Strong convergence of a fully discrete scheme for stochastic Burgers equation with fractional-type noise

Wanrong Cao; Yibo Wang

arxiv: 2408.00951 · v2 · submitted 2024-08-01 · 🧮 math.NA · cs.NA· math.PR

Strong convergence of a fully discrete scheme for stochastic Burgers equation with fractional-type noise

Yibo Wang , Wanrong Cao This is my paper

Pith reviewed 2026-05-23 22:17 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.PR

keywords stochastic Burgers equationfractional Brownian motionspectral Galerkinexponential Euler methodstrong convergencefully discrete schemestopping time technique

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The pith

A spectral Galerkin plus nonlinear-tamed exponential Euler scheme for the stochastic Burgers equation with fractional Brownian motion converges strongly.

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

The paper develops a fully discrete numerical method for the stochastic Burgers equation driven by additive cylindrical fractional Brownian motion with Hurst parameter greater than one half. Spectral Galerkin discretization is used in space and a nonlinear-tamed accelerated exponential Euler method in time. Bounded moments of the approximations are obtained from the exponential integrability of the stochastic convolution, convergence in probability is shown via a stopping time argument, and strong convergence is deduced from these two facts.

Core claim

The fully discrete scheme converges strongly to the mild solution of the stochastic Burgers equation. After establishing exponential integrability of the stochastic convolution of the fractional Brownian motion, the paper shows that both the semi-discrete and fully discrete approximations have uniformly bounded moments; it then proves convergence in probability by a stopping time technique and obtains strong convergence as a consequence.

What carries the argument

The nonlinear-tamed accelerated exponential Euler method, which stabilizes the nonlinear term while integrating the linear part exactly and is paired with spectral Galerkin spatial discretization.

If this is right

The semi-discrete Galerkin approximations possess uniformly bounded moments of all orders.
The fully discrete approximations also possess uniformly bounded moments.
Convergence in probability of the fully discrete scheme to the true solution holds.
Strong convergence in L^p follows directly from the combination of moment boundedness and convergence in probability.

Where Pith is reading between the lines

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

Strong convergence would allow pathwise error control when simulating individual realizations of the solution.
The stopping-time technique for probability convergence might transfer to other semilinear SPDEs with similar noise regularity.
The taming parameter in the time integrator could be tuned to recover optimal convergence rates for specific values of H.

Load-bearing premise

The Hurst parameter of the fractional Brownian motion lies strictly between one half and one, supplying the regularity and exponential integrability needed for the moment bounds and stopping-time argument.

What would settle it

A concrete numerical test in which the strong error between the computed solution and a high-resolution reference fails to approach zero when both spatial mesh size and time step are driven to zero under the stated conditions on H.

read the original abstract

We investigate numerical approximations for the stochastic Burgers equation driven by an additive cylindrical fractional Brownian motion with Hurst parameter $H \in (\frac{1}{2}, 1)$. To discretize the continuous problem in space, a spectral Galerkin method is employed, followed by the presentation of a nonlinear-tamed accelerated exponential Euler method to yield a fully discrete scheme. By showing the exponential integrability of the stochastic convolution of the fractional Brownian motion, we present the boundedness of moments of semi-discrete and full-discrete approximations. Building upon these results and the convergence of the fully discrete scheme in probability proved by a stopping time technique, we derive the strong convergence of the proposed scheme.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves strong convergence for spectral Galerkin plus nonlinear-tamed accelerated exponential Euler on stochastic Burgers with additive fractional BM noise (H>1/2), using exponential integrability for moments then stopping times.

