Stability and Error estimates of the SAV Fourier-spectral method for the Phase Field Crystal Equation

Jie Shen; Xiaoli Li

arxiv: 1907.07462 · v1 · pith:UGMWT5KXnew · submitted 2019-07-17 · 🧮 math.AP

Stability and Error estimates of the SAV Fourier-spectral method for the Phase Field Crystal Equation

Xiaoli Li , Jie Shen This is my paper

Pith reviewed 2026-05-24 20:37 UTC · model grok-4.3

classification 🧮 math.AP

keywords phase field crystal equationscalar auxiliary variableFourier spectral methodenergy stabilityerror estimatesunconditional stabilityfully discrete schemes

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The pith

Second-order SAV Fourier-spectral schemes for the phase field crystal equation are unconditionally energy stable and converge at rate O(Δt² + N^{-m}).

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

The paper develops fully discrete schemes for the phase field crystal equation that combine the scalar auxiliary variable approach in time with Fourier spectral discretization in space. It proves unconditional energy stability for both first- and second-order versions, so the discrete energy decreases at every step without any limit on the size of the time step. It also derives a rigorous error bound showing that the second-order scheme converges at rate O(Δt² + N^{-m}), where the second term reflects how well the Fourier series approximates the solution when the solution is sufficiently smooth. A reader would care because these guarantees allow long-time simulations of crystal formation to run efficiently and without artificial numerical damping from stability constraints.

Core claim

The authors establish that both first- and second-order SAV and stabilized SAV schemes with Fourier-spectral spatial discretization for the phase field crystal equation satisfy unconditional energy stability. They further derive an error estimate showing that the second-order scheme converges to the exact solution at the rate O(Δt² + N^{-m}), where Δt is the time step, N is the number of Fourier modes per direction, and m is the spatial regularity index of the exact solution.

What carries the argument

The scalar auxiliary variable (SAV) reformulation, which introduces an auxiliary scalar variable to rewrite the nonlinear energy dissipation law so that the time discretization remains linear and energy stable.

If this is right

Simulations can use arbitrarily large time steps while still obeying a discrete energy law that mirrors the continuous equation.
The computed solutions remain physically consistent because energy cannot increase at any step.
Accuracy improves predictably when more Fourier modes are added, provided the solution smoothness is known.
Stabilized SAV variants inherit the same unconditional stability property as the basic SAV schemes.

Where Pith is reading between the lines

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

The same SAV-plus-Fourier framework could be applied to other phase-field models that share an energy-dissipation structure.
The proven bounds suggest that adaptive time-step control is unnecessary for stability reasons alone.
In long-time materials simulations the method could reduce overall cost by allowing larger steps without loss of the energy property.

Load-bearing premise

The exact solution of the phase field crystal equation has enough spatial regularity for the Fourier truncation error to decay as N to the power of minus m.

What would settle it

A numerical experiment on a manufactured solution with known exact regularity where the measured error fails to decrease quadratically in Δt or fails to follow the predicted power of N.

read the original abstract

We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order $O(\Delta t^2+N^{-m})$, where $\Delta t$, $N$ and $m$ are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes.

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

SAV plus Fourier spectral on the PFC equation yields unconditional stability and an error bound O(Δt² + N^{-m}), but the bound rests on an unverified high-regularity assumption.

read the letter

This paper applies the scalar auxiliary variable approach to the phase field crystal equation, pairs it with Fourier spectral discretization in space, and proves unconditional energy stability for both first- and second-order time schemes. It also supplies an error analysis that delivers the rate O(Δt² + N^{-m}) under a Sobolev regularity assumption on the exact solution. The specific combination and the accompanying error statement for this model are not in the earlier SAV literature cited in the abstract, so that part is new. The stability proofs follow the usual SAV energy argument and look standard but correctly adapted. The numerical experiments are included to check the predicted rates, which is the right thing to do for a methods paper. The main limitation is the regularity hypothesis needed for the spatial error term. The PFC equation with its cubic nonlinearity yields only H¹-type control from the basic energy, and higher norms are not automatically bounded uniformly in time. The paper states the convergence result conditional on ||u||_{H^m} remaining bounded, yet supplies no additional a-priori estimates or smoothing argument to justify that the assumption holds for typical initial data. If the solution only lives in lower Sobolev spaces, the Fourier truncation error does not deliver the claimed rate. This is a genuine but contained weakness rather than a fatal flaw; the stability claim stands independently. The work is aimed at people who need provably stable schemes for phase-field crystal simulations and who are willing to accept the regularity caveat. It is the sort of solid, incremental numerical analysis paper that belongs in a specialized journal and deserves a serious referee.

