{"id":"4fb5ee2c-61c8-43cf-9d59-983a7a59ced8","arxiv_id":"2505.15985","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"FWSW-SDC time integration is shown to work for the compressible Euler equations on standard NWP benchmarks, with convergence up to order 8 in an advection test and stable treatment of moisture and baroclinic waves.","lead":"This paper applies a high-order time-stepping method called FWSW-SDC to the compressible Euler equations used in weather and climate models, and tests it on standard atmospheric benchmarks. It shows the method can reach arbitrary accuracy in time while remaining stable on gravity waves, moist rising bubbles, and baroclinic waves.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The gravity-wave convergence test in §3.2 coarsens Δx and Δt together at fixed CFL, so the observed slopes track the spatial discretisation order rather than the temporal SDC order; the arbitrary-order claim for compressible Euler is therefore supported only by the linear advection test, not by a…","rationale":"I read the paper as claiming both viability and arbitrary-order temporal accuracy. The viability claim is reasonably supported by the four benchmarks, and that is real evidence. The load-bearing gap is the arbitrary-order claim for the nonlinear compressible Euler equations. In the gravity-wave convergence test, Δt and Δx are coarsened together at fixed CFL, so the observed slopes are consistent with spatial-order limits rather than temporal SDC order. For example, p=1 with SDC(2,3) should be third order in time but converges at second order, and p=5 with SDC(4,7) should be seventh order in time but converges at better than fifth order — in both cases the spatial error is the likely bottleneck. This means the experiment never isolates the temporal order of FWSW-SDC on the nonlinear system. The only clean temporal-order demonstration is the advection test, which is linear and explicit and thus does not test the fast/slow splitting or the nonlinear implicit solves that are central to the paper's stated method. The reader's weakest assumption focused on solver tolerance and splitting order preservation; my concern is more basic, namely that the experimental protocol in the nonlinear setting cannot even measure the temporal order. This is not an internal inconsistency, but it is a missing verification of the strongest claim in the abstract. It does not change the conditional verdict; instead it sharpens what the condition should be: before accepting arbitrary-order accuracy on the compressible Euler equations, a temporal-only convergence study at fixed, sufficiently high spatial resolution is required.","tokens_in":17174,"tokens_out":5142,"duration_ms":46918,"concrete_test":"Run a temporal-only self-convergence study for the §3.2 gravity-wave case at p=5 with SDC(4,7): fix Δx = 2.5 km, Δz = 1000 m, and vary Δt over 1.875, 0.9375, 0.46875, and 0.234375 s, measuring error against a reference at the same spatial resolution with Δt = 0.15 s. If the L2 slope approaches 7, the nonlinear high-order claim is supported; if it saturates around 2–5, the current evidence for arbitrary order on the compressible Euler equations is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is that the paper never actually demonstrates high temporal order for FWSW-SDC on the compressible Euler equations. In §3.2 (Table 2) the gravity-wave convergence study keeps CFL = 0.06, so Δt is reduced in lockstep with Δx. The observed slopes are 2 for p=1 SDC(2,3), better than 4 for p=3 SDC(3,5), and better than 5 for p=5 SDC(4,7) — essentially the spatial accuracy of the compatible finite-element pairs (second, fourth, sixth order), not the nominal temporal orders 3, 5, and 7. Because spatial and temporal errors are refined together, the asymptotic rate is dominated by the lower spatial order, and the experiment cannot distinguish the SDC temporal order from the spatial one. The one clean temporal self-convergence test (§3.1), which reaches order 8, is an explicit linear tracer-advection problem with a prescribed wind; it exercises neither the fast/slow implicit splitting nor the nonlinear Euler system. The moist-bubble and baroclinic cases are single-resolution qualitative matches. Thus the central claim of arbitrary temporal order on the target equations rests on an extrapolation, and the loose implicit-solve tolerance (10^-4, §2.4) is a plausible additional mechanism for order reduction that this experiment would not expose.