A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability

1 Introduction

• The paper targets the no-slope-selection equation, whose mound slopes may grow unbounded on an infinite domain, in contrast to the slope-selection model's pyramidal structures with slopes near one.Both models have up-down symmetry.Source: §1 · p. 2
• The scheme combines Fourier pseudo-spectral spatial approximation with ETD time integration, using exact integration of the linear part and explicit multistep approximation of the nonlinear integral.An alternate splitting combines surface diffusion with auxiliary linear diffusion and revises the explicit extrapolation.Source: §1 · p. 3
• Prior work established energy-stable first- and second-order schemes, while theoretical analysis of third-order accurate gradient-flow schemes remained open.The NSS nonlinear term has automatically bounded higher-order derivatives in the L∞ norm, a property important to nonlinear energy-stability analysis.Source: §1 · pp. 2–3
• A third-order Douglas-Dupont regularization and Fourier eigenvalue analysis establish stability for a modified energy and a uniform-in-time bound for the original energy.The modified energy includes the original energy and nonnegative numerical correction terms.Source: §1 · p. 3
• The proposed third-order ETD scheme has a theoretically proved optimal-rate convergence estimate, obtained through two-stage error analysis, operator eigenvalue bounds, and aliasing-error control.The analysis combines linear exponential integration with explicit multistep extrapolation for the nonlinear part.Source: §1 · p. 3
• The convergence estimate is unconditional, requiring no scaling law between the time step and spatial resolution, and is reported as the first such result for a third-order gradient-flow scheme.The analysis uses aliasing-error control and estimates for operators and their compositions.Source: §1 · p. 3
• Long-time simulations indicate logarithmic energy decay and power-law growth of surface roughness and mound width; their growth power indices are more accurate than those from certain second-order schemes at the same resolution.The simulations examine the coarsening process.Source: §1 · pp. 3–4

2.1 Review of the Fourier pseudo-spectral approximation

• To address aliasing in the nonlinear term, the analysis introduces periodic extension and Fourier collocation interpolation.The collocation coefficients may differ from the function’s Fourier coefficients because of truncation and aliasing errors.Source: §2.1 · pp. 5–6

2.2 The proposed third order ETD-based numerical scheme

• The proposed third-order ETD scheme adds a Douglas-Dupont regularization term and preserves mass.The added term is −A∆t^2∆_N^2(un+1 − un); the paper states it enables a theoretical energy-stability justification.Source: §2.2 · p. 7

2.3 Some preliminary estimates

• Preliminary operator and Fourier analyses establish estimates used in the later stability and convergence analyses, with constants independent of f and N.The constants C_i depend only on Ω.Source: §2.3 · pp. 7–8

2.4 The energy stability analysis

• Under condition (2.57), the scheme’s modified energy is nonincreasing, yielding a uniform-in-time bound on its numerical energy.The theorem states a modified-energy inequality, and the corollary bounds the numerical energy for all k ≥ 0.Source: §2.4 · pp. 12, 14
• Artificial regularization is needed to theoretically justify energy stability for higher-order ETD schemes, whose original third-order version lacked an energy bound.The original second-order scheme's alternate energy inequality also did not provide a uniform-in-time bound for the original energy because of positive terms on its right-hand side.Source: §2.4 · p. 14
• The theoretical condition A = O(ε^-2) uses surface-diffusion stability to offset stability loss from explicit multistep treatment of nonlinear terms.The regularization must be O(∆t^3) to preserve third-order temporal accuracy, so it is insufficient by itself to ensure stability.Source: §2.4 · pp. 14–15
• In practical computations, choosing A = O(1) has not caused energy stability loss, despite the theoretical requirement A = O(ε^-2).Source: §2.4 · p. 14
• The NSS nonlinearity has automatically bounded higher-order derivatives in L∞, enabling an energy estimate with much reduced artificial regularization for higher-order schemes.Source: §2.4 · p. 15

3 The convergence analysis for the third order ETD-based scheme

• The convergence theorem assumes a sufficiently regular exact solution and periodic boundary conditions, with ∆t and h sufficiently small.The stated regularity class includes temporal and spatial Sobolev regularity.Source: §3 · pp. 15–16

3.1 The error evolutionary equation

• The error equation is rewritten as a two-stage system to separate exponential integration of the linear part from explicit multi-step treatment of nonlinear error.The split is introduced because the original error evolution does not make the interaction between linear and nonlinear terms clear.Source: §3.1 · p. 16, §3.2 · p. 17

