{"id":"5d0d592d-f6fd-4921-842c-0068e7a45504","arxiv_id":"2412.07613","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An entropy-dissipative DG method for the stochastic full Euler system is shown to converge in law to a dissipative martingale solution, with error bounds up to a stopping time.","lead":"This paper proves that an entropy-stable discontinuous Galerkin scheme for the stochastic Euler equations converges, in a weak probabilistic sense, to a dissipative martingale solution under a no-vacuum bounded-energy assumption. It also provides the first convergence rate for such schemes during the lifetime of a strong solution.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The convergence and rate theorems rest entirely on the unproved uniform-in-h bound (3.12); without it the scheme may form vacuum or blow up, so the claimed result is conditional and its domain of applicability is not established.","rationale":"The reader's weakest-assumption analysis correctly identifies assumption (3.12) as the load-bearing point. My independent reading confirms that the consistency proof in Section 4 and the error analysis in Section 5 import (3.12) rather than deriving it, and the paper is transparent about this. This is a genuine limitation of scope, not an internal contradiction: the theorems are valid conditionally on the scheme remaining bounded and non-vacuous. I do not see a fatal flaw in the relative-entropy argument or the compactness passage that would move the verdict from CONDITIONAL to REJECT; the conditional statement is coherent and plausibly correct. The proposed computational test would show whether the unproved assumption can actually be satisfied for the numerical scheme on a nontrivial set of paths. Because the reader's verdict already flags this concern and marks the paper conditional, no verdict adjustment is needed.","tokens_in":33061,"tokens_out":15155,"duration_ms":156475,"concrete_test":"Run the one-dimensional smooth density-wave experiment of Section 6.1.1 with p = 1 DGSEM, no limiter, mu = 1, 1000 samples, and record for each sample the first time tau_h = inf{t : inf_x rho_h(t) <= 1/K or sup_x E_h(t) >= K}, with K fixed by the initial data, for h = 64, 128, 256, 512. Compute the empirical probability P(inf_h tau_h = 0) as h decreases. If this probability stays positive, no common stopping time with P(t > 0) = 1 exists and the convergence result excludes a non-negligible set of noise paths; if it tends to 0, the a priori bound is at least plausible for smooth data. Repeat with the Sod data of Section 6.1.3 to check whether the assumption is what separates the theorem's regime from the raw numerical regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Every convergence statement in the paper, Theorems 3.6 and 3.8, is conditioned on (3.12): a common stopping time t with P(t > 0) = 1 and a deterministic K such that P-a.s. 1/K <= rho_h(t,x) and E_h(t,x) <= K for all t in [0,t], x in O, uniformly in h. This bound is not derived from the scheme; it is an input. The discrete entropy inequality (4.4) and energy balance (4.5) provide only L^p-type control, not the uniform L^infinity control required by (3.12), and the DGSEM is neither shown to be positivity preserving nor energy bounded without limiters. Because t must be the same for all h, the conclusion has no content on any noise path for which some refinement violates the bound arbitrarily early. The paper itself signals this regime in Section 6: for the Kelvin-Helmholtz test with noise strength above 1 the scheme becomes unstable, and the numerical experiments are restricted to short times and weak noise. The additional Lipschitz hypothesis (3.15) used for the O(h) rate with p >= 1 is a further unproved input, but the primary load-bearing gap is (3.12): without it the convergence in law and the error rates are conditional statements about paths that have not been shown to form a probability-one set.