Strong error estimates for a fully discrete SAV scheme for the stochastic Allen--Cahn equation with multiplicative noise

Stefan Metzger

REVIEW 2 major objections 4 minor 57 references

Strong error estimates for a fully discrete SAV scheme for the stochastic Allen--Cahn equation with multiplicative noise

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A linear, unconditionally stable SAV scheme attains the optimal strong rates for the stochastic Allen–Cahn equation with multiplicative noise.

desk verdict Solid new rate theorem for a linear SAV scheme, with a load-bearing carry-over claim that a referee should check; numerics use a coefficient outside the stated assumptions. read the letter →

arxiv 2501.04618 v1 pith:2ONIXQTG submitted 2025-01-08 math.NA cs.NA

Stefan Metzger This is my paper · ORCID

classification math.NAcs.NA MSC 60H3565M6060H1565M12

keywords stochasticAllen-Cahnequationmultiplicativenoisefiniteelementsstrongrateofconvergencescalarauxiliaryvariablemethodperiodicboundaryconditionsunconditionalstability

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

The paper proves that the augmented scalar auxiliary variable (SAV) method—a fully discrete, linear finite element scheme—approximates solutions of the stochastic Allen–Cahn equation with multiplicative noise at optimal strong rates: for every $\delta>0$, the expected $L^2(O)$ error at the final time together with the accumulated $H^1(O)$ error is bounded by $C_\delta(\tau^{1-\delta}+h^2)$. This matches the rates previously established for a nonlinear, implicit scheme, but the SAV scheme solves only linear systems at each step and is unconditionally stable. The essential device is a modification of the SAV auxiliary-variable update that compensates for the poor temporal regularity of stochastic solutions, so the auxiliary variable tracks $\sqrt{E_h(\varphi_h^n)}$ to order $\tau^{1/2-\delta/2}$. The result makes linear, energy-stable SAV-type discretizations provably reliable alternatives to implicit methods for phase-field SPDEs.

What carries the argument

The key mechanism is the augmented SAV update. The auxiliary variable $r_h^n$ is meant to approximate $\sqrt{E_h(\varphi_h^n)}$, where $E_h(\zeta):=\int_{\mathcal{O}} I_h\{F(\zeta)\}\,\mathrm{d}x$ and $F(\varphi)=\tfrac14(\varphi^2-1)^2+\gamma$ is the shifted double-well potential with $\gamma>0$. In a standard SAV scheme the update of $r_h^n$ is a first-order Taylor approximation of $\sqrt{E_h(\varphi_h^n)}$, leaving an error of order $|\varphi(t_n)-\varphi(t_{n-1})|^2$ per step; Brownian solutions are only Hölder continuous with exponent below $1/2$, so that error does not vanish fast enough. The augmentation adds two terms that linearly reproduce the second-order Taylor terms involving $\Phi_h(\varphi_h^{n-1})\Delta_n\xi^\tau$, where $\Delta_n\xi^\tau:=W(t_n)-W(t_{n-1})$, lowering the remaining auxiliary-variable error to $\tau^{1/2-\delta/2}$ (Lemma 4.2(4.4b)). That bound, together with the uniform discrete regularity estimates (4.4a), is what allows the error analysis to reach $\tau^{1-\delta}+h^2$.

What would settle it

Compute, on the periodic domain and with the finite-dimensional noise of Section 6, the experimental order of convergence in $\tau$ at fixed small $h$ (scaling $\tau\sim h^2$); if the temporal EOC stabilizes below $1/2-\delta$ for some fixed $\delta>0$, Theorem 3.1 is false. A more direct falsifier is to test the imported Lemma 4.2: check numerically that $E[\max_{0\le m\le N}|r_h^m-\sqrt{E_h(\varphi_h^m)}|^p]\le C\tau^{p(1/2-\delta/2)}$ and that the bounds (4.4a) hold for periodic boundary conditions; since the proof assumes rather than proves this transfer, failure of either bound would collapse the argument.