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper applies fast-wave slow-wave spectral deferred correction (FWSW-SDC) time stepping to the moist compressible Euler equations discretised with compatible finite elements in the Gusto/Firedrake framework. The method is formulated as a preconditioned Richardson iteration on a collocation problem, with an implicit treatment of fast-wave terms and explicit treatment of slow-wave terms. The authors demonstrate the scheme on four benchmarks: tracer advection on the sphere (temporal convergence up to order 8), a non-hydrostatic gravity wave (combined space-time convergence), a moist rising bubble, and a dry baroclinic wave in a channel. The central claim is that FWSW-SDC provides runtime-selectable, arbitrarily high temporal order for a modern dynamical-core equation set, and that it is a promising alternative to existing semi-implicit and IMEX methods for NWP and climate modelling.","tokens_in":17459,"tokens_out":4773,"duration_ms":43627,"significance":"If fully substantiated, the paper would provide a useful demonstration that SDC-type time integrators can be coupled to a compatible finite element compressible Euler model, with the practical advantage of choosing temporal order at runtime and the potential for parallel-in-time variants. The work has clear strengths: the benchmarks are standard and are compared against external reference solutions (Williamson, Skamarock and Klemp, Bryan and Fritsch, Ullrich), the code and run scripts are publicly available, and the comparison between classical LU-trick preconditioning and the newer diagonal MIN-SR preconditioners addresses an actively developing part of the SDC literature. The main gap is that the evidence for the headline 'arbitrary temporal order' claim on the target equations is indirect: the only clean temporal self-convergence test is a linear explicit advection problem, while the compressible Euler convergence study refines space and time together and therefore cannot distinguish temporal from spatial order.","major_comments":[{"comment":"The gravity-wave convergence study refines Δx and Δt simultaneously at fixed CFL = 0.06, so the observed slopes cannot isolate the temporal order of the SDC scheme. The p=1, p=3, and p=5 spatial discretisations have nominal spatial orders 2, 4, and 6, respectively, and the observed slopes of approximately 2, better than 4, and better than 5 are exactly what one would expect from spatial error alone when temporal errors are of higher order. The statement in §3.2 that the combination 'can provide methods with variable and arbitrarily high order of accuracy' therefore needs support from a temporal self-convergence experiment on the compressible Euler equations, i.e. fixed fine spatial resolution with Δt varying alone, ideally over several Δt values at the same Δx.","section":"§3.2, Table 2"},{"comment":"The nonlinear and linear solver tolerances are set to 10^-4, a level comparable to the error magnitudes needed to verify 5th- to 7th-order convergence on the reported meshes. Since SDC order statements assume exact solution of the implicit sweeps, it is load-bearing to demonstrate that the inexact Newton solves with Eisenstat-Walker forcing do not cap the observed order or produce the kind of error plateau that could be mistaken for superconvergence. A concrete test would be to repeat the proposed temporal convergence study with tolerances 10^-4 and 10^-8 and report both the measured slopes and the absolute error levels.","section":"§2.4"},{"comment":"The only clean high-order temporal convergence test in the paper is linear tracer advection on the sphere with a prescribed wind and an explicit scheme; it does not exercise the fast-wave implicit solve, the FWSW splitting, or the nonlinear pressure and gravity dynamics that define the compressible Euler target. Consequently, the evidence for 'arbitrary order accuracy' on the target application currently rests on extrapolation from a linear test to the nonlinear benchmarks. The authors should either add a temporal convergence test for the actual FWSW-SDC Euler scheme (for example, the gravity wave at fixed spatial resolution, or a small-amplitude acoustic/gravity-wave problem with a known reference) or substantially soften the arbitrary-order wording in the abstract and conclusion.","section":"§3.1 vs §3.2"}],"minor_comments":[{"comment":"The sentence 'Finally, a baroclinic wave in simulated in a Cartesian domain' contains a grammatical typo ('in simulated' should be 'is simulated').","section":"Abstract"},{"comment":"Internal cross-references are broken: 'Section 22.2' and 'Section 22.3' appear instead of Sections 2.2 and 2.3; the same issue occurs with 'Section 33.2', '33.3', and '33.4' in the results and conclusion.","section":"§2.3"},{"comment":"The perturbation radius is defined as r = sqrt((x-x_c)^2 - (y-y_x)^2), which contains a minus sign and an undefined y_x; this should presumably be sqrt((x-x_c)^2 + (y-y_c)^2) as in Ullrich et al. (2015).","section":"Eq. (32a)"},{"comment":"The Coriolis parameter