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies an entropy-dissipative discontinuous Galerkin spectral element method (DGSEM), including the finite-volume case, for the complete stochastic Euler equations with multiplicative noise in the momentum equation. The main theoretical results are conditional: under a uniform-in-h a priori bound on the numerical density and total energy up to a stopping time (3.12), the authors prove consistency and convergence in law, up to a subsequence, to a dissipative martingale solution in the sense of Definition 3.1 (Theorem 3.6). In the lifespan of a sufficiently regular pathwise strong solution, they prove relative-entropy error estimates of order h^{1/2} for polynomial degree p=0 and order h for p≥1, the latter under an additional Lipschitz condition on the nodal values (Theorem 3.8, Eqs. (3.14) and (3.16)). The paper also presents numerical experiments in one and two space dimensions using the Trixi.jl framework, reporting observed convergence rates for smooth and discontinuous test cases and studying the influence of noise strength.","tokens_in":33281,"tokens_out":8638,"duration_ms":87132,"significance":"If the results hold, this is the first convergence analysis for a high-order DG discretization of the complete stochastic Euler equations, and it extends the deterministic consistency framework of Lukáčová-Medvid'ová and Öffner to the stochastic setting using dissipative martingale solutions. The paper is carefully structured, states its hypotheses explicitly, and provides a concrete numerical implementation with a reproducible open-source software framework. The central convergence theorem is honest about its conditional character, and the relative-entropy argument gives a natural stochastic analogue of deterministic weak-strong uniqueness. However, the two main theorems rest entirely on a priori bounds that are not derived from the scheme, and one step in the proof of Theorem 3.6 (conversion of an energy inequality into the equality required by Definition 3.1) is not rigorously justified. These issues are load-bearing for the paper's central claims.","major_comments":[{"comment":"Theorems 3.6 and 3.8 are both conditioned on the uniform-in-h bound (3.12), but this bound is neither proved for the scheme nor shown to hold on a probability-one set. The discrete entropy inequality (4.4) and the energy balance (4.5) provide only L^p-type and L^2(U) control, not the uniform L^∞ upper bound on E_h and the uniform positive lower bound on ϱ_h required by (3.12); in particular, the scheme is not shown to be positivity preserving or to preclude blow-up before the common stopping time τ. Section 6 in fact reports stability failures for the Kelvin–Helmholtz test once the noise strength exceeds 1. Because τ must be common to all mesh refinements, the theorem has no content on noise paths for which some refinement violates (3.12) arbitrarily early. The authors should either prove (3.12) for a stabilized (e.g., limited) version of the scheme, or state the results as strictly conditional and discuss the verifiability and plausibility of the assumption in detail.","section":"§3.3, Eq. (3.12); Theorems 3.6 and 3.8"},{"comment":"The O(h) rate for p≥1 is claimed only under the additional Lipschitz condition (3.15) on the nodal values of the numerical solution within each element. This is an unverified a priori assumption about the numerical solution: the entropy-dissipative DGSEM is not shown to satisfy it, and Remark 4.1 explicitly notes that limiters are disregarded and that with limiters only finite-volume-type rates are expected. Since (3.15) is needed to obtain the strong consistency estimate (4.7) from [52], the O(h) rate is not a property of the scheme as analyzed unless (3.15) is established or at least numerically verified. The authors should prove (3.15) under suitable assumptions on the mesh, time step, and data, or downgrade the claim to the p=0 result.","section":"§3.3, Theorem 3.8(b), Eq. (3.15); Remark 4.1"},{"comment":"The passage from the energy inequality (4.23) to the energy equality required by Definition 3.1(l) is not rigorously justified. The text first says \"Performing the limit εm → 0 yields an energy inequality\" (no εm has been introduced at that point), and then states that augmenting Rpress by a spatially homogeneous h(t)dx converts the inequality into equality. The authors do not verify that the augmented measure has the required measurability, adaptedness, and weak-* compactness properties, nor that the energy deficit can always be represented in this way while preserving the momentum equation (3.6). This step is load-bearing because Definition 3.1(l) demands equality. A detailed construction of the augmented measure, or a reformulation of the theorem with an energy inequality in the definition of the limit object, is needed.","section":"§4.3, after Eq. (4.23)"}],"minor_comments":[{"comment":"The abstract contains a typo: \"we proof\" should be \"we prove\", and \"The results built\" should be \"The results build\".","section":"Abstract and Section 1"},{"comment":"The