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Extended reading notes

Core claim

The central claim (Theorem 3.1) is that, under assumptions (S), (T), (C), (I), (W1), (W2), and (Z), for every $\delta>0$ there is a constant $C_\delta$ independent of $\tau$ and $h$ such that for all sufficiently small $\tau$, $$\max_{0\le M\le N}\mathbb{E}[\|u(t_M)-\varphi_h^M\|_{$L^{2}$(O)}^2]+\tau\sum_{n=1}^N\mathbb{E}[\|\nabla u(t_n)-\nabla\varphi_h^n\|_{$L^{2}$(O)}^2]\le C_\delta(\$tau^{{1-\delta}}$+$h^{2}$).$$ The estimate controls the maximal expected $L^2$ error over all time levels and the expected accumulated $H^1$ error, giving temporal order $1/2-\delta$ and spatial order $1$. The scheme is linear in the unknowns $(\varphi_h^n,r_h^n)$ and unconditionally stable with respect to the SAV modified energy, and the proof passes through a one-step error decomposition in which the only temporal loss comes from the auxiliary-variable mismatch at order $\tau^{1/2-\delta/2}$.

Load-bearing premise

The load-bearing premise is Lemma 4.2: the uniform discrete regularity bounds (4.4a) and the auxiliary-variable error bound (4.4b), which are imported from the earlier analysis for homogeneous Neumann boundary conditions and asserted, without proof, to remain valid for the periodic boundary conditions of this paper, together with the improved remainder estimate for the polynomial double-well potential.

Editorial extensions

If this is right

A linear, unconditionally stable scheme now provably attains the same strong convergence order as the nonlinear implicit scheme of [43] for the stochastic Allen–Cahn equation.
The error bound suggests the practical scaling $\tau\sim h^2$, which is exactly the scaling used in the paper's numerical experiments; at this scaling the observed temporal order is close to $1/2$ even on the coarsest meshes tested.
The auxiliary-variable error is controlled at rate $\tau^{1/2-\delta/2}$, the specific property that lets the SAV update survive the low Hölder regularity of stochastic solutions.
For multiplicative noise of the type considered here, the paper's numerical experiments indicate that the augmented SAV scheme reproduces the accuracy of the implicit scheme of [43] while requiring about half the computational time.

Reading between the lines

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

The same augmentation mechanism should transfer to other gradient-flow SPDEs whose solutions have Hölder regularity below $1/2$; in particular, this rate analysis provides a template for upgrading the convergence-along-sequences result for stochastic Cahn–Hilliard equations with dynamic boundary conditions to genuine strong rates.
A sharper bound in Lemma 4.2(4.4b) would plausibly remove the $\delta$-loss from the final temporal rate, since the paper's error decomposition isolates the auxiliary-variable mismatch as the source of the $\tau^{1-\delta}$ factor; this upgrade is not claimed in the paper.
A practical diagnostic suggested by the proof is to monitor $|r_h^n-\sqrt{E_h(\varphi_h^n)}|$ along sample paths: the theory predicts this quantity should exhibit the same experimental order of convergence as the phase-field error.

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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

Short version: this is a genuine result, not a rehash. Metzger proves strong convergence rate tau^{1-delta}+h^2 for the augmented SAV scheme for stochastic Allen-Cahn with multiplicative noise. The scheme was introduced in his earlier IMA paper [47], which only got convergence along subsequences; the rate theorem is new, and it matches the optimal rates Majee and Prohl proved for a nonlinear implicit scheme. The proof is a careful energy argument. The new part I can verify — the improved remainder estimate for polynomial double-well potentials — checks out. The numerical section, while short, gives EOCs consistent with the theory.

The soft spots are real but not fatal. Lemma 4.2 is load-bearing: the discrete regularity bounds in (4.4a) and the auxiliary-variable error (4.4b) are used at every absorption step in Section 5. Those bounds are not proved here; they're imported from [47], which was written for homogeneous Neumann data. The paper says the results 'carry over to the periodic case' without showing the discrete boundary terms behave. For periodic FE with mass lumping that is plausible, but I would want a referee to check it explicitly. If the transfer fails, the proof doesn't close.

