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An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation
Kelong Cheng, Wenqiang Feng, Cheng Wang, Steven M. Wise
TL;DR
The paper addresses the need for an energy-stable, high-order finite-difference method for the Cahn–Hilliard equation. It combines long-stencil fourth-order spatial discretization with a modified second-order BDF scheme and Fourier-based truncation analysis, proving unique solvability, energy stability, and optimal convergence. The analysis also identifies author-stated scope boundaries concerning regularity and iterative-solver convergence.
Problem
Existing Cahn–Hilliard finite-difference methods have largely used second-order spatial differences, while fourth-order approaches lacked a theoretically proved energy stability or required additional solver cost.
Method
The paper combines a long-stencil fourth-order finite-difference spatial operator with a modified second-order BDF method, explicit concave-term extrapolation, Douglas–Dupont regularization, and discrete Fourier truncation analysis.
Results
The scheme is uniquely solvable, mass conservative, energy stable, and achieves optimal O(∆t^2+h^4) convergence in ℓ∞(0,T;ℓ2) ∩ ℓ2(0,T;H_h^2).
Takeaways & Limitations
The construction provides a theoretically justified fourth-order spatial approximation for Cahn–Hilliard dynamics without the additional Poisson solver associated with compact differences.
Takeaways & Limitations
The geometric convergence rate of the preconditioned steepest-descent solver is expected by analogy but is not derived, with details left to future readers.
Abstract
from arXiv · showhide
In this paper we propose and analyze an energy stable numerical scheme for the Cahn-Hilliard equation, with second order accuracy in time and the fourth order finite difference approximation in space. In particular, the truncation error for the long stencil fourth order finite difference approximation, over a uniform numerical grid with a periodic boundary condition, is analyzed, via the help of discrete Fourier analysis instead of the the standard Taylor expansion. This in turn results in a reduced regularity requirement for the test function. In the temporal approximation, we apply a second order BDF stencil, combined with a second order extrapolation formula applied to the concave diffusion term, as well as a second order artificial Douglas-Dupont regularization term, for the sake of energy stability. As a result, the unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the $\ell^\infty (0,T; \ell^2) \cap \ell^2 (0,T; H_h^2)$ norm. A few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.
1 Introduction
The paper develops an energy-stable Cahn–Hilliard scheme combining second-order temporal accuracy with a long-stencil fourth-order spatial approximation. Its analysis establishes solvability, energy stability, and optimal convergence under reduced regularity requirements for truncation estimates.
- Fourth-order spatial discretization is pursued because existing finite-difference work largely uses second-order centered differences, while higher-order accuracy can capture finer structure at reduced computational cost.
- The scheme combines a long-stencil fourth-order difference operator with a second-order energy-stable temporal algorithm based on modified BDF treatment.
- Explicit treatment of the concave diffusion term ensures unique solvability, while Douglas–Dupont regularization guarantees energy stability when A ≥ 1/16.
- Energy stability provides a uniform-in-time H1 bound for the numerical solution and supports the discrete analysis.
- Discrete Fourier and aliasing analysis yields an optimal O(∆t^2+h^4) convergence rate in the ℓ∞(0,T;ℓ2) ∩ ℓ2(0,T;H_h^2) norm.
- The paper establishes unique solvability, energy stability, and mass conservation for the fully discrete scheme, with the analysis organized around truncation estimates, scheme construction, proofs, and numerical results.
2 The long stencil difference operator and the local truncation error estimate
This section analyzes long-stencil fourth-order finite differences through discrete Fourier methods, obtaining ℓ2 truncation-error estimates in periodic one-, two-, and three-dimensional settings. The estimates achieve fourth-order accuracy with H6 regularity of the test function.
- The long-stencil fourth-order operators approximate first- and second-order derivatives on uniform grids and are extended to multidimensional periodic grids.
- Standard Taylor analysis gives classical truncation estimates requiring C6 regularity, motivating weaker regularity conditions for discrete ℓ2 convergence analysis.
- Integral-form Taylor expansion improves the one-dimensional estimate by requiring only H6 regularity of the test function.
- The two-dimensional truncation error satisfies ∥τ∥2 ≤ C h4 ∥f∥H6, with C depending only on L.
- Discrete Fourier expansions, Parseval identities, eigenvalue comparisons, and aliasing estimates establish the multidimensional truncation bounds.
- The analogous three-dimensional estimate is ∥τ∥2 ≤ C h4 ∥f∥H6, again with a constant depending only on L.
3 The numerical scheme for the Cahn-Hilliard equation
The scheme combines a long-stencil fourth-order spatial discretization with a modified second-order BDF temporal method for the Cahn–Hilliard equation. It is designed to preserve mass, admit unique solutions, provide modified energy decay, and support optimal convergence under stated regularity assumptions.