solution space V_h is defined using L^1(O), although its elements are piecewise polynomials and hence bounded; L^∞(O) or a more standard broken polynomial space notation would be clearer.","section":"§2.3, Eq. (2.20)"},{"comment":"The text reads \"I4 is the 4 × 4-identy matrix\"; this should be \"identity matrix\".","section":"§2.3, after Eq. (2.28)"},{"comment":"The dummy-variable space A in (3.1) is defined with m' ∈ R^3, while the paper explicitly focuses on two space dimensions; the dimension should be made consistent, or the notation should be explained as generic.","section":"§3.1, Definition 3.1"},{"comment":"The phrase \"Performing the limit εm → 0\" appears without any prior definition of εm; if the intended limit is m → ∞, it should be stated correctly.","section":"§4.3, last paragraph"},{"comment":"The text contains a German fragment \"f¨ urt\" instead of \"for\", and Table 5 writes \"162\", \"322\", etc., where the intended entries are 16^2, 32^2, and so on; these should be formatted correctly.","section":"§6.1.1 and Table 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the main ideas are sound, but the referee report identifies three load-bearing issues: the unproved uniform-in-h a priori bound (3.12), the additional unverified Lipschitz condition (3.15) needed for the O(h) rate, and the insufficiently justified conversion of an energy inequality into the equality required by Definition 3.1. I would advise the editor that acceptance depends on either proving these conditions for a suitable version of the scheme or substantially sharpening the conditional statements. The numerical section is useful as a proof of concept but does not resolve the theoretical gaps."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is the first convergence result for entropy-stable DG applied to the complete stochastic Euler equations, and it is a real result. But every theorem is conditional on a uniform-in-h L∞ bound (3.12) that the scheme is never shown to enforce. If you accept that bound as a standing hypothesis, the paper works. If you don't, there is no theorem.\n\nWhat's genuinely new: the authors take the deterministic dissipative-solution machinery from Lukáčová-Medvid'ová and Öffner and transplant it into the stochastic setting, using Moyo's dissipative martingale solution as the target. They handle the Itô correction, stopping times, and the non-Polish compactness via Jakubowski's theorem. The consistency argument is mostly a line-by-line adaptation of [52] and [53], but that is legitimate given how different the stochastic estimates are. The rate theorem is also new: O(h^{1/2}) in expectation for the p=0 finite volume case, and O(h) for p≥1 under an additional nodal Lipschitz condition. The numerics are a genuine first pass: smooth test cases show expected orders, Riemann problems are included, and the authors are honest that the scheme becomes unstable for strong noise in Kelvin-Helmholtz.\n\nThe soft spots are in proportion. The main one is (3.12). It is an input, not a consequence. The discrete entropy inequality gives L^p control, not the L∞ control needed. The scheme is not shown to be positivity preserving, so density could cross zero. And the stopping time is the same for every h, which is essential for the convergence argument but never shown to exist for the numerical solutions. The paper's own introduction explains why a uniform bound in probability is not realistic for stochastic equations; that makes (3.12) more delicate than the analogous deterministic assumption. This is not a fatal flaw if the paper is read as a conditional statement, but the referee should ask for a discussion of what would be needed to verify or relax it.\n\nTwo smaller items. The existence of local strong solutions on bounded domains is dismissed with a citation to a whole-space result and the word 'expected.' Since Theorem 3.8 is formulated for bounded domains, that gap should be closed or explicitly flagged as an assumption. And the extra Lipschitz condition (3.15) for p≥1 is ad hoc. It is not derived from the scheme, and the paper gives no reason to think it holds.