Also worth flagging: the numerical experiments use rho(phi) = max(1-phi^2,0), which is not C^2 — it violates assumption (C) by a mile. That doesn't make the theorem wrong, but it means the numerics are outside the stated assumptions. Either replace rho in the test or relax (C). And there's no code or data shipped, so the rate table is not independently reproducible; that's minor but worth mentioning.

The citation pattern is fine: heavy self-citation, but the self-cited papers actually contain the imported estimates and the scheme, so it's not gratuitous.

Bottom line: central theorem is likely correct and is a useful step — linear, unconditionally stable, optimal rates for Monte Carlo. I'd send it to a serious referee, asking specifically for a careful check of Lemma 4.2 and the periodic-carry-over. The author should also fix the numerical/assumption mismatch. Not a takedown; it's a solid paper that needs minor to moderate tightening.

Referee Report

2 major / 4 minor

Summary. The paper analyzes a fully discrete linear finite element scheme based on an augmented scalar auxiliary variable (SAV) method for the stochastic Allen–Cahn equation with multiplicative noise on a periodic domain. The main result, Theorem 3.1, states that under assumptions (S), (T), (C), (I), (W1), (W2), and (Z), for every δ > 0 and all sufficiently small τ, max_M E[||u(t_M)-φ^M_h||²_{L²}] + τ Σ E[||∇u(t_n)-∇φ^n_h||²_{L²}] ≤ C_δ (τ^{1-δ}+h²). The proof combines continuous regularity (Lemma 4.1) and discrete regularity/auxiliary-variable estimates (Lemma 4.2) with an energy-error argument that estimates six remainder terms in Section 5. The numerical section reports experimental orders of convergence consistent with the theorem.

Significance. If the proof is fully substantiated, the result is significant: it shows that a linear, unconditionally stable augmented SAV scheme matches the optimal strong convergence rates τ^{1/2-δ}+h² previously established for nonlinear implicit schemes. The proof is a careful and detailed energy-error estimate with explicit treatment of the nonlinearity, the SAV error, the augmentation terms, and the stochastic term; the improved estimate for the polynomial double-well potential in Lemma 4.2 is a concrete contribution. Numerical experiments support the theory. The main caveat is that the load-bearing discrete regularity estimates are imported from a companion Neumann-boundary paper, so the periodic boundary-condition transfer needs to be justified.

major comments (2)

[Section 4, Lemma 4.2] Lemma 4.2 is the load-bearing pillar of Theorem 3.1: the uniform bounds (4.4a) and the auxiliary-variable bound (4.4b) are used in the estimates of R2 (5.7), R3 (5.15), R4 (5.16), R5 (5.17), and R6 (5.24), and they are essential for the absorption argument that produces (5.25). The manuscript does not prove these bounds; it states that the results established in [47] for homogeneous Neumann boundary conditions 'carry over to the periodic case' (Section 4, paragraph before Lemma 4.2). This transfer is not demonstrated. The periodic setting changes the function-space and boundary-term structure in the discrete Laplacian (2.5), the mass-lumped estimates (2.2), and the lag-difference bounds in (4.4a), so the carry-over is not a formality. Please include a proof of (4.4a)-(4.4b) for the periodic spaces, or at least a rigorous reduction to the Neumann results that specifies exactly which boundary terms vanish and why no new terms appear.
[Section 4, Eq. (4.4b)] The proof of (4.4b) is only completed for the term R2 in (4.8)-(4.9); for j∈{1,3,4,5,6} the paper refers to [47] for estimates of the form E[(Σ|Rj|)^p] ≤ Cτ^{p/2}. Because (4.4b) is applied in Section 5 with arbitrary p and δ (see (5.16)), the reader needs to know that these estimates hold under the current polynomial potential and with constants independent of the Neumann-to-periodic transfer. Please identify the exact lemmas in [47] used for each Rj or reproduce the short arguments.

minor comments (4)