appears as 'f0 = 2.0 −6 Km−1', which seems to be missing the factor 10^-6; please correct the typesetting.","section":"Eq. (29b)"},{"comment":"The state vector x is first defined as [u, ρ, θ_vd, m_v]^T but the governing equations and Algorithm 1 also include m_c; the definition should be [u, ρ, θ_vd, m_v, m_c]^T for consistency.","section":"§2.3"},{"comment":"The caption says 'SDC(2,3) and SDC(3,5) achieve the expected order of convergence of 4, 6 and 8 respectively' but three schemes are listed in the text; please clarify which scheme corresponds to which order and correct the apparent omission of SDC(4,7).","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is suitable in scope for a numerical-methods or geophysical-modelling journal and the code-availability policy is a strength. In my view the key issue is evidentiary: the headline arbitrary-temporal-order claim for the compressible Euler equations is not yet supported by an experiment that isolates the temporal order from the spatial order or that quantifies the effect of the loose implicit-solve tolerance. These are fixable with additional experiments, so I recommend major revision rather than rejection. I would be willing to review a revised version focused on a clean temporal convergence study for the actual FWSW-SDC Euler scheme."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is real: this is the first time FWSW-SDC is applied to the moist compressible Euler equations with compatible finite elements on standard NWP benchmarks. That is not a routine parameter sweep—it requires getting the fast/slow splitting, the nonlinear solve, and the moisture extension to work in a modern dynamical core framework. The paper also does several things well. The advection self-convergence test reaching order 8 is clean, the comparisons to published reference solutions for all four benchmarks are appropriate, the code and plotting scripts are public, and the model description is detailed enough to reproduce. I believe the authors are honest and the central viability claim—that FWSW-SDC can run these problems stably and accurately at practical timesteps—is supported.\n\nThe soft spot is exactly where the stress-test note lands, and it is load-bearing for the \"arbitrary order\" claim. The gravity-wave convergence study in §3.2 fixes CFL=0.06 and refines Δx and Δt together. The observed slopes (2, better than 4, better than 5) match the spatial orders of the p=1, p=3, p=5 finite element pairs. That experiment cannot distinguish temporal SDC order from spatial order, so it does not demonstrate high temporal order for the compressible Euler equations. The only clean temporal self-convergence test is the linear explicit advection problem, which exercises neither the fast/slow splitting nor the nonlinearity. The conclusion's claim that the method achieves \"variable and arbitrary high order of accuracy\" is therefore overstated for the target equations. This is a fixable problem—run a temporal-only refinement at fixed high-resolution spatial mesh, or use a manufactured solution—but it needs to be done before the claim is accepted.\n\nThe loose implicit-solve tolerance (10^-4) is a plausible additional contributor to order loss, though the paper does not check this. The SUPG parameter is hand-tuned, and there is no cost comparison against a production semi-implicit scheme. Those are minor in comparison. None of this sinks the paper; it is a solid engineering contribution with reproducible code and fair benchmarking. But the abstract and conclusion oversell what the convergence tests actually show.\n\nMy recommendation: send it to peer review. A serious referee should ask for a temporal-order demonstration on a nonlinear test case and a softened abstract. After that, it will be a useful reference for anyone working on high-order timestepping for dynamical cores. I would bring it to a reading group, and I would cite it for the application and code, while noting the convergence-test caveat.","headline":"A genuinely useful application paper whose high-order claim for the Euler equations is not actually demonstrated by the convergence tests as presented.","tokens_in":18006,"tokens_out":2249,"would_cite":true,"duration_ms":21664,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M60","65M70","65L60","86A10"],"pacs":[],"model":"deepseek-v4-flash","headline":"FWSW-SDC can time-step the compressible Euler equations at arbitrarily high order, and it reproduces four standard atmospheric benchmark solutions.","keywords":["spectral deferred correction","fast-wave slow-wave splitting","compressible Euler equations","compatible finite elements","arbitrary order time integration","numerical weather prediction","moist atmosphere","baroclinic wave"],"falsifier":"Run the non-hydrostatic