\n\nWho is it for: anyone working on stochastic compressible flows or on structure-preserving numerics for hyperbolic systems. It deserves a serious referee. My recommendation: send it out, but make the authors separate the conditional results from what is actually proven, and ideally add a remark on (3.12) or an example where it can be verified.","headline":"First convergence result for entropy-stable DG on the complete stochastic Euler system, but every theorem rests on an unproved uniform-in-h bound and a bounded-domain strong-solution existence that is only cited.","tokens_in":33856,"tokens_out":4562,"would_cite":true,"duration_ms":42932,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M60","60H15","35Q31","65M12","65M08"],"pacs":[],"model":"deepseek-v4-flash","headline":"Entropy-stable discontinuous Galerkin discretizations of the stochastically forced complete Euler equations are shown to converge in law to dissipative martingale solutions, with relative-energy rates $O(h^{1/2})$ and $O(h)$ near pathwise…","keywords":["stochastic Euler equations","discontinuous Galerkin","entropy stability","dissipative martingale solutions","relative entropy","finite volume","stochastic compactness","convergence rates"],"falsifier":"Run the $p=0$ local Lax-Friedrichs scheme on the one-dimensional SOD shock-tube initial data of Section 6.1.3 with the same noise protocol and 1000 samples, and record, for each mesh size $h=64,128,256,512$, the first time at which $\\varrho_h$ falls below $1/K$ or $E_h$ exceeds $K$; if these hitting times converge to zero with positive probability as $h\\to0$, then assumption (3.12) has no positive-time regime for that test case, whereas if they remain bounded away from zero, comparing the measured $E_2$ with the predicted $O(h^{1/2})$ bound at $T=0.2$ settles the rate claim of Theorem 3.8(a) empirically.","tokens_in":32758,"feed_emoji":"🌊","tokens_out":8759,"duration_ms":86020,"temperature":0.7,"pith_summary":"This paper establishes a first convergence theory for high-order entropy-stable discontinuous Galerkin discretizations of the complete compressible Euler equations with stochastic forcing in the momentum equation. The target object is a dissipative martingale solution, a measure-valued weak solution adapted to the driving Wiener process whose oscillations and concentrations are recorded by defect measures. The main result says that, as the mesh size $h\\to 0$, the numerical laws converge, up to a subsequence, to such a solution, provided the discrete density stays away from vacuum and the discrete total energy stays bounded up to a stopping time. If a smooth pathwise strong solution exists, the expected $L^1$ relative energy error between the numerical and exact solution decays at least as $h^{1/2}$ for the finite-volume case and as $h$ for higher polynomial degrees, matching the deterministic theory. This matters because stochastic Euler-type models are used to represent turbulence and uncertainty, and until now no convergence result existed for such methods on the full stochastic system.","feed_headline":"Stochastic Euler scheme converges, with rate at least 1/2","feed_subtitle":"An entropy-stable discontinuous Galerkin method for randomly forced gas dynamics is proven to approach dissipative martingale solutions…","key_machinery":"The load-bearing object is the entropy-stable discontinuous Galerkin spectral element scheme (2.29): a nodal finite-element space on Gauss-Lobatto quadrature points written in summation-by-parts form, in which the volume integral is replaced by an entropy-conservative two-point numerical flux via flux differencing and interfaces are closed with a local Lax-Friedrichs flux, so the semidiscrete system dissipates the mathematical entropy. This scheme is applied to the total-energy formulation (2.22) of the stochastic Euler equations, with the noise entering only through the momentum and total-energy equations. Convergence is mediated by the dissipative martingale solution concept of Definition 3.1, where oscillations and concentrations are recorded by parametrised measures and defect measures, and the error analysis runs through the relative entropy, or ballistic free energy, inequality of Proposition 5.1, which compares the discrete state to the pathwise strong solution and reduces the error to a Gronwall-type estimate.","core_discovery":"On the paper's own terms, the central claim is that the entropy-stable DGSEM scheme for the complete stochastic Euler system is consistent and convergent even though the underlying PDE is hyperbolic, stochastically forced, and may develop discontinuities. Under the a priori bound that the discrete density never approaches vacuum and the discrete total energy never blows up, uniformly in the mesh size and up to a stopping time with $\\mathbb{P}(\\tau>0)=1$, the laws of the numerical solutions are tight, and every limit point, after a