[Section 4, Lemma 4.1] The statement 'for all s,t∈[0,t]' should read 'for all s,t∈[0,T]'.
[Section 5, Eq. (5.18)] After adding and subtracting P_U_h e_h^{n-1}, the first stochastic integral is a martingale difference with zero expectation; this should be stated explicitly, since the displayed inequality alone does not show how that term is removed.
[Section 5, Eq. (5.25)] The absorption step is terse; please state explicitly how the (1+6α)Σ_{n=1}^M τ E||e^n_h||² term is split between the e^M term and the Gronwall sum, so that the coefficient (1/4-(1+6α)τ) is transparent.
[Section 6, Table 1] The notation 'eh = 2^{-8}' appears to be a typo; it should probably read 'ĥ = 2^{-8}'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.1's rate is proved from stability and regularity bounds, not assumed or fitted; the only flagged issue is an asserted Neumann-to-periodic transfer of cited regularity results, which is a rigor risk but not circularity.

full rationale

The central estimate tau^{1-delta} + h^2 in Theorem 3.1 is not an input. Section 5 derives it from the error identity (5.1), Young/Hoelder/Itô estimates, and an absorption-plus-Gronwall argument yielding (5.25). The only externally imported ingredients are the continuous regularity facts in Lemma 4.1 and the discrete bounds in Lemma 4.2. The discrete bounds (4.4a) are cited to the author's earlier paper [47], and (4.4b) is proved here by a Taylor expansion using (4.4a). These are stability and regularity estimates for the scheme, not equivalent to the target rate: they do not contain the conclusion tau^{1-delta} + h^2, and the proof genuinely reduces the error to them rather than assuming the error itself. No parameter is fitted and then renamed a prediction; all constants are explicit and independent of tau and h. The Section 4 sentence "Although [47] dealt with the case of homogeneous Neumann boundary conditions, the established results carry over to the periodic case" is an asserted transfer rather than a demonstrated proof, which creates a correctness risk if the periodic finite-element setting changes the discrete Laplacian or mass-lumped estimates; however, a missing or risky dependency is not circular. The numerical section compares against an independent reference solution and reports an experimental order near 1/2, providing external falsifiable evidence. Verdict: no significant circularity.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

All assumptions are stated transparently in Section 2. The proof imports several stability and regularity results from the author's earlier papers [47] and [46]. There are no fitted parameters; the only hand-chosen constant is the positivity shift gamma > 0 used to keep F positive.

free parameters (1)

gamma (positivity shift in F) = arbitrary gamma > 0; gamma = 1e-5 in simulations
Introduced by hand to guarantee F > 0 and Eh(phi_h) >= gamma > 0, which the proof uses to control terms with 1/sqrt(Eh). It does not change the PDE dynamics and is not fitted to data.

assumptions (3)

standard math Finite element interpolation, inverse and Gagliardo-Nirenberg estimates for the quasi-uniform mesh family (S)
Used throughout Section 5, e.g. Lemma 2.1 and estimate (5.13), to convert spatial interpolation errors into powers of h.
domain assumption Assumptions (T), (S), (I), (C), (W1), (W2), (Z) on time grid, mesh, initial data, noise coefficient, Wiener process and mode truncation
These are the hypotheses of Theorem 3.1; they restrict the result to periodic domains, colored trace-class noise with smooth eigenfunctions, and finite mode truncations.
domain assumption Discrete regularity and stability bounds (4.4a)-(4.4b) from [47], carried over from Neumann to periodic boundary conditions
The proof of Lemma 4.2 asserts the carry-over without proof; Theorem 3.1 depends on these bounds for the Gronwall argument.