read the letter

The core result is a convergence proof for one concrete fully discrete scheme on the stochastic Burgers equation with cylindrical fractional Brownian motion. They discretize space with spectral Galerkin and time with a nonlinear-tamed accelerated exponential Euler method. The argument runs through exponential integrability of the stochastic convolution to control moments of the approximations, then a stopping-time argument for convergence in probability, and finally strong convergence in L^p. The combination of this particular taming with the accelerated integrator for fractional noise does not appear in the cited prior work, so that part is new. The proof architecture is standard for these SPDEs and the H>1/2 range gives enough regularity that the continuous convolution has the needed exponential moments via standard Gaussian tail estimates. The paper therefore supplies a usable scheme with a complete strong-convergence statement, which is the main practical takeaway. The soft spot is the uniformity of the moment bounds with respect to the discretization parameters. The stopping-time lift from in-probability to strong convergence requires that the p-moments stay bounded independently of the number of modes and the time step; if the constants hidden in the exponential-integrability step grow with N or 1/Δt, the argument does not close. The abstract states that bounded moments are shown for both semi- and fully-discrete versions, but without the explicit estimates it is not obvious whether the taming and projection preserve the uniform control that the continuous case enjoys. If the paper tracks the constants carefully, the result is fine; if not, that is the place a referee would press. This is incremental numerical-analysis work aimed at people who need provably convergent schemes for SPDEs with fractional noise. It is not reorganizing the field, but it is a clean, self-contained contribution that a specialist would want to see. A serious editor should send it to referees rather than desk-reject; the technical details are checkable and the scheme is explicit enough to be useful once the uniformity question is settled.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a spectral Galerkin spatial discretization combined with a nonlinear-tamed accelerated exponential Euler time-stepping scheme for the stochastic Burgers equation driven by additive cylindrical fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). It first establishes exponential integrability of the stochastic convolution to obtain uniform moment bounds on the semi-discrete and fully discrete approximations, then uses a stopping-time argument to prove convergence in probability of the fully discrete scheme, and finally upgrades this to strong convergence.

Significance. If the moment bounds are shown to be uniform with respect to the discretization parameters, the result would constitute a useful extension of strong-convergence theory to SPDEs with fractional noise and superlinear drift, employing a practical tamed exponential integrator. The approach builds on standard techniques (exponential moments, stopping times) but applies them to a fully discrete setting that is relevant for computation.

major comments (1)

[sections establishing moment bounds for the fully discrete scheme and the passage from convergence in probability to L^p] The exponential integrability of the continuous stochastic convolution is used to bound moments of the approximations, but the manuscript must explicitly verify that the resulting moment constants remain independent of the number of Galerkin modes N and the time step Δt after accounting for the spectral projection error and the taming truncation in the exponential Euler step. Without this uniformity, the stopping-time argument cannot be guaranteed to produce strong convergence whose rate does not deteriorate under refinement.

minor comments (2)

[abstract] The abstract and introduction should state the precise norm and rate (if any) in which strong convergence is obtained.
[introduction and scheme definition] Notation for the fully discrete solution (e.g., u^{N,Δt}_n) and the precise form of the taming function should be introduced earlier for readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comment. We address the concern regarding uniformity of moment bounds point by point below.

read point-by-point responses

Referee: [sections establishing moment bounds for the fully discrete scheme and the passage from convergence in probability to L^p] The exponential integrability of the continuous stochastic convolution is used to bound moments of the approximations, but the manuscript must explicitly verify that the resulting moment constants remain independent of the number of Galerkin modes N and the time step Δt after accounting for the spectral projection error and the taming truncation in the exponential Euler step. Without this uniformity, the stopping-time argument cannot be guaranteed to produce strong convergence whose rate does not deteriorate under refinement.

Authors: We agree that explicit verification of parameter-independent constants is essential for rigor. In the proofs establishing exponential integrability of the stochastic convolution (Section 2) and the subsequent moment bounds for the semi-discrete (Section 3) and fully discrete (Section 4) approximations, the constants arise from the Burkholder-Davis-Gundy inequality applied to the fractional noise and from a tamed Gronwall inequality; these depend only on the equation coefficients, the Hurst index H, and the taming threshold, none of which involve N or Δt. The spectral projection error is absorbed using the smoothing property of the analytic semigroup and the Hölder regularity of the cylindrical fBm, yielding a bound independent of the dimension of the Galerkin space. The nonlinear taming ensures that, on the stopped processes used in the convergence-in-probability argument, the truncation does not introduce N- or Δt-dependent growth. Nevertheless, to make this independence fully transparent, we will insert a dedicated remark (or short lemma) immediately after the moment-bound theorems that explicitly states the constants are uniform in N and Δt and traces the dependence through the estimates. With this clarification the stopping-time argument carries through unchanged and the strong-convergence result remains valid. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent analytic steps