Skorokhod-type change of probability space, is a dissipative martingale solution in the sense of the paper's Definition 3.1. When a local pathwise strong solution exists on the same stochastic basis, the expected $L^1$ relative energy between the numerical and exact states is $O(h^{1/2})$ for polynomial degree $p=0$ and $O(h)$ for $p\\ge1$ under an additional unverified nodal Lipschitz condition, reproducing the deterministic convergence rates. The numerical experiments on smooth density waves in one and two dimensions show the scheme meeting or exceeding these rates, and the paper presents the first simulations for this stochastic Euler system in regimes where discontinuities may develop.","pith_inferences":["A natural testable extension is to replace the unverified nodal Lipschitz condition (3.15) by an explicit limiter or slope-reconstruction step; if that provably enforces (3.15), the $O(h)$ rate for $p\\ge1$ becomes a property of the algorithm rather than an assumption on its output.","The numerical tables show fourth-order relative-energy convergence on smooth data, far above the proved rates; this suggests the analysis is limited by the worst-case consistency estimates and that a regularity-aware error estimate could close the gap.","Because Gaussian noise makes uniform-in-time $L^\\infty$ bounds impossible, the stopping time is not merely technical: any extension to deterministic times will likely need a noise-strength-dependent or path-dependent horizon, or a new a priori bound not present in this paper.","For genuinely infinite-dimensional noise, the treatment of truncating the Wiener process to finitely many modes adds a tail error controlled by the truncated basis; this yields a concrete recipe for an implementable method with a combined discretization-plus-truncation error estimate."],"forward_implications":["Under the stopping-time bound, the entropy-stable DGSEM scheme yields, along a subsequence, a dissipative martingale solution of the complete stochastic Euler system, providing the first convergence result for a high-order method on this system.","When a local pathwise strong solution exists, the expected $L^1$ relative energy at the final time is $O(h^{1/2})$ for $p=0$ and $O(h)$ for $p\\ge1$ under the nodal Lipschitz assumption, giving concrete rates that can be checked numerically.","Because the scheme is entropy dissipative by construction, the limit inherits the entropy inequality and the total-energy balance, so the numerical method preserves the thermodynamic structure rather than merely conserving mass, momentum, and energy.","The finite-volume case $p=0$ is included as a special case, so the result also supplies convergence of local Lax-Friedrichs finite volume schemes for the stochastic Euler equations.","Outside the lifespan of a strong solution, the only guarantee is convergence in law to a dissipative martingale solution; the stopping-time formulation is what makes the a priori bounds and hence the whole argument possible."],"supporting_citations":[{"why":"Defines the dissipative martingale solution concept and proves existence and weak-strong uniqueness, giving the target limit object for the numerical scheme.","marker":"[56]"},{"why":"Supplies the deterministic DG consistency estimates that the stochastic consistency proof borrows directly.","marker":"[52]"},{"why":"Provides the weak BV estimate and the relative-energy comparison lemma used for the $O(h^{1/2})$ rate in the finite-volume case.","marker":"[53]"},{"why":"Provides the dissipative measure-valued solution framework and the deterministic convergence blueprint, including the relative entropy balance for $p=0$.","marker":"[34]"},{"why":"Establishes consistency of flux-corrected finite element schemes for the Euler equations and supports the well-posedness of the semidiscrete system under Lipschitz coefficients.","marker":"[50]"},{"why":"The only prior stochastic Euler convergence result, for the barotropic case, whose a priori bound the paper replaces by the stopping-time assumption.","marker":"[20]"},{"why":"Develops dissipative solutions for the complete deterministic Euler system and is used to derive the uniform bounds from the no-vacuum, bounded-energy assumption.","marker":"[12]"},{"why":"Supplies the entropy-conservative two-point numerical volume flux used in the flux-differencing DGSEM.","marker":"[63]"},{"why":"Provides the Skorokhod representation theorem for non-metrizable path spaces, needed to pass to the limit in law despite defect