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arxiv: 1907.02234 · v1 · pith:GBMC5ZM6new · submitted 2019-07-04 · 🧮 math.NA · cs.NA

A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection

Wenbin Chen , Weijia Li , Zhiwen Luo , Cheng Wang , Xiaoming Wang This is my paper

Pith reviewed 2026-05-25 09:27 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords thin film growthexponential time differencingenergy stabilitymultistep methodFourier collocationslope selectionerror analysis
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The pith

A stabilized ETD multistep scheme achieves second-order accuracy and long-time energy stability for the thin film growth model.

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

The paper develops a second-order linear exponential time differencing multistep scheme for the thin film growth equation without slope selection. An artificial term A τ² ∂Δ²u/∂t is added to the PDE to enable unconditional energy stability while the time integral is treated by ETD multistep formulas and space is discretized by Fourier collocation. Energy-method analysis then yields long-time stability of the discrete energy together with an ℓ^∞(0,T; ℓ²) error bound that carefully controls the aliasing contribution. Numerical experiments confirm monotonic energy decay and the predicted convergence rate.

Core claim

The authors present a second-order linear ETD multistep method stabilized by the term A τ² ∂Δ²u/∂t, which permits an energy stability proof and an ℓ^∞(0,T; ℓ²) error analysis for the no-slope-selection thin film equation, using Fourier spectral discretization in space.

What carries the argument

The artificial stabilizing term A τ² ∂Δ²u/∂t added to the model, combined with the ETD multistep time integration, which enables unconditional energy stability via the energy method.

If this is right

  • The discrete energy decreases monotonically for any time step size.
  • The approximation error remains bounded by a constant times τ² plus spatial discretization error over any time interval.
  • The scheme can be implemented efficiently using FFT due to the spectral method.
  • Energy decay is observed in simulations consistent with the continuous model.

Where Pith is reading between the lines

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

  • This stabilization technique may apply to other fourth-order nonlinear evolution equations.
  • Further analysis could examine how the parameter A affects the accuracy for fixed time steps.
  • Comparison with other implicit-explicit methods for the same model would test relative efficiency.

Load-bearing premise

Adding the artificial stabilizing term does not invalidate the energy stability or approximation properties of the original thin film model.

What would settle it

A simulation where the numerical energy increases for sufficiently small time steps or where the observed temporal convergence order is less than two would contradict the claims.

Figures

Figures reproduced from arXiv: 1907.02234 by Cheng Wang, Weijia Li, Wenbin Chen, Xiaoming Wang, Zhiwen Luo.

Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
read the original abstract

In this paper, a stabilized second order in time accurate linear exponential time differencing (ETD) scheme for the no-slope-selection thin film growth model is presented. An artificial stabilizing term $A\tau^2\frac{\partial\Delta^2 u}{\partial t}$ is added to the physical model to achieve energy stability, with ETD-based multi-step approximations and Fourier collocation spectral method applied in the time integral and spatial discretization of the evolution equation, respectively. Long time energy stability and detailed $\ell^{\infty}(0,T; \ell^2)$ error analysis are provided based on the energy method, with a careful estimate of the aliasing error. In addition, numerical experiments are presented to demonstrate the energy decay and convergence rate.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a second-order linear ETD multistep scheme for the no-slope-selection thin-film equation. An artificial term A τ² ∂_t Δ² u is added to the PDE to enable unconditional energy stability. The time integrals are discretized via ETD multistep formulas and space via Fourier collocation. Long-time energy stability is proved by the energy method; an ℓ^∞(0,T; ℓ²) error bound is derived that includes a careful aliasing-error estimate. Numerical tests confirm energy decay and second-order convergence.

Significance. If the perturbation induced by the artificial term can be controlled uniformly in time, the paper supplies a rare combination of unconditional energy stability, rigorous ℓ^∞ error analysis, and aliasing control for a fourth-order nonlinear PDE. The explicit parameter A and the multistep ETD construction are technically positive features.