Referee Report

1 major / 0 minor

Summary. The paper develops first- and second-order SAV (and stabilized SAV) time discretizations combined with Fourier spectral spatial discretization for the phase field crystal equation. It proves unconditional energy stability for the fully discrete schemes and derives an error bound of O(Δt² + N^{-m}) for the second-order scheme, assuming the exact solution possesses uniform H^m regularity. Numerical experiments are included to illustrate the theoretical rates and robustness.

Significance. If the stability proofs and error analysis hold under the stated assumptions, the work supplies a practical, provably stable method with explicit convergence rates for a fourth-order nonlinear PDE arising in materials science. The unconditional stability result is a concrete strength for long-time integration.

major comments (1)

[§4 (error estimate theorem)] The error estimate (abstract and the theorem in §4) states convergence of order O(Δt² + N^{-m}) under the assumption that the exact solution satisfies ||u(t)||_{H^m(Ω)} ≤ C uniformly in time. Standard energy estimates for the PFC equation with f(u) = u³ - u yield only H¹ control; the manuscript supplies no additional a-priori bounds or smoothing arguments to close the higher-norm estimate. This assumption is load-bearing for the spatial truncation term and must be justified or the theorem restated with the minimal regularity actually available from typical initial data.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the error analysis. We address the point below.

read point-by-point responses

Referee: [§4 (error estimate theorem)] The error estimate (abstract and the theorem in §4) states convergence of order O(Δt² + N^{-m}) under the assumption that the exact solution satisfies ||u(t)||_{H^m(Ω)} ≤ C uniformly in time. Standard energy estimates for the PFC equation with f(u) = u³ - u yield only H¹ control; the manuscript supplies no additional a-priori bounds or smoothing arguments to close the higher-norm estimate. This assumption is load-bearing for the spatial truncation term and must be justified or the theorem restated with the minimal regularity actually available from typical initial data.

Authors: We agree that the uniform H^m regularity assumption is essential to obtain the stated spatial truncation bound for the Fourier spectral method and is not derived from the basic energy estimates available for the PFC equation. The theorem is stated under this assumption, which is standard when seeking high-order spatial accuracy with spectral discretizations. We will revise the manuscript to make the assumption more prominent both in the theorem statement and in the abstract, and we will add a short remark acknowledging that a rigorous a-priori H^m bound lies beyond the scope of the present work (which centers on the stability and convergence analysis of the schemes). The unconditional energy stability results remain valid without this higher-regularity assumption. revision: yes

Circularity Check

0 steps flagged

No circularity: standard conditional error analysis from scheme definitions

full rationale

The paper defines SAV-based time discretizations and Fourier-spectral spatial discretization, then proves unconditional energy stability via direct energy estimates on the discrete scheme. The error analysis in §4 proceeds by standard consistency + stability arguments that bound the truncation error under an explicit a-priori assumption that the exact solution lies in H^m; this assumption is stated up-front and is not derived from the numerical method. No parameter is fitted to data and then relabeled a prediction, no self-citation supplies a uniqueness theorem that forces the result, and the claimed rate O(Δt² + N^{-m}) is the direct output of the approximation properties of the Fourier projector plus the second-order time truncation, not a tautology. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard functional-analysis assumptions (existence of a sufficiently regular solution to the continuous PFC equation) and on the algebraic properties of the SAV reformulation; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption The exact solution of the PFC equation belongs to a Sobolev space of index m sufficient for the Fourier truncation error to be O(N^{-m}).
Invoked to obtain the spatial contribution to the error bound stated in the abstract.

