Discontinuous Galerkin methods for the complete stochastic Euler equations

Dominic Breit; Philipp \"Offner; Thamsanqa Castern Moyo

REVIEW 3 major objections 6 minor 73 references

Discontinuous Galerkin methods for the complete stochastic Euler equations

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2412.07613 v1 pith:LIJ263JQ submitted 2024-12-10 math.NA cs.NAmath.APmath.PR

Dominic Breit , Thamsanqa Castern Moyo , Philipp Öffner This is my paper · ORCID

classification math.NAcs.NAmath.APmath.PR MSC 65M6060H1535Q3165M1265M08

keywords stochasticEulerequationsdiscontinuousGalerkinentropystabilitydissipativemartingalesolutionsrelativefinitevolumecompactnessconvergencerates

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What'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.

The 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.

Two 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.

Who 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.

Referee Report

3 major / 6 minor

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.

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 (3)

[§3.3, Eq. (3.12); Theorems 3.6 and 3.8] 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.
[§3.3, Theorem 3.8(b), Eq. (3.15); Remark 4.1] 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.
[§4.3, after Eq. (4.23)] 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.

minor comments (6)

[Abstract and Section 1] The abstract contains a typo: "we proof" should be "we prove", and "The results built" should be "The results build".
[§2.3, Eq. (2.20)] 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.
[§2.3, after Eq. (2.28)] The text reads "I4 is the 4 × 4-identy matrix"; this should be "identity matrix".
[§3.1, Definition 3.1] 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.
[§4.3, last paragraph] 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.
[§6.1.1 and Table 5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence and rate theorems are conditional on an explicit a priori bound, but they do not reduce to their inputs by construction.

full rationale

The paper's central claim is that, under the explicit uniform-in-h bound (3.12) (positive density and bounded total energy up to a stopping time), the entropy-stable DGSEM solutions converge in law to a dissipative martingale solution of the stochastic Euler system, and that the relative-energy error against a strong solution is O(h^{1/2}) for p=0 or O(h) for p>=1 under the additional nodal-Lipschitz assumption (3.15). This is a conditional theorem: (3.12) and (3.15) are hypotheses, not consequences, and the paper says so explicitly ('We work under the following hypothesis...'). A conditional statement is not circular merely because its hypothesis is unverified; the convergence proof supplies consistency estimates (4.2)-(4.8), compactness via Jakubowski-Skorokhod, and a relative-entropy inequality (Proposition 5.1) that are not the same as the assumed bounds. The consistency estimates and the dissipative-solution concept are imported from the authors' own prior works [52] and [56], but those are published, parameter-free results with stated assumptions that do not include the stochastic convergence target; therefore, by the independence rule, they are real evidence rather than circularity. The step of 'augmenting' Rpress by a spatially homogeneous measure to convert the energy inequality (4.23) into the equality required by Definition 3.1 is a standard defect-measure completion and does not alter the momentum or entropy equations, so it is not a self-definitional reduction. The paper also flags genuine scope limitations: (3.12) is not shown to hold for any fixed deterministic horizon, the scheme is not shown to be positivity-preserving, and the numerical section reports instability for large noise in the Kelvin-Helmholtz test. These are correctness and robustness gaps, not circular derivations; no equation is defined in terms of the result it is used to prove.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims are conditional on several domain assumptions rather than on fitted parameters. There are no newly invented physical entities: the dissipative martingale solution with defect measures Rconv, Rpress and Young measure V was introduced in Moyo [56]. The scheme itself is a standard entropy-stable DGSEM with a stochastic forcing term. The free parameter list is empty because the constant K and the stopping time t are assumed inputs, not fitted values.