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Pith. "Pith review of Strong error estimates for a fully discrete SAV scheme for the stochastic Allen--Cahn equation with multiplicative noise." pith.science (2026). https://pith.science/paper/2ONIXQTG

@misc{pith2026250104618,
  author       = {Pith},
  title        = {Pith review of: Strong error estimates for a fully discrete SAV scheme for the stochastic Allen--Cahn equation with multiplicative noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2ONIXQTG}},
  note         = {Machine review of arXiv:2501.04618}
}

read the original abstract

We investigate the numerical approximation of the stochastic Allen--Cahn equation with multiplicative noise on a periodic domain. The considered scheme uses a recently proposed augmented variant of scalar auxiliary variable method for the discretization with respect to time. While scalar auxiliary variable methods in general allow for the construction of unconditionally stable, efficient linear schemes, the considered augmented version (cf. [S. Metzger, 2024, IMA J. Numer. Anal.]) additionally compensates for the typically poor temporal regularity of solutions to stochastic partial differential equations and hence extends the range of applicability of the scheme. In this work, we establish strong rates of convergence and show that the proposed linear scheme exhibits the same optimal rates of convergence that were established in [A. K. Majee & A. Prohl, 2018, Comput. Methods Appl. Math.] for a nonlinear structure preserving scheme. Finally, we provide numerical simulations verifying our theoretical findings.

Figures

Figures reproduced from arXiv: 2501.04618 by the authors.

**Figure 1.** Evolution of a droplet computed with τ = 1 · 10−5 and h = 2−8 . The left column depicts the expected value, while the remaining columns illustrate three individual sample paths. The orange line indicates the zero level set of the deterministic evolution. 7. Conclusion We analyzed an augmented SAV discretization for the stochastic Allen–Cahn equation with multiplicative noise. This scheme has the advantage of being l… view at source ↗

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Works this paper leans on

57 extracted references · 57 canonical work pages

[47]

, A convergent stochastic scalar auxiliary variable method, IMA J. Numer. Anal. (2024), 66 pages

work page 2024
[1]

Antonopoulou, L

D. Antonopoulou, L. Baňas, R. Nürnberg, and A. Prohl,Numerical approximation of the stochastic Cahn–Hilliard equation near the sharp interface limit, Numer. Math.147 (2021), no. 3, 505–551

work page 2021
[2]

Bauke and S

H. Bauke and S. Mertens, Random numbers for large-scale distributed Monte Carlo simulations, Phys. Rev. E75 (2007), 066701

work page 2007
[3]

Beccari, M

M. Beccari, M. Hutzenthaler, A. Jentzen, R. Kurniawan, F. Lindner, and D. Salimova,Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differ- ential equations with superlinearly growing nonlinearities, arxiv preprint (2019), 65 pages

work page 2019
[4]

Becker, B

S. Becker, B. Gess, A. Jentzen, and P. E. Kloeden, Strong convergence rates for explicit space- time discrete numerical approximations of stochastic Allen–Cahn equations, Stoch. Partial Dif- fer. Equ.: Anal. Comput.11 (2023), no. 1, 211–268

work page 2023
[5]

Becker and A

S. Becker and A. Jentzen,Strong convergence rates for nonlinearity-truncated Euler-type approxima- tions of stochastic Ginzburg–Landau equations, Stochastic Process. Appl.129 (2019), no. 1, 28–69

work page 2019
[6]

Blömker and A

D. Blömker and A. Jentzen, Galerkin approximations for the stochastic Burgers equation, SIAM J. Numer. Anal.51 (2013), no. 1, 694–715

work page 2013
[7]

Bouchriti, M

A. Bouchriti, M. Pierre, and N. E. Alaa,Remarks on the asymptotic behavior of scalar auxiliary variable (SAV) schemes for gradient-like flows, J. Appl. Anal. Comput.10 (2020), no. 5, 2198–2219

work page 2020

Show all 57 references

[8]

Bréhier, J

C.-E. Bréhier, J. Cui, and J. Hong,Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation, IMA J. Numer. Anal.39 (2019), no. 4, 2096–2134

2019
[9]

Bréhier and L

C.-E. Bréhier and L. Goudenége,Analysis of some splitting schemes for the stochastic Allen-Cahn equation, Discrete Contin. Dyn. Syst. B24 (2019), no. 8, 4169–4190

2019
[10]