- Spatial discretization and notation: The discrete setting uses periodic grid functions, a fourth-order discrete Laplacian, discrete inner products, and an inverse-Laplacian operator on the mean-zero space.The analysis focuses on a two-dimensional periodic domain, with extension to three dimensions expected to be straightforward.
- Fully discrete scheme: The method uses a long-stencil fourth-order finite difference approximation in space and a modified second-order BDF discretization in time.The concave diffusion term is treated explicitly, and a Douglas–Dupont-type regularization is added to the chemical potential.
- Temporal initialization: The ghost value required by the two-step method is extrapolated to preserve second-order temporal accuracy, with approximation accuracy O(∆t^2 + h^4).The extrapolation supplies the missing previous-time value φ^-1 needed by the BDF scheme.
- Theoretical properties: The scheme has a unique solution and conserves mass at every time step.Strong convexity on the fixed-mean hyperplane supports unique solvability, while the theorem states φ^k ≡ φ^0.
- Energy stability: For A ≥ 1/16, the scheme satisfies a modified energy-decay property and yields a uniform-in-time discrete H_h^1 bound.The energy-stability result makes the uniform bound available under sufficiently regular initial data.
- Convergence analysis: Assuming exact-solution regularity class R and sufficiently small ∆t and h, the analysis establishes convergence with constant C independent of ∆t and h.The stated regularity is R = H^3(0,T;C^0) ∩ H^2(0,T;H^4) ∩ L∞(0,T;H^8).
4 The detailed proof
The proof establishes solvability, mass conservation, energy bounds, and optimal convergence for the fully discrete scheme. Discrete Fourier, Sobolev, truncation-error, and Gronwall estimates connect stability to the final error bound.
- Unique solvability: Strong convexity of the discrete energy functional implies a unique numerical solution on the fixed-mean hyperplane.
- Mass conservation: Discrete summation shows that the scheme preserves the initial mean value, hence is mass conservative at every time step.
- A priori bounds: Energy estimates provide a uniform-in-time H_h^1 bound, which is then used with discrete Sobolev embedding to obtain an ℓ6 estimate.
- Consistency: The long-stencil truncation error satisfies ||τ^{k+1}||_2 ≤ C(∆t^2+h^4).This estimate supplies the temporal and spatial consistency orders used in the convergence proof.
5 Numerical results
The numerical section implements the nonlinear scheme with a preconditioned steepest descent solver and tests convergence and spinodal decomposition. The experiments support the theoretical rates and reproduce the predicted one-third-power energy decay.
- PSD solver: The preconditioned steepest descent solver uses a linearized nonlinear operator as a preconditioner and solves the search-direction problem efficiently with FFT.
- Spinodal decomposition: The spinodal-decomposition simulation uses a uniform Cartesian grid, periodic boundaries, and time steps ∆t=0.01 followed by ∆t=0.04.
- Convergence test: Table 1 confirms the theoretical convergence results for the computed solution under the proposed scheme.
- Energy dissipation: The simulated discrete energy decays like t^-1/3, with fitted coefficients a_e=2.3670 and b_e=0.3332.The reported regression is based on computed data and is described as an almost perfect one-third-power law.
6 Concluding remarks
The paper concludes that a long-stencil fourth-order spatial discretization and modified second-order BDF time integrator produce a provably energy-stable scheme. The analysis establishes solvability, uniform bounds, optimal convergence, and numerical agreement with one-third-power energy dissipation.
- Contributions: The proposed method combines fourth-order finite differences in space with second-order temporal accuracy for the Cahn-Hilliard equation.
- Spatial analysis: Discrete Fourier analysis reduces the regularity requirement for the long-stencil truncation-error estimate on periodic uniform grids.
- Temporal scheme: A modified BDF method, explicit concave-term extrapolation, and Douglas-Dupont regularization ensure discrete energy stability and unique solvability.
- Convergence: The scheme achieves optimal convergence in ℓ∞(0,T;ℓ2) ∩ ℓ2(0,T;H_h^2), supported by numerical experiments.
- Numerical behavior: The spinodal-decomposition simulations indicate an energy dissipation law proportional to t^-1/3.
C Proof of Lemma 3.2
The proof of Lemma 3.2 uses discrete Fourier representations of periodic grid functions and the fourth-order Laplacian. Parseval identities and discrete Cauchy-Schwarz estimates then establish the required norm relations.
- Fourier representation: A periodic grid function is represented through a discrete Fourier transformation.
- Operator analysis: Applying the fourth-order discrete Laplacian and its inverse is analyzed mode by mode in Fourier space.
- Norm identities: Orthogonality and Parseval equality convert Fourier coefficients into discrete norm identities.
- Lemma conclusion: A discrete Cauchy-Schwarz estimate completes the proof of the first norm inequality, while the second follows similarly.