Nolte, M., Ohlberger, M.: A Generic Interface for Parallel and Adaptive Scientific Computing: abstraction Principles and the DUNE-FEM Module. Computing 90(3–4), 165–196 (2010) https://doi.org/10. 1007/s00607-010-0110-3 A Modified Newton-Cotes formul...

    2010

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.