Bréhier and L

C.-E. Bréhier and L. Goudenège,Weak convergence rates of splitting schemes for the stochastic Allen– Cahn equation, BIT Numerical Mathematics60 (2020), no. 3, 543–582

2020
[11]

Breit and A

D. Breit and A. Prohl,Weak error analysis for the stochastic Allen–Cahn equation, Stoch. Partial Dif- fer. Equ.: Anal. Comput.12 (2024), no. 4, 2181–2245

2024
[12]

S. C. Brenner and L. R. Scott,The Mathematical Theory of Finite Element Methods, Springer, New York, 2002

2002
[13]

M. Cai, S. Gan, and X. Wang,Weak Convergence Rates for an Explicit Full-Discretization of Sto- chastic Allen–Cahn Equation with Additive Noise, J. Sci. Comput.86 (2021), no. 3, 34

2021
[14]

Campillo-Funollet, G

E. Campillo-Funollet, G. Grün, and F. Klingbeil,On modeling and simulation of electrokinetic phe- nomena in two-phase flow with general mass densities, SIAM J. Appl. Math. 72 (2012), no. 6, 1899–1925. 20 S. METZGER

2012
[15]

Q.Cheng, C.Liu, andJ.Shen, A new Lagrange multiplier approach for gradient flows, Comput.Meth- ods Appl. Mech. Eng.367 (2020), 113070

2020
[16]

J. Cui, J. Hong, and L. Sun,Semi-implicit energy-preserving numerical schemes for stochastic wave equation via SAV approach, arxiv preprint, 24 pages (2022)

2022
[17]

de Bouard and A

A. de Bouard and A. Debussche,A semi-discrete scheme for the stochastic nonlinear Schrödinger equation, Numer. Math.96 (2004), no. 4, 733–770

2004
[18]

C. M. Elliott and A. M. Stuart,The Global Dynamics of Discrete Semilinear Parabolic Equations, SIAM J. Numer. Anal.30 (1993), no. 6, 1622–1663

1993
[19]

Ern and J.-L

A. Ern and J.-L. Guermond,Theory and Practice of Finite Elements, Springer Series in Applied Mathematical Sciences, vol. 159, Springer, New York, US-NY, 2004

2004
[20]

D. J. Eyre,Unconditionally gradient stable time marching the Cahn–Hilliard equation, MRS Online Proceedings Library 529 (1998), 39–46

1998
[21]

Feireisl and A

E. Feireisl and A. Novotný,Singular Limits in Thermodynamics of Viscous Fluids, Springer-Verlag GmbH, 2017

2017
[22]

X. Feng, Y. Li, and Y. Zhang,Finite Element Methods for the Stochastic Allen–Cahn Equation with Gradient-type Multiplicative Noise, SIAM J. Numer. Anal.55 (2017), no. 1, 194–216

2017
[23]

Grün, F Guillén-González, and S

G. Grün, F Guillén-González, and S. Metzger,On fully decoupled, convergent schemes for diffuse interface models for two-phase flow with general mass densities, Commun. Comput. Phys.19 (2016), no. 5, 1473–1502

2016
[24]

Grün and F

G. Grün and F. Klingbeil,Two-phase flow with mass density contrast: stable schemes for a ther- modynamic consistent and frame-indifferent diffuse-interface model, J. Comput. Phys.257, Part A (2014), 708–725

2014
[25]

Gyöngy and A

I. Gyöngy and A. Millet,Rate of Convergence of Space Time Approximations for Stochastic Evolution Equations, Potential Anal.30 (2009), no. 1, 29–64

2009
[26]

Gyöngy, S

I. Gyöngy, S. Sabanis, and D. Šiška,Convergence of tamed Euler schemes for a class of stochastic evolution equations, Stoch. Partial Differ. Equ.: Anal. Comput.4 (2016), no. 2, 225–245

2016
[27]

Hausenblas, Numerical analysis of semilinear stochastic evolution equations in Banach spaces, J

E. Hausenblas, Numerical analysis of semilinear stochastic evolution equations in Banach spaces, J. Comput. Appl. Math.147 (2002), no. 2, 485–516