major comments (2)
  1. [error analysis section / abstract] The error analysis (abstract and §4) is performed on the augmented PDE that contains the term A τ² ∂_t Δ² u. No estimate is given for the difference between solutions of the original thin-film equation and the modified equation. Without a uniform-in-time bound on this perturbation (of size O(τ²) or better), the claimed ℓ^∞(0,T; ℓ²) error bound does not transfer to the physical model.
  2. [stability analysis] The stability proof treats A as a fixed positive constant chosen large enough for the discrete energy to be nonincreasing. The dependence of the error constant on A is not tracked; if A must grow with 1/τ to maintain stability, the second-order accuracy claim is compromised.
minor comments (2)
  1. Notation for the multistep coefficients and the precise definition of the ETD operators should be collected in a single preliminary section for readability.
  2. The numerical experiments section should include a table or plot that isolates the effect of A on both stability and observed convergence rate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We appreciate the referee's insightful comments. We address each major comment below and will revise the manuscript to incorporate the necessary additions and clarifications.

read point-by-point responses

  1. Referee: [error analysis section / abstract] The error analysis (abstract and §4) is performed on the augmented PDE that contains the term A τ² ∂_t Δ² u. No estimate is given for the difference between solutions of the original thin-film equation and the modified equation. Without a uniform-in-time bound on this perturbation (of size O(τ²) or better), the claimed ℓ^∞(0,T; ℓ²) error bound does not transfer to the physical model.

    Authors: We agree that the error analysis is performed directly on the modified PDE. In the revised manuscript we will add a lemma establishing a uniform-in-time bound on the difference between solutions of the original and augmented equations. Under the regularity assumptions already used in §4, this difference is O(τ²) uniformly for t ∈ [0,T], so the ℓ^∞ error bound transfers to the physical model with an additional term of the same order that preserves second-order accuracy. revision: yes

  2. Referee: [stability analysis] The stability proof treats A as a fixed positive constant chosen large enough for the discrete energy to be nonincreasing. The dependence of the error constant on A is not tracked; if A must grow with 1/τ to maintain stability, the second-order accuracy claim is compromised.

    Authors: A is chosen as a fixed constant independent of τ (depending only on the domain, initial data, and a priori solution bounds). The stability condition does not require A to grow with 1/τ. While the current text does not explicitly display the polynomial dependence of the error constants on A, this dependence can be verified from the estimates in §4 and does not affect the convergence order in τ. We will add a remark in the revised version that explicitly states the A-dependence and confirms its independence from τ. revision: partial

Circularity Check

0 steps flagged

No circularity: explicit stabilization term and energy-method proof are independent of fitted inputs or self-citation chains

full rationale

The paper explicitly introduces the artificial term Aτ² ∂Δ²u/∂t to the PDE, then derives energy stability and ℓ^∞(0,T;ℓ²) error bounds for the resulting modified equation via the energy method plus aliasing estimates. This construction does not reduce any claimed result to a fitted parameter renamed as prediction, nor does it rely on a self-citation whose content is itself unverified; the stabilization parameter is stated openly rather than hidden inside a fit. The derivation chain therefore remains self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the addition of an artificial term whose coefficient A is not derived from the physics and on the assumption that the modified equation still permits a clean energy-method proof.

free parameters (1)
  • A
    Coefficient of the artificial stabilizing term A τ² ∂Δ²u/∂t; introduced by hand to obtain unconditional energy stability.
axioms (1)
  • domain assumption The underlying thin-film equation without slope selection is the correct physical model.
    The PDE is taken as given; the stabilization is added on top of it.
invented entities (1)
  • artificial stabilizing term A τ² ∂Δ²u/∂t no independent evidence
    purpose: To enforce discrete energy decay for the numerical scheme.
    New term introduced solely to make the time-stepping method stable; no independent physical justification or falsifiable prediction is supplied.

pith-pipeline@v0.9.0 · 5667 in / 1494 out tokens · 25123 ms · 2026-05-25T09:27:58.181252+00:00 · methodology

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages · 1 internal anchor

  1. [1]

    R. A. Adams and J. J. F. Fournier, Sobolev spaces, 2nd ed., Academic press, Singapore, 2003

    work page 2003

  2. [2]

    Agmon, Lectures on elliptic boundary value problems, vol

    S. Agmon, Lectures on elliptic boundary value problems, vol. 369, American Mathematical Soc., 2010

    work page 2010

  3. [3]

    Benesova, C

    B. Benesova, C. Melcher, and E. Suli, An implicit midpoint spectral approximation of nonlocal Cahn-Hilliard equations, SIAM J. Numer. Anal., 52 (2014), 1466–1496

    work page 2014

  4. [4]

    Beylkin, J

    G. Beylkin, J. M. Keiser, and L. Vozovoi, A new class of time discretization schemes for the solution of nonlinear PDEs , J. Comput. Phys., 147 (1998), 362–387

    work page 1998

  5. [5]