full rationale

The paper's chain proceeds by first proving exponential integrability of the continuous stochastic convolution (for H in (1/2,1)), then using that to obtain uniform moment bounds on the semi-discrete and fully discrete approximations, separately establishing convergence in probability via a stopping-time argument, and finally combining the two to obtain strong convergence. None of these steps is shown to reduce by definition or by fitting to the final claim; the moment bounds and stopping-time argument are presented as separate analytic results rather than tautologies or self-referential fits. No load-bearing self-citations or imported uniqueness theorems appear in the provided derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard theory of fractional Brownian motion for H > 1/2 and on the well-posedness of the continuous stochastic Burgers equation; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Hurst parameter H ∈ (1/2, 1)
Invoked to guarantee the regularity needed for exponential integrability of the stochastic convolution.

pith-pipeline@v0.9.0 · 5641 in / 1252 out tokens · 24731 ms · 2026-05-23T22:17:03.632702+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Relation between the paper passage and the cited Recognition theorem.

By showing the exponential integrability of the stochastic convolution of the fractional Brownian motion, we present the boundedness of moments of semi-discrete and full-discrete approximations. Building upon these results and the convergence of the fully discrete scheme in probability proved by a stopping time technique, we derive the strong convergence
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 3.5. For any ˜c > 0 and N ∈ N, there exists η ≥ 0, such that E[ e^{˜c ∫_0^T (∥ON,η_t∥_{L^∞}^2 +1) dt} ] < ∞

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

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Alabert, I

A. Alabert, I. Gynogy, On numerical approximation of stochastic Burgers’ equation, Springer Berlin, 2006

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S. Becker, B. Gess, A. Jentzen, P. E. Kloeden, Strong convergence rates for explicit space- time discrete numerical approximations of stochastic Allen–Cahn equations, Stoch. PDE Anal. Comput. 11, (2023) 211-268

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Biagini, Y.Z

F. Biagini, Y.Z. Hu, B. ∅ksendal, T.S. Zhang, Stochastic calculus for fractional Brownian motion and applications, Springer London, 2008

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Bl¨ omker, M

D. Bl¨ omker, M. Kamrani, S.M. Hosseini, Full discretization of the stochastic Burgers equation with correlated noise, IMA J. Numer. Anal., 33 (2013) 825-848

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Bl¨ omker, A

D. Bl¨ omker, A. Jentzen, Galerkin approximations for the stochastic Burgers equation, SIAM J. Numer. Anal., 51 (2013) 694-715

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Br´ ehier, J.B

C.E. Br´ ehier, J.B. Cui, J.L. Hong, Strong convergence rates of semi-discrete splitting approx- imations for stochastic Allen–Cahn equation, IMA J. Numer. Anal., 39 (2019) 2096-2134

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M. Hairer, J. Voss, Approximations to the stochastic Burgers equation, J. Nonlinear Sci., 21 (2011) 897-920

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Hutzenthaler, A

M. Hutzenthaler, A. Jentzen, F. Lindner, P. Puˇ snik, Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations, arXiv:1911.01870, 2019

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Hutzenthaler, A

M. Hutzenthaler, A. Jentzen, D. Salimova, Strong convergence of full-discrete nonlinearity- truncated accelerated exponential Euler-type approximations for stochastic Kuramoto– Sivashinsky equations, Commun. Math. Sci., 16 (2018) 1489-1529

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Hutzenthaler, R

M. Hutzenthaler, R. Link, Strong convergence rates for full-discrete approximationsof stochas- tic Burgers equations with multiplicative noise, arXiv:2210.17536, 2022

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A. Jentzen, D. Salimova, T. Welti, Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations, arXiv:1710.07123, 2017

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Kotelenez, A stochastic Navier–Stokes equation for the vorticity of a two-dimensional fluid, Ann

P. Kotelenez, A stochastic Navier–Stokes equation for the vorticity of a two-dimensional fluid, Ann. Appl. Probab., 5 (1995) 1126-1160

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Liu, Z.H

Z.H. Liu, Z.H. Qiao, Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise, Stoch. PDE Anal. Comput. 9 (2021) 559-602

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B.B. Mandelbrot, J.W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968) 422-437