2002
[28]

18 (2003), no

, Approximation for Semilinear Stochastic Evolution Equations, Potential Anal. 18 (2003), no. 2, 141–186

2003
[29]

D. Hou, M. Azaiez, and C. Xu,A variant of scalar auxiliary variable approaches for gradient flows, J. Comput. Phys.395 (2019), 307–332

2019
[30]

Huang and J

C. Huang and J. Shen,Stability and convergence analysis of a fully discrete semi-implicit scheme for stochastic Allen–Cahn equations with multiplicative noise, Math. Comp. 92 (2023), no. 344, 2685– 2713

2023
[31]

Hutzenthaler and A

M. Hutzenthaler and A. Jentzen,Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc.236 (2015), no. 1112

2015
[32]

Probab.48 (2020), no

, On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients, Ann. Probab.48 (2020), no. 1, 53–93

2020
[33]

Hutzenthaler, A

M. Hutzenthaler, A. Jentzen, and P. E. Kloeden,Strong and weak divergence in finite time of Eu- ler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.467 (2011), no. 2130, 1563–1576

2011
[34]

Jentzen, P

A. Jentzen, P. Kloeden, and G. Winkel, Efficient simulation of nonlinear parabolic SPDEs with additive noise, Ann. Appl. Probab.21 (2011), no. 3, 908–950

2011
[35]

Jentzen and P

A. Jentzen and P. Pušnik,Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities, IMA J. Nu- mer. Anal. 40 (2020), no. 2, 1005–1050

2020
[36]

Kovács, S

M. Kovács, S. Larsson, and F. Lindgren,On the backward Euler approximation of the stochastic Allen–Cahn equation, J. Appl. Probab.52 (2015), no. 2, 323–338

2015
[37]

, On the discretisation in time of the stochastic Allen–Cahn equation, Math. Nachr. 291 (2018), no. 5–6, 966–995

2018
[38]

Kruse and R

R. Kruse and R. Weiske,The BDF2-Maruyama method for the stochastic Allen–Cahn equation with multiplicative noise, J. Comput. Appl. Math.419 (2023), 114634. Error estimates for a fully discrete SA V scheme 21

2023
[39]

K. F. Lam and R. Wang,Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn–Hilliard systems with bounded mass source, J. Numer. Math.32 (2024), no. 3, 233–255

2024
[40]

Liu and X

Z. Liu and X. Li, The Exponential Scalar Auxiliary Variable (E-SAV) Approach for Phase Field Models and Its Explicit Computing, SIAM J. Sci. Comput.42 (2020), no. 3, B630–B655

2020
[41]

Liu and Z

Z. Liu and Z. Qiao,Strong approximation of monotone stochastic partial differential equations driven by white noise, IMA J. Numer. Anal.40 (2019), no. 2, 1074–1093

2019
[42]

Partial Differ

, Strong approximation of monotone stochastic partial differential equations driven by multi- plicative noise, Stoch. Partial Differ. Equ.: Anal. Comput.9 (2021), no. 3, 559–602

2021
[43]

A. K. Majee and A. Prohl,Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen–Cahn Equation with Multiplicative Noise, Comput. Methods Appl. Math.18 (2018), no. 2, 297–311

2018
[44]

Metzger,A convergent finite element scheme for a fourth-order liquid crystal model, IMA J

S. Metzger,A convergent finite element scheme for a fourth-order liquid crystal model, IMA J. Nu- mer. Anal. 42 (2022), no. 1, 440–486, published online 2020

2022
[45]

, A convergent SAV scheme for Cahn–Hilliard equations with dynamic boundary conditions, IMA J. Numer. Anal.43 (2023), no. 6, 3593–3627, drac078

2023
[46]

, A convergent augmented SAV scheme for stochastic Cahn–Hilliard equations with dynamic boundary conditions describing contact line tension, arxiv preprint (accepted in Interfaces and Free Boundaries) (2024), 60 pages

2024
[48]