gravity-wave convergence test at SDC(4,7) with the same timesteps but tighten the nonlinear and linear solver tolerances from $10^{-4}$ to $10^{-10}$. If the measured temporal order stays below the expected value at the tighter tolerance, the loose solver tolerance is limiting the arbitrary-order claim; if the order is unchanged, the tolerance is not the bottleneck.","tokens_in":16939,"feed_emoji":"🌦️","tokens_out":9797,"duration_ms":74664,"temperature":0.7,"pith_summary":"The paper tries to show that fast-wave slow-wave spectral deferred correction, an iterative time integrator whose order is set at runtime by the number of correction sweeps, can solve the moist compressible Euler equations used in atmospheric dynamical cores. If true, weather and climate models would no longer be locked into the fixed low order of conventional semi-implicit or IMEX Runge-Kutta time stepping; the same software could select low-order or higher-order accuracy depending on the application. The evidence is a set of standard benchmarks: tracer advection on a sphere, a nonlinear gravity wave, a moist rising bubble, and a dry baroclinic wave in a channel, with convergence tests showing expected orders up to eight.","feed_headline":"Time integrator for atmosphere reaches 8th order on standard tests","feed_subtitle":"A runtime-selectable high-order scheme passes advection, gravity-wave, moist-bubble, and baroclinic-wave benchmarks.","key_machinery":"The central object is the FWSW-SDC sweep, a preconditioned Richardson iteration on the collocation problem $(I-\\Delta t Q f)(X)=X_n$, in which the fast wave terms $F$ and slow wave terms $S$ are assigned different lower-triangular preconditioning matrices $Q^\\mathrm{imp}_\\Delta$ and $Q^\\mathrm{exp}_\\Delta$. Each correction sweep raises the order by one until the underlying Gauss-Legendre or Gauss-Radau collocation order is reached, so accuracy is controlled by runtime parameters $M$ (number of collocation nodes) and $K$ (number of sweeps). The time integrator is coupled to a compatible finite element spatial discretisation that mimics C-grid staggering, and the implicit solves in each sweep use a Newton-GMRES method with an additive Schwarz preconditioner and a relative tolerance of $10^{-4}$.","core_discovery":"The paper demonstrates that FWSW-SDC, which splits the right-hand side into fast terms treated implicitly and slow transport terms treated explicitly, can be coupled to a compatible finite element spatial discretisation of the compressible Euler equations. A convergence test for advection on the sphere produces the expected fourth, sixth, and eighth order rates for SDC(2,3), SDC(3,5), and SDC(4,7). In the nonlinear gravity-wave test the measured convergence is second order, better than fourth order, and better than fifth order as spatial and temporal order increase together, and the rising-bubble and baroclinic-wave simulations reproduce reference fields. The paper concludes that SDC offers a viable alternative to IMEX Runge-Kutta and other semi-implicit temporal discretisations used in numerical weather prediction and climate modelling.","pith_inferences":["Beyond the paper: the measured order for the highest-order schemes on nonlinear tests could be limited by the $10^{-4}$ solver tolerance, so a tolerance study would separate algebraic error from space-time discretisation error.","Beyond the paper: the demonstrated stability of diagonal preconditioners suggests a concrete parallel-in-time implementation with one thread per collocation node, though the paper only lists time-parallel SDC as future work.","Beyond the paper: if arbitrary-order temporal accuracy persists over climate-length integrations, the accumulated temporal error could be reduced without changing the dynamical core's time-stepping scheme, but the benchmarks here cover days, not decades.","Beyond the paper: the moisture extension covers vapour and cloud water with a simple two-phase scheme, so adding rain and testing a cold-pool or precipitating case would test the splitting's behaviour under stronger source terms."],"forward_implications":["The temporal order of a dynamical core can be raised without re-engineering the discretisation, because $M$ and $K$ are runtime parameters.","A single unified model could use low-order time stepping for short forecasts and higher-order time stepping for climate integrations, where temporal error accumulates over many steps.","Diagonal preconditioners for the implicit and explicit sweeps produce nearly identical rising-bubble results to standard LU and explicit-Euler preconditioners, which is a step toward parallelising SDC across collocation nodes.","On the gravity-wave benchmark, the combined space-time convergence improves from about second order at $p=1$ with SDC(2,3) to better than fifth order at $p=5$ with SDC(4,7), showing that both orders can be increased together.","The method remains stable for the benchmark time steps used in the literature, matching reference baroclinic-wave structures at 12 days with a 1800-second time step."],"supporting_citations":[{"why":"Introduces spectral deferred correction methods, the iterative collocation foundation on which the paper's time integrator is built.","marker":"Dutt et al. [2000]"},{"why":"Supplies the fast-wave slow-wave splitting and the linear stability results that this paper extends to nonlinear atmospheric equations.","marker":"Ruprecht and Speck [2016]"},{"why":"Provides the compatible finite element discretisation and the two-phase moisture source terms used in the model.","marker":"Bendall et al. [2020]"},{"why":"Supplies the Newton-GMRES with additive Schwarz strategy that the implicit sweeps rely on for the nonlinear solves.","marker":"Cotter and Shipton [2023]"},{"why":"Supplies the MIN-SR-NS and MIN-SR-FLEX preconditioners used for diagonal, parallelisable SDC sweeps.","marker":"Caklovic et al. [2025]"},{"why":"Defines the advection-on-the-sphere test used for the self-convergence study of temporal order.","marker":"Williamson et al. [1992]"},{"why":"Defines the non-hydrostatic gravity-wave benchmark used to measure combined space-time convergence and wave structure.","marker":"Skamarock and Klemp [1994]"},{"why":"Defines the moist rising-bubble benchmark used to test the moisture extension.","marker":"Bryan and Fritsch [2002]"},{"why":"Defines the dry baroclinic-wave channel test used to test synoptic-scale behaviour.","marker":"Ullrich et al. [2015]"}],"fun_headline_variants":["Fast-slow SDC reaches 8th order on Euler test suite","8th-order time precision from fast-slow wave splitting","FWSW-SDC scheme passes weather benchmarks up to 8th order","Arbitrary-order time integrator for compressible Euler"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the order and stability proven for FWSW-SDC on linear model problems carry over to the nonlinear moist compressible Euler equations even though each implicit fast-wave solve is stopped at a relative tolerance of $10^{-4}$.","fun_headline_variants_meta":{"raw":{"variants":["Fast-slow SDC reaches 8th order on Euler test suite","8th-order time precision from fast-slow wave splitting","FWSW-SDC scheme passes weather benchmarks up to 8th order","Arbitrary-order time integrator for compressible Euler"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001431,"raw_usage":{"total_tokens":5724,"prompt_tokens":847,"completion_tokens":4877,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":4803}},"tokens_in":463,"tokens_out":4877,"duration_ms":31858,"temperature":1.0,"reasoning_tokens":4803,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:08:36.930296+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the non-hydrostatic gravity-wave convergence test at SDC(4,7) with the same timesteps but tighten the nonlinear and linear solver tolerances from $10^{-4}$ to $10^{-10}$. If the measured temporal order stays below the expected value at the tighter tolerance, the loose solver tolerance is limiting the arbitrary-order claim; if the order is unchanged, the tolerance is not the bottleneck.","supporting_citations":[{"cited_title":"Greengard, and V","cited_arxiv_id":null,"evidence_quote":"Introduces spectral deferred correction methods, the iterative collocation foundation on which the paper's time integrator is built."},{"cited_title":"Speck, 2016: Spectral deferred corrections with fast-wave slow-wave splitting","cited_arxiv_id":null,"evidence_quote":"Supplies the fast-wave slow-wave splitting and the linear stability results that this paper extends to nonlinear atmospheric equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the compatible finite element discretisation and the two-phase moisture source terms used in the model."},{"cited_title":"J., and J","cited_arxiv_id":null,"evidence_quote":"Supplies the Newton-GMRES with additive Schwarz strategy that the implicit sweeps rely on for the nonlinear solves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the advection-on-the-sphere test used for the self-convergence study of temporal order."},{"cited_title":"C., and J","cited_arxiv_id":null,"evidence_quote":"Defines the non-hydrostatic gravity-wave benchmark used to measure combined space-time convergence and wave structure."},{"cited_title":"H., and J","cited_arxiv_id":null,"evidence_quote":"Defines the moist rising-bubble benchmark used to test the moisture extension."},{"cited_title":"Reed, and C","cited_arxiv_id":null,"evidence_quote":"Defines the dry baroclinic-wave channel test used to test synoptic-scale behaviour."}],"review_version":1}