assumptions (6)

domain assumption A priori bound (3.12): inf_{[0,t]×O} ϱh ≥ 1/K and sup_{[0,t]×O} E_h ≤ K P-a.s., uniformly in h, for some stopping time t with P(t>0)=1.
Central to Theorem 3.6 and Theorem 3.8; not derived from the scheme. The paper argues it is plausible only for short random times, but the stopping time is not quantified.
domain assumption Existence of a local strong pathwise solution [r, Θ, v] on bounded domain O with no-flux boundary conditions up to stopping time s (Definition 3.3).
Required in Theorem 3.8. The paper says 'we expect' existence and cites [47] for the whole space, but does not provide a proof for the bounded-domain case.
ad hoc to paper Lipschitz continuity of nodal values within each element, (3.15): |U_h(x_j,t∧t) - U_h(x_i,t∧t)| ≤ ch for all x_i, x_j in each element.
Introduced specifically to obtain the O(h) rate for p≥1 in Theorem 3.8(b). No justification is given that realistic DG solutions satisfy it; the authors note limiters are not considered.
domain assumption Deterministic consistency estimates (4.7) and (4.8) from [52] and [53] hold for the stochastic scheme.
The paper asserts 'line by line' following [52] and [53] without re-deriving them. These estimates control the consistency error terms e_h in the stochastic setting.
domain assumption Weak-strong uniqueness principle for dissipative martingale solutions, Theorem 3.4 from [56].
Used in the proof of Theorem 3.8 to control the relative entropy production Q. The result is from an author's prior paper and is not re-proved here.
standard math Itô formula, martingale representation, and Jakubowski's extension of the Skorokhod representation theorem.
Standard tools in stochastic PDE analysis, used without proof in Section 4.

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Pith. "Pith review of Discontinuous Galerkin methods for the complete stochastic Euler equations." pith.science (2026). https://pith.science/paper/LIJ263JQ

@misc{pith2026241207613,
  author       = {Pith},
  title        = {Pith review of: Discontinuous Galerkin methods for the complete stochastic Euler equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LIJ263JQ}},
  note         = {Machine review of arXiv:2412.07613}
}

abstract

In recent years, stochastic effects have become increasingly relevant for describing fluid behaviour, particularly in the context of turbulence. The most important model for inviscid fluids in computational fluid dynamics are the Euler equations of gas dynamics which we focus on in this paper. To take stochastic effects into account, we incorporate a stochastic forcing term in the momentum equation of the Euler system. To solve the extended system, we apply an entropy dissipative discontinuous Galerkin spectral element method including the Finite Volume setting, adjust it to the stochastic Euler equations and analyze its convergence properties. Our analysis is grounded in the concept of dissipative martingale solutions, as recently introduced by Moyo (J. Diff. Equ. 365, 408-464, 2023). Assuming no vacuum formation and bounded total energy, we proof that our scheme converges in law to a dissipative martingale solution. During the lifespan of a pathwise strong solution, we achieve convergence of at least order 1/2, measured by the expected $L^1$ norm of the relative energy. The results built a counterpart of corresponding results in the deterministic case. In numerical simulations, we show the robustness of our scheme, visualise different stochastic realizations and analyze our theoretical findings.

Figures

Figures reproduced from arXiv: 2412.07613 by the authors.

**Figure 1.** Different noise (left) and different noise strength(right) 6.2. Two dimension. We consider the two-dimensional stochastic compressible Euler equations (2.23) with the right hand side h(U) = µ 2 ϱ   0 0 0 1   , g(U)dW = µ   0 ϱ dβ1 ϱ dβ2 m1 dβ1 + m2 dβ2   . where µ denotes again a stochastic forcing factor. The domain is divided into n × n uniform quads for the elements (n ∈ N) [PITH_FULL_IMAGE:figur… view at source ↗

**Figure 2.** Kelvin-Helmholtz and influence of the noise references for instance in [44, 46, 51, 60]. However, our analysis is relying on the consistency estimation where no regularity have been assumed. In the future, we plan to derive new error estimates using the consistency analysis and transfer it also from the purely deterministic setting to the stochastic one. Meanwhile, we have numerically analyzed the stochastic Euler e… view at source ↗

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