    Canuto, M

    C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods: fundamentals in single domains , Springer, 2006

    work page 2006

  6. [6]

    Canuto, M

    C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods: evolution to complex geometries and applications to fluid dynamics, Springer Science & Business Media, 2007

    work page 2007

  7. [7]

    Canuto and A

    C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in sobolev spaces, Math. Comput., 38 (1982), no. 157, 67–86

    work page 1982

  8. [8]

    W. Chen, S. Conde, C. Wang, X. Wang, and S. M. Wise, A linear energy stable scheme for a thin film model without slope selection , J. Sci. Comput., 52 (2012), no. 3, 546–562

    work page 2012

  9. [9]

    W. Chen, C. Wang, X. Wang, and S. M. Wise, A linear iteration algorithm for a second-order 24 W. CHEN, W. LI, Z. LUO, C. WANG AND X. WANG energy stable scheme for a thin film model without slope selection , J. Sci. Comput., 59 (2014), no. 3, 574–601

    work page 2014

  10. [10]

    Chen and Y

    W. Chen and Y. Wang, A mixed finite element method for thin film epitaxy , Numer. Math., 122 (2012), no. 4, 771–793

    work page 2012

  11. [11]

    Cheng, Z

    K. Cheng, Z. Qiao, and C. Wang, A third order exponential time differencing numeri- cal scheme for no-slope-selection epitaxial thin film model with energy stability , J. Sci. Com- put., submitted and in review, 2018

    work page 2018

  12. [12]

    S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems , J. Comput. Phys., 176 (2002), 430–455

    work page 2002

  13. [13]

    Ehrlich and F

    G. Ehrlich and F. G. Hudda, Atomic view of surface self-diffusion: Tungsten on tungsten , J. Chem. Phys., 44 (1966), no. 3, 1039–1049

    work page 1966

  14. [14]

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

    work page 1998

  15. [15]

    W. Feng, C. Wang, S. M. Wise, and Z. Zhang, A second-order energy stable backward differ- entiation formula method for the epitaxial thin film equation with slope selection , arXiv preprint, arXiv:1706.01943 (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  16. [16]

    Golubovi´ c,Interfacial coarsening in epitaxial growth models without slope selection , Phys

    L. Golubovi´ c,Interfacial coarsening in epitaxial growth models without slope selection , Phys. Rev. Lett., 78 (1997), 90–93

    work page 1997

  17. [17]

    Gottlieb and S

    D. Gottlieb and S. A. Orszag, Numerical analysis of spectral methods: theory and applications , vol. 26, SIAM, 1977

    work page 1977

  18. [18]

    Gottlieb and C

    S. Gottlieb and C. Wang, Stability and convergence analysis of fully discrete fourier collocation spectral method for 3-D viscous burgers’ equation , J. Sci. Comput., 53 (2012), no. 1, 102– 128

    work page 2012

  19. [19]

    Hochbruck and A

    M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209–286

    work page 2010

  20. [20]

    Hochbruck and A

    M. Hochbruck and A. Ostermann, Exponential multistep methods of Adams-type , BIT Numer. Math., 51 (2011), 889–908

    work page 2011

  21. [21]

    L. Ju, X. Li, Z. Qiao, and H. Zhang, Energy stability and convergence of exponential time differencing schemes for the epitaxial growth model without slope selection, Math. Comput., 87 (2018), no. 312, 1859–1885

    work page 2018

  22. [22]

    L. Ju, X. Liu, and W. Leng, Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations, Discrete Cont. Dyn-B., 19 (2014), 1667–1687

    work page 2014

  23. [23]

    L. Ju, J. Zhang, and Q. Du, Fast and accurate algorithms for simulating coarsening dynamics of Cahn-Hilliard equations , Comp. Mater. Sci., 108 (2015), 272–282

    work page 2015

  24. [24]

    L. Ju, J. Zhang, L. Zhu, and Q. Du, Fast explicit integration factor methods for semilinear parabolic equations, J. Sci. Comput., 62 (2015), 431–455

    work page 2015

  25. [25]

    V Kohn and X

    R. V Kohn and X. Yan, Upper bound on the coarsening rate for an epitaxial growth model , Commun. Pur. Appl. Math., 56 (2003), no. 11, 1549–1564

    work page 2003

  26. [26]

    Li, High-order surface relaxation versus the Ehrlich-Schwoebel effect , Nonlinearity, 19 (2006), no