pith-pipeline@v0.9.0 · 5654 in / 1391 out tokens · 18910 ms · 2026-05-24T20:37:01.608535+00:00 · methodology

discussion (0)

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Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

[1]

Ainsworth and Z

M. Ainsworth and Z. Mao , Analysis and approximation of a fractional Cahn-Hilliard equation , SIAM Journal on Numerical Analysis, 55 (2017), pp. 1689–1718

work page 2017
[2]

Elder and M

K. Elder and M. Grant , Modeling elastic and plastic deformations in nonequilibrium processing using phase ﬁeld crystals, Physical Review E, 70 (2004), p. 051605

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Elder, M

K. Elder, M. Katakowski, M. Haataja, and M. Grant , Modeling elasticity in crystal growth , Physical review letters, 88 (2002), p. 245701

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Gomez and X

H. Gomez and X. Nogueira , An unconditionally energy-stable method for the phase ﬁeld crystal equation , Computer Methods in Applied Mechanics and Engineering, 249 (2012), pp. 52–61

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[5]

Guo and Y

R. Guo and Y. Xu , Local discontinuous Galerkin method and high order semi-implicit scheme for the phase ﬁeld crystal equation, SIAM Journal on Scientiﬁc Computing, 38 (2016), pp. A105–A127

work page 2016
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Q. Li, L. Mei, X. Yang, and Y. Li , Eﬃcient numerical schemes with unconditional energy stabilities for the modiﬁed phase ﬁeld crystal equation , Advances in Computational Mathematics, (2019), pp. 1–30

work page 2019
[7]

X. Li, J. Shen, and H. Rui , Energy stability and convergence of SAV block-centered ﬁnite diﬀerence method for gradient ﬂows, Mathematics of Computation, (2019)

work page 2019
[8]

Li and J

Y. Li and J. Kim , An eﬃcient and stable compact fourth-order ﬁnite diﬀerence scheme for the phase ﬁeld crystal equation, Computer Methods in Applied Mechanics and Engineering, 319 (2017), pp. 194–216

work page 2017
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J. Ramos , C. canuto, my hussaini, a. quarteroni, ta zang, spectral methods in ﬂuid dynamics, springer-verlag, new york (1988), dm 162 , 1991

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[11]

Shen and J

J. Shen and J. Xu , Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient ﬂows , SIAM Journal on Numerical Analysis, 56 (2018), pp. 2895–2912

work page 2018
[12]

J. Shen, J. Xu, and J. Yang , The scalar auxiliary variable (SAV) approach for gradient ﬂows , Journal of Computational Physics, 353 (2018), pp. 407–416

work page 2018
[13]

Shen and X

J. Shen and X. Yang , Numerical approximations of Allen-Cahn and Cahn-Hilliard equations , Discrete Contin. Dyn. Syst, 28 (2010), pp. 1669–1691

work page 2010
[14]

Swift and P

J. Swift and P. C. Hohenberg , Hydrodynamic ﬂuctuations at the convective instability , Physical Review A, 15 (1977), p. 319

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S. M. Wise, C. W ang, and J. S. Lowengrub , An energy-stable and convergent ﬁnite-diﬀerence scheme for the phase ﬁeld crystal equation , SIAM Journal on Numerical Analysis, 47 (2009), pp. 2269–2288

work page 2009
[16]

X. Yang , Eﬃcient schemes with unconditionally energy stability for the anisotropic Cahn-Hilliard equation using the stabilized-Scalar Augmented Variable (S-SAV) approach, arXiv preprint arXiv:1804.02619, (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018
[17]

Yang and D

X. Yang and D. Han , Linearly ﬁrst-and second-order, unconditionally energy stable schemes for the phase ﬁeld crystal model, Journal of Computational Physics, 330 (2017), pp. 1116–1134

work page 2017
[18]

Zhang, Y

Z. Zhang, Y. Ma, and Z. Qiao , An adaptive time-stepping strategy for solving the phase ﬁeld crystal model , Journal of Computational Physics, 249 (2013), pp. 204–215. This manuscript is for review purposes only

work page 2013

[1] [1]