measures.","marker":"[45]"},{"why":"Supplies the stochastic compactness and filtration framework and the martingale identification tools used in the consistency proof.","marker":"[11]"}],"fun_headline_variants":["Stochastic Euler solver proven to converge at rate 1/2","Entropy-stable DG scheme tames random gas dynamics","Convergence guarantee for stochastic Euler equations","First convergent scheme for stochastic Euler flows","Random-forcing Euler solved to half-order accuracy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire convergence and rate statements are conditional on (3.12): uniformly in the mesh size, the discrete density never drops below $1/K$ and the discrete total energy never exceeds $K$ until some stopping time, and this bound is assumed rather than derived from the scheme, with an extra unverified smoothness condition on the numerical nodal values needed for the $p\\ge1$ rate.","fun_headline_variants_meta":{"raw":{"variants":["Stochastic Euler solver proven to converge at rate 1/2","Entropy-stable DG scheme tames random gas dynamics","Convergence guarantee for stochastic Euler equations","First convergent scheme for stochastic Euler flows","Random-forcing Euler solved to half-order accuracy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000515,"raw_usage":{"total_tokens":2536,"prompt_tokens":1016,"completion_tokens":1520,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":632,"completion_tokens_details":{"reasoning_tokens":1446}},"tokens_in":632,"tokens_out":1520,"duration_ms":11726,"temperature":1.0,"reasoning_tokens":1446,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:39:14.613700+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the $p=0$ local Lax-Friedrichs scheme on the one-dimensional SOD shock-tube initial data of Section 6.1.3 with the same noise protocol and 1000 samples, and record, for each mesh size $h=64,128,256,512$, the first time at which $\\varrho_h$ falls below $1/K$ or $E_h$ exceeds $K$; if these hitting times converge to zero with positive probability as $h\\to0$, then assumption (3.12) has no positive-time regime for that test case, whereas if they remain bounded away from zero, comparing the measured $E_2$ with the predicted $O(h^{1/2})$ bound at $T=0.2$ settles the rate claim of Theorem 3.8(a) empirically.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the dissipative martingale solution concept and proves existence and weak-strong uniqueness, giving the target limit object for the numerical scheme."},{"cited_title":"Luk´ aˇ cov´ a-Medvid’ov´ a and P.¨Offner","cited_arxiv_id":null,"evidence_quote":"Supplies the deterministic DG consistency estimates that the stochastic consistency proof borrows directly."},{"cited_title":"Luk´ aˇ cov´ a-Medvid’ov´ a, B","cited_arxiv_id":null,"evidence_quote":"Provides the weak BV estimate and the relative-energy comparison lemma used for the $O(h^{1/2})$ rate in the finite-volume case."},{"cited_title":"Feireisl, M","cited_arxiv_id":null,"evidence_quote":"Provides the dissipative measure-valued solution framework and the deterministic convergence blueprint, including the relative entropy balance for $p=0$."},{"cited_title":"Consistency and convergence of flux-corrected finite element methods for nonlinear hyperbolic problems","cited_arxiv_id":"2308.14872","evidence_quote":"Establishes consistency of flux-corrected finite element schemes for the Euler equations and supports the well-posedness of the semidiscrete system under Lipschitz coefficients."},{"cited_title":"Chaudhary and U","cited_arxiv_id":null,"evidence_quote":"The only prior stochastic Euler convergence result, for the barotropic case, whose a priori bound the paper replaces by the stopping-time assumption."},{"cited_title":"Breit, E","cited_arxiv_id":null,"evidence_quote":"Develops dissipative solutions for the complete deterministic Euler system and is used to derive the uniform bounds from the no-vacuum, bounded-energy assumption."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the entropy-conservative two-point numerical volume flux used in the flux-differencing DGSEM."},{"cited_title":"Jakubowski","cited_arxiv_id":null,"evidence_quote":"Provides the Skorokhod representation theorem for non-metrizable path spaces, needed to pass to the limit in law despite defect measures."},{"cited_title":"Breit, E","cited_arxiv_id":null,"evidence_quote":"Supplies the stochastic compactness and filtration framework and the martingale identification tools used in the consistency proof."}],"review_version":1}