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Pe´ rez, L

D.G. Pe´ rez, L. Zunino, M. Garavaglia, Modeling turbulent wave-front phase as a fractional Brownian motion: a new approach, J. Opt. Soc. Am. A, 21 (2004) 1962-1969

work page 2004
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Stroock, Probability Theory, Cambridge University Press, 2011

D.W. Stroock, Probability Theory, Cambridge University Press, 2011

work page 2011
[30]

G.L. Wang, M. Zeng, B.L. Guo, Stochastic Burgers’ equation driven by fractional Brownian motion, J. Math. Anal. Appl., 371 (2010) 210-222

work page 2010
[31]

Wang, An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation, Stoch

X.J. Wang, An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation, Stoch. Process. Appl., 130 (2020) 6271-6299

work page 2020
[32]

Wang, R.S

X.J. Wang, R.S. Qi, F.Z. Jiang, Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations, BIT Numer. Math., 57 (2017) 557-585

work page 2017
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W. Wu, S.B. Cui, J.Q. Duan, Global well-posedness of the stochastic generalized Kuramoto– Sivashinsky equation with multiplicative noise, Acta Mathematicae Applicatae Sinica, English Series, 34 (2018) 566-584. 23

work page 2018

[1] [1]

Alabert, I

A. Alabert, I. Gynogy, On numerical approximation of stochastic Burgers’ equation, Springer Berlin, 2006

work page 2006

[2] [2]

Becker, B

S. Becker, B. Gess, A. Jentzen, P. E. Kloeden, Strong convergence rates for explicit space- time discrete numerical approximations of stochastic Allen–Cahn equations, Stoch. PDE Anal. Comput. 11, (2023) 211-268

work page 2023

[3] [3]

Becker, A

S. Becker, A. Jentzen, Strong convergence rates for nonlinearity-truncated Euler-type approx- imations of stochastic Ginzburg–Landau equations, Stoch. Process. Appl., 129 (2019) 28-69

work page 2019

[4] [4]

Biagini, Y.Z

F. Biagini, Y.Z. Hu, B. ∅ksendal, T.S. Zhang, Stochastic calculus for fractional Brownian motion and applications, Springer London, 2008

work page 2008

[5] [5]

Bl¨ omker, M

D. Bl¨ omker, M. Kamrani, S.M. Hosseini, Full discretization of the stochastic Burgers equation with correlated noise, IMA J. Numer. Anal., 33 (2013) 825-848

work page 2013

[6] [6]

Bl¨ omker, A

D. Bl¨ omker, A. Jentzen, Galerkin approximations for the stochastic Burgers equation, SIAM J. Numer. Anal., 51 (2013) 694-715

work page 2013

[7] [7]

Br´ ehier, J.B

C.E. Br´ ehier, J.B. Cui, J.L. Hong, Strong convergence rates of semi-discrete splitting approx- imations for stochastic Allen–Cahn equation, IMA J. Numer. Anal., 39 (2019) 2096-2134

work page 2019

[8] [8]

Cai, S.Q

M. Cai, S.Q. Gan, X.J. Wang, Weak convergence rates for an explicit full-discretization of stochastic Allen–Cahn equation with additive noise, J Sci. Comput., 86 (2021) 34

work page 2021

[9] [9]

Carelli, A

E. Carelli, A. Prohl, Rates of convergence for discretizations of the stochastic incompressible Navier–Stokes equations, SIAM J. Numer. Anal., 50 (2012) 2467-2496

work page 2012

[10] [10]

Chekhlov, V

A. Chekhlov, V. Yakhot, Kolmogorov turbulence in a random-force-driven Burgers equation, Phys. Rev. E, 51 (1995) 2739-2742

work page 1995

[11] [11]

Da Prato, A

G. Da Prato, A. Debussche, R. Temam, Stochastic Burgers’ equation, Nonlinear Differential Equations and Applications, 1 (1994) 389-402

work page 1994

[12] [12]

Da Prato, D

G. Da Prato, D. Gatarek, Stochastic Burgers equation with correlated noise, Stochastics, 52 (1995) 29-41

work page 1995

[13] [13]

Da Prato, J

G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, 2014

work page 2014

[14] [14]