Ainsworth and Z

M. Ainsworth and Z. Mao , Analysis and approximation of a fractional Cahn-Hilliard equation , SIAM Journal on Numerical Analysis, 55 (2017), pp. 1689–1718

work page 2017

[2] [2]

Elder and M

K. Elder and M. Grant , Modeling elastic and plastic deformations in nonequilibrium processing using phase ﬁeld crystals, Physical Review E, 70 (2004), p. 051605

work page 2004

[3] [3]

Elder, M

K. Elder, M. Katakowski, M. Haataja, and M. Grant , Modeling elasticity in crystal growth , Physical review letters, 88 (2002), p. 245701

work page 2002

[4] [4]

Gomez and X

H. Gomez and X. Nogueira , An unconditionally energy-stable method for the phase ﬁeld crystal equation , Computer Methods in Applied Mechanics and Engineering, 249 (2012), pp. 52–61

work page 2012

[5] [5]

Guo and Y

R. Guo and Y. Xu , Local discontinuous Galerkin method and high order semi-implicit scheme for the phase ﬁeld crystal equation, SIAM Journal on Scientiﬁc Computing, 38 (2016), pp. A105–A127

work page 2016

[6] [6]

Q. Li, L. Mei, X. Yang, and Y. Li , Eﬃcient numerical schemes with unconditional energy stabilities for the modiﬁed phase ﬁeld crystal equation , Advances in Computational Mathematics, (2019), pp. 1–30

work page 2019

[7] [7]

X. Li, J. Shen, and H. Rui , Energy stability and convergence of SAV block-centered ﬁnite diﬀerence method for gradient ﬂows, Mathematics of Computation, (2019)

work page 2019

[8] [8]

Li and J

Y. Li and J. Kim , An eﬃcient and stable compact fourth-order ﬁnite diﬀerence scheme for the phase ﬁeld crystal equation, Computer Methods in Applied Mechanics and Engineering, 319 (2017), pp. 194–216

work page 2017

[9] [9]

Ramos , C

J. Ramos , C. canuto, my hussaini, a. quarteroni, ta zang, spectral methods in ﬂuid dynamics, springer-verlag, new york (1988), dm 162 , 1991

work page 1988

[10] [10]

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

work page 2011

[11] [11]

Shen and J

J. Shen and J. Xu , Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient ﬂows , SIAM Journal on Numerical Analysis, 56 (2018), pp. 2895–2912

work page 2018

[12] [12]

J. Shen, J. Xu, and J. Yang , The scalar auxiliary variable (SAV) approach for gradient ﬂows , Journal of Computational Physics, 353 (2018), pp. 407–416

work page 2018

[13] [13]

Shen and X

J. Shen and X. Yang , Numerical approximations of Allen-Cahn and Cahn-Hilliard equations , Discrete Contin. Dyn. Syst, 28 (2010), pp. 1669–1691

work page 2010

[14] [14]

Swift and P

J. Swift and P. C. Hohenberg , Hydrodynamic ﬂuctuations at the convective instability , Physical Review A, 15 (1977), p. 319

work page 1977

[15] [15]

S. M. Wise, C. W ang, and J. S. Lowengrub , An energy-stable and convergent ﬁnite-diﬀerence scheme for the phase ﬁeld crystal equation , SIAM Journal on Numerical Analysis, 47 (2009), pp. 2269–2288

work page 2009

[16] [16]

X. Yang , Eﬃcient schemes with unconditionally energy stability for the anisotropic Cahn-Hilliard equation using the stabilized-Scalar Augmented Variable (S-SAV) approach, arXiv preprint arXiv:1804.02619, (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018

[17] [17]

Yang and D

X. Yang and D. Han , Linearly ﬁrst-and second-order, unconditionally energy stable schemes for the phase ﬁeld crystal model, Journal of Computational Physics, 330 (2017), pp. 1116–1134

work page 2017

[18] [18]

Zhang, Y

Z. Zhang, Y. Ma, and Z. Qiao , An adaptive time-stepping strategy for solving the phase ﬁeld crystal model , Journal of Computational Physics, 249 (2013), pp. 204–215. This manuscript is for review purposes only

work page 2013