3.2 Some preliminary nonlinear error inequalities

• The nonlinear error analysis uses preliminary estimates and a discrete Gronwall inequality to obtain the convergence estimate and close the induction argument.The proof recovers its a-priori assumption at the next time step before concluding that no ∆t–h scaling law is needed.Source: §3.2 · pp. 19–20
• The scheme’s error is bounded in ℓ∞(0, T; H1) ∩ ℓ2(0, T; H3), with no scaling law required between ∆t and h.The theorem assumes sufficiently small ∆t and h, and the constant in its estimate is independent of both.Source: §3 · p. 16, §3.2 · pp. 19–20

4.1 Convergence test for the numerical scheme

• L1, L2, and L∞ errors show clear third-order accuracy when ∆t = 0.5h, ε = 0.5, and the error is measured at T = 1.The test uses grid sizes N = 64 to 192 and regularization parameter A = 1.Source: §4.1 · p. 20, Figure 1 · p. 21

4.2 Coarsening, energy dissipation and other physical quantities

• The ε = 0.02 simulation reaches t = 300000, where a one-hill-one-valley structure emerges and further coarsening is not possible.The computation uses L = 12.8 and N = 512, with increasing time steps across three intervals.Source: §4.2 · pp. 21–22
• For ε = 0.02, energy decreases like −ln(t) until saturation, while average height grows like t1/2 and average width like t1/4.The reported least-squares fits use data only up to about t = 400.Source: §4.2 · pp. 22–23, Figure 3 · p. 23
• For the proposed scheme, mr = 0.5001 and bm = 0.2545, versus mr = 0.510 and bm = 0.258 for ETDMs2; costs about 3 to 5 times more per step.The simulations otherwise produce similar long-time curves for energy, standard deviation, and mound width; the comparison with reports mr = 0.5132 and mr = 0.2607 for the respective growth exponents.Source: §4.2 · pp. 23–24

5 Concluding remarks

• The third-order ETD scheme combines Fourier pseudo-spectral spatial discretization with a Douglas-Dupont regularization term, yielding modified energy stability and a uniform-in-time numerical energy bound.The analysis uses Fourier eigenvalue estimates to justify the regularization coefficient.Source: §5 · p. 24
• The paper derives an optimal-rate convergence estimate for the scheme, the first such result for a third-order accurate scheme for a gradient flow.The analysis uses a two-stage decomposition, eigenvalue estimates, nonlinear analysis, aliasing-error control, and scaling arguments between Δt and h.Source: §5 · p. 24
• For ε=0.02 up to T=3 × 10^5, long-time simulations with the proposed third-order ETD-based scheme show surface-roughness and mound-width growth indices more accurate than those from certain second-order energy-stable schemes.Source: §5 · p. 24

A Proof of Lemma 2.4

• The proof states that H(x) is decreasing over (0, +∞), using a comparison of H(x1) and H(x2) for 0 < x1 < x2.The comparison uses bounds on f(x) relative to g(x) over the intervals around x1 and x2.Source: A Proof of Lemma 2.4 · pp. 24–25
• g0(x), g1(x), and g2(x) are each shown to be decreasing over (0, +∞).The proof rewrites each function and applies Lemma A.1.Source: A Proof of Lemma 2.4 · p. 25
• The decreasing property gives g0(x) ≤ 1 and g1(x) ≤ 1.Source: A Proof of Lemma 2.4 · p. 25

B Proof of Proposition 2.5

• Parseval’s equality applied to discrete Fourier expansions supports the estimates, with −λk,ℓ, Λk,ℓ, and e^−ΔtΛk,ℓ nonnegative.Source: B Proof of Proposition 2.5 · pp. 26–27
• The proof establishes (2.38) by taking C1 = 1.Source: B Proof of Proposition 2.5 · p. 26
• The proof establishes the first inequality of (2.42) with C4 = 1 and the second with C5 = 1.Source: B Proof of Proposition 2.5 · p. 27

C Proof of Proposition 3.2

• The proof establishes (3.13) using an expansion and a constant C6 associated with elliptic regularity.Source: C Proof of Proposition 3.2 · pp. 27–28
• Grid-point interpolation and Lemma 2.2 are used to control the aliasing error in comparing gN(UNk) and gN(uk).Source: C Proof of Proposition 3.2 · p. 27
• The estimates also use repeated Hölder and Sobolev inequalities together with the a-priori assumption (3.12).Source: C Proof of Proposition 3.2 · p. 28

• phi6K.jpg, phi20K.jpg, and phi40K.jpg are available in JPG format.Source: http://arxiv.org/ps/1903.03296v1 · pp. 33–35
• phi80K.jpg and phi200.jpg are available in JPG format.Source: http://arxiv.org/ps/1903.03296v1 · pp. 36–37
• phi300K.jpg and phi400.jpg are available in JPG format.Source: http://arxiv.org/ps/1903.03296v1 · pp. 38–39