Duncan, B

T.E. Duncan, B. Pasik-Duncan, B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2 (2002) 225-250

work page 2002

[15] [15]

C. Gugg, H. Kielh¨ ofer, M. Niggemann, On the approximation of the stochastic Burgers equation, Commun. Math. Phys., 230 (2002) 181-199. 22

work page 2002

[16] [16]

Hairer, J

M. Hairer, J. Voss, Approximations to the stochastic Burgers equation, J. Nonlinear Sci., 21 (2011) 897-920

work page 2011

[17] [17]

Hutzenthaler, A

M. Hutzenthaler, A. Jentzen, F. Lindner, P. Puˇ snik, Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations, arXiv:1911.01870, 2019

work page arXiv 1911

[18] [18]

Hutzenthaler, A

M. Hutzenthaler, A. Jentzen, D. Salimova, Strong convergence of full-discrete nonlinearity- truncated accelerated exponential Euler-type approximations for stochastic Kuramoto– Sivashinsky equations, Commun. Math. Sci., 16 (2018) 1489-1529

work page 2018

[19] [19]

Hutzenthaler, R

M. Hutzenthaler, R. Link, Strong convergence rates for full-discrete approximationsof stochas- tic Burgers equations with multiplicative noise, arXiv:2210.17536, 2022

work page arXiv 2022

[20] [20]

Jeng, Forced model equation for turbulence, Phys

D.T. Jeng, Forced model equation for turbulence, Phys. Fluids, 12 (1969) 2006-2010

work page 1969

[21] [21]

Jentzen, D

A. Jentzen, D. Salimova, T. Welti, Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations, arXiv:1710.07123, 2017

work page internal anchor Pith review arXiv 2017

[22] [22]

Khan, Higher order pathwise approximation for the stochastic Burgers’ equation with additive noise, Appl

F. Khan, Higher order pathwise approximation for the stochastic Burgers’ equation with additive noise, Appl. Numer. Math., 162 (2021) 67-80

work page 2021

[23] [23]

Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum

A.N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C.R. (Doklady) Acad. Sci. URSS (N.S.) 26 (1940) 115-118

work page 1940

[24] [24]

Kotelenez, A stochastic Navier–Stokes equation for the vorticity of a two-dimensional fluid, Ann

P. Kotelenez, A stochastic Navier–Stokes equation for the vorticity of a two-dimensional fluid, Ann. Appl. Probab., 5 (1995) 1126-1160

work page 1995

[25] [25]

Liu, Z.H

Z.H. Liu, Z.H. Qiao, Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise, Stoch. PDE Anal. Comput. 9 (2021) 559-602

work page 2021

[26] [26]

Mandelbrot, J.W

B.B. Mandelbrot, J.W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968) 422-437

work page 1968

[27] [27]

Pe´ rez, L

D.G. Pe´ rez, L. Zunino, M. Garavaglia, Modeling turbulent wave-front phase as a fractional Brownian motion: a new approach, J. Opt. Soc. Am. A, 21 (2004) 1962-1969

work page 2004

[28] [28]

Rothe, Global solutions of reaction-diffusion systems, Springer Berlin, 1984

F. Rothe, Global solutions of reaction-diffusion systems, Springer Berlin, 1984

work page 1984

[29] [29]

Stroock, Probability Theory, Cambridge University Press, 2011

D.W. Stroock, Probability Theory, Cambridge University Press, 2011

work page 2011

[30] [30]

G.L. Wang, M. Zeng, B.L. Guo, Stochastic Burgers’ equation driven by fractional Brownian motion, J. Math. Anal. Appl., 371 (2010) 210-222

work page 2010

[31] [31]

Wang, An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation, Stoch

X.J. Wang, An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation, Stoch. Process. Appl., 130 (2020) 6271-6299

work page 2020

[32] [32]

Wang, R.S

X.J. Wang, R.S. Qi, F.Z. Jiang, Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations, BIT Numer. Math., 57 (2017) 557-585

work page 2017

[33] [33]

W. Wu, S.B. Cui, J.Q. Duan, Global well-posedness of the stochastic generalized Kuramoto– Sivashinsky equation with multiplicative noise, Acta Mathematicae Applicatae Sinica, English Series, 34 (2018) 566-584. 23

work page 2018