Qi and X

R. Qi and X. Wang,Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Allen–Cahn Equation with Additive Noise, J. Sci. Computing80 (2019), no. 2, 1171–1194

2019
[49]

X. Qi, M. Azaiez, C. Huang, and C. Xu,An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise, AIMS Math.7 (2022), no. 12, 20684–20710

2022
[50]

Shen and J

J. Shen and J. Xu,Convergence and Error Analysis for the Scalar Auxiliary Variable (SAV) Schemes to Gradient Flows, SIAM J. Numer. Anal.56 (2018), no. 5, 2895–2912

2018
[51]

J. Shen, J. Xu, and J. Yang,The scalar auxiliary variable (SAV) approach for gradient flows, J. Com- put. Phys. 353 (2018), 407–416

2018
[52]

O. Sieber, Analysis and Numerics of Two-Phase Flows of Active Liquid Crystals with Willmore- type Interfacial Energy: A Micro-Macro Approach, doctoral thesis, Friedrich–Alexander–Universität Erlangen–Nürnberg, Erlangen, 2021, p. 255

2021
[53]

Thomée,Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol

V. Thomée,Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer, Berlin, Heidelberg, DE, 2006

2006
[54]

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

X. Wang,An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation, Stochastic Process. Appl.130 (2020), no. 10, 6271–6299

2020
[55]

Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J

X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys.327 (2016), 294–316

2016
[56]

Yang and H

X. Yang and H. Yu,Efficient Second Order Unconditionally Stable Schemes for a Phase Field Moving Contact Line Model Using an Invariant Energy Quadratization Approach, SIAM J. Sci. Comput.40 (2018), no. 3, B889–B914

2018
[57]

Yang and G.-D

X. Yang and G.-D. Zhang, Convergence Analysis for the Invariant Energy Quadratization (IEQ) Schemes for Solving the Cahn–Hilliard and Allen–Cahn Equations with General Nonlinear Potential, J. Sci. Comput.82 (2020), no. 3, 55

2020

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [47]

, A convergent stochastic scalar auxiliary variable method, IMA J. Numer. Anal. (2024), 66 pages

work page 2024

[2] [1]

Antonopoulou, L

D. Antonopoulou, L. Baňas, R. Nürnberg, and A. Prohl,Numerical approximation of the stochastic Cahn–Hilliard equation near the sharp interface limit, Numer. Math.147 (2021), no. 3, 505–551

work page 2021

[3] [2]

Bauke and S

H. Bauke and S. Mertens, Random numbers for large-scale distributed Monte Carlo simulations, Phys. Rev. E75 (2007), 066701

work page 2007

[4] [3]

Beccari, M

M. Beccari, M. Hutzenthaler, A. Jentzen, R. Kurniawan, F. Lindner, and D. Salimova,Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differ- ential equations with superlinearly growing nonlinearities, arxiv preprint (2019), 65 pages

work page 2019

[5] [4]

Becker, B

S. Becker, B. Gess, A. Jentzen, and P. E. Kloeden, Strong convergence rates for explicit space- time discrete numerical approximations of stochastic Allen–Cahn equations, Stoch. Partial Dif- fer. Equ.: Anal. Comput.11 (2023), no. 1, 211–268

work page 2023

[6] [5]

Becker and A

S. Becker and A. Jentzen,Strong convergence rates for nonlinearity-truncated Euler-type approxima- tions of stochastic Ginzburg–Landau equations, Stochastic Process. Appl.129 (2019), no. 1, 28–69

work page 2019

[7] [6]

Blömker and A

D. Blömker and A. Jentzen, Galerkin approximations for the stochastic Burgers equation, SIAM J. Numer. Anal.51 (2013), no. 1, 694–715

work page 2013

[8] [7]

Bouchriti, M

A. Bouchriti, M. Pierre, and N. E. Alaa,Remarks on the asymptotic behavior of scalar auxiliary variable (SAV) schemes for gradient-like flows, J. Appl. Anal. Comput.10 (2020), no. 5, 2198–2219

work page 2020