    B. Li, High-order surface relaxation versus the Ehrlich-Schwoebel effect , Nonlinearity, 19 (2006), no. 11, 2581–2603

    work page 2006

  27. [27]

    Li and J

    B. Li and J. Liu, Thin film epitaxy with or without slope selection , Eur. J. Appl. Math., 14 (2003), no. 6, 713–743

    work page 2003

  28. [28]

    Li and J

    B. Li and J. Liu, Epitaxial growth without slope selection: energetics, coarsening, and dynamic scaling, J. Nonlinear Sci., 14 (2004), no. 5, 429–451

    work page 2004

  29. [29]

    Li and Z

    D. Li and Z. Qiao, On the stabilization size of semi-implicit Fourier-spectral methods for 3D Cahn-Hilliard equations, Commun. Math. Sci., 15 (2017), 1489–1506

    work page 2017

  30. [30]

    D. Li, Z. Qiao, and T. Tang, Characterizing the stabilization size for semi-implicit Fourier- spectral method to phase field equations , SIAM J. Numer. Anal., 54 (2016), no. 3, 1653– 1681

    work page 2016

  31. [31]

    W. Li, W. Chen, C. Wang, Y. Yan, and R. He, A second order energy stable linear scheme for a thin film model without slope selection , J. Sci. Comput., 76 (2018), no. 3, 1905–1937

    work page 2018

  32. [32]

    X. Li, Z. Qiao, and H. Zhang, Convergence of a fast explicit operator splitting method for the epitaxial growth model with slope selection , SIAM J. Numer. Anal., 55 (2017), no. 1, 265–285

    work page 2017

  33. [33]

    Moldovan and L

    D. Moldovan and L. Golubovic, Interfacial coarsening dynamics in epitaxial growth with slope selection, Phys. Rev. E, 61 (2000), no. 6, 6190-6214

    work page 2000

  34. [34]

    Nirenberg, On elliptic partial differential equations , IL Principio Di Minimo E Sue Appli- cazioni Alle Equazioni Funzionali, 13 (1959), no

    L. Nirenberg, On elliptic partial differential equations , IL Principio Di Minimo E Sue Appli- cazioni Alle Equazioni Funzionali, 13 (1959), no. 1, 1–48

    work page 1959

  35. [35]

    Z. Qiao, T. Tang, and H. Xie, Error analysis of a mixed finite element method for the molecular beam epitaxy model, SIAM J. Numer. Anal., 53 (2015), no. 1, 184–205

    work page 2015

  36. [36]

    Z. Qiao, C. Wang, S. M. Wise, and Z. Zhang, Error analysis of a finite difference scheme for the epitaxial thin film model with slope selection with an improved convergence constant , Inter. J. Numer. Anal. Mod., 14 (2017), 283–305. sETDMs2 for no-slope-selection thin film model 25

    work page 2017

  37. [37]

    R. L. Schwoebel, Step motion on crystal surfaces. II , J. Appl. Phys., 40 (1969), no. 2, 614–618

    work page 1969

  38. [38]

    J. Shen, T. Tang, and L. Wang, Spectral methods: algorithms, analysis and applications, vol. 41, Springer Science & Business Media, 2011

    work page 2011

  39. [39]

    J. Shen, C. Wang, X. Wang, and S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich–Schwoebel type energy: application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), no. 1, 105–125

    work page 2012

  40. [40]

    J. Shen, J. Xu, and J. Yang, The scalar auxiliary variable (sav) approach for gradient flows , J. Comput. Phys., 353 (2018), 407–416

    work page 2018

  41. [41]

    C. Wang, X. Wang, and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst., 28 (2010), no. 1, 405–423

    work page 2010

  42. [42]

    X. Wang, L. Ju, and Q. Du, Efficient and stable exponential time differencing Runge-Kutta methods for phase field elastic bending energy models , J. Comput. Phys., 316 (2016), 21– 38

    work page 2016

  43. [43]

    Xu and T

    C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2006), no. 4, 1759–1779

    work page 2006

  44. [44]

    X. Yang, J. Zhao, and Q. Wang, Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method, J. Comput. Phys., 333 (2017), 104–127

    work page 2017

  45. [45]

    L. Zhu, L. Ju, and W. Zhao, Fast high-order compact exponential time differencing Runge- Kutta methods for second-order semilinear parabolic equations, J. Sci. Comput., 67 (2016), 1043–1065

    work page 2016