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Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential
Wenbin Chen, Cheng Wang, Xiaoming Wang, Steven M. Wise
TL;DR
Logarithmic Flory–Huggins potentials require numerical phase variables to remain inside (−1,1), but preserving this property alongside stability is difficult. The paper proposes first- and second-order finite-difference schemes whose singular logarithmic terms support unique solvability and positivity, with optimal-rate convergence and efficient multigrid implementation.
Problem
Numerical schemes for logarithmic Flory–Huggins Cahn–Hilliard models must preserve positivity of 1+φ and 1−φ despite singularities at −1 and 1.
Method
The paper analyzes first- and second-order finite-difference schemes using implicit logarithmic terms, explicit expansive-term treatment, convexity-based arguments, and multigrid solution methods.
Results
The schemes have unique solutions, preserve positivity, are unconditionally energy stable, and achieve optimal-rate error estimates for both temporal orders.
Takeaways & Limitations
The singular logarithmic structure can be used directly to construct well-defined, positivity-preserving schemes without restricting the time-step size.
Takeaways & Limitations
For Cahn–Hilliard flow, the lack of a maximum principle and mass conservation make uniform pointwise bounds more difficult, with an estimate that becomes singular as Δt→0.
Abstract
from arXiv · showhide
We present and analyze finite difference numerical schemes for the Allen Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both the first order and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that, this numerical algorithm has a unique solution such that the positivity is always preserved for the logarithmic arguments. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis, which gives the full order error estimate. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes.
1. Introduction.
The paper develops positivity-preserving, energy-stable finite-difference schemes for the Cahn–Hilliard model with logarithmic Flory–Huggins potential. It addresses singularities at φ=−1 and 1 while providing first- and second-order temporal accuracy, convergence analysis, and efficient implementation.
- Motivation: Logarithmic Flory–Huggins energies are physically motivated but singular as the phase variable approaches −1 or 1.The PDE phase variable is expected to remain pointwise in (−1,1), creating a central numerical challenge.
- Motivation: Polynomial approximations avoid these singularities but may allow the PDE solution to leave the interval (−1,1).The paper instead focuses on preserving the logarithmic model’s positivity property.
- Research gap: Existing logarithmic-potential schemes lacked a general theoretical guarantee that 1+φ and 1−φ remain positive and the scheme stays unconditionally well-defined.Earlier positivity analysis was available, but unconditional energy stability was not, and a time-step constraint could hinder computations for small ε and large θ0.
- Approach: The proposed first-order scheme treats nonlinear logarithmic and surface-diffusion terms implicitly while treating the expansive term and mobility explicitly.Centered finite differences are used in space.
- Theoretical results: The schemes establish unique solvability, positivity preservation, and unconditional energy stability through the singular logarithmic term and convex discrete-energy structure.The analysis uses a strictly convex discrete minimization formulation and shows the singular values cannot be reached.
2. The first order numerical scheme.
The first-order scheme uses centered finite differences, convexity splitting, and explicit mobility updates. Its theoretical results guarantee a well-defined unique solution while preserving the pointwise bounds required by the logarithmic terms.
- Discrete setting: The spatial discretization uses uniform periodic grids with cell-centered and face-centered function spaces for finite-difference operators.The framework defines discrete gradients, divergences, Laplacians, inner products, norms, and summation-by-parts identities.
- Discrete operators: Centered difference and averaging operators define discrete gradients, divergences, Laplacians, mobility-weighted operators, and associated norms.These operators support the discrete energy and H−1 analysis.
- Scheme construction: The scheme applies convexity splitting in a semi-implicit, fully discrete formulation.For Cahn–Hilliard flow, mobility is defined at face centers from averaged phase values.
- Positivity requirement: The logarithmic arguments require −1<φn+1_i,j,k<1 for the numerical scheme to be well-defined.The analysis establishes this bound rather than assuming it only formally.
- Theoretical results: For constant mobility, the first-order Allen–Cahn and Cahn–Hilliard schemes have unique numerical solutions under bounded previous-step data.The Cahn–Hilliard update also preserves the mean-zero increment condition.
- Theoretical results: The Allen–Cahn solution remains uniformly separated from the singular values when the initial data satisfy a corresponding bound.The separation parameter depends on the initial bound but is independent of ε and the time index.
3. Theoretical justification of the positivity-preserving properties.
The positivity analysis formulates each numerical update as a strictly convex minimization problem and shows its minimizer lies strictly inside the admissible set. This establishes existence and uniqueness while preventing the phase variable from reaching the logarithmic singularities.
- Boundary exclusion: A directional-derivative contradiction excludes minimizers on the boundary of the restricted admissible set.A direction into the interior decreases the functional, so a boundary point cannot be a global minimizer.
- Existence and uniqueness: The numerical update is represented as minimization of a strictly convex discrete energy over a bounded admissible set.A compact, convex restriction is used to establish a minimizer before excluding boundary minimizers.
- Existence and uniqueness: The resulting first-order Cahn-Hilliard scheme has an interior numerical solution, and strict convexity gives uniqueness.The solution remains pointwise within the interval required by the logarithmic arguments.
- Positivity mechanism: The singular logarithmic terms prevent the numerical solution from reaching −1 or 1 when the previous solution is bounded and the initial average lies between them.This avoids the cut-off energy construction used in earlier analysis.
- Analytical challenge: Unlike Allen-Cahn, the Cahn-Hilliard analysis cannot directly use a maximum principle because the flow is H−1-based and mass-conserving.The lack of a maximum principle makes the pointwise-bound argument more involved.
- Non-constant mobility: For non-constant mobility, positivity analysis assumes pointwise mobility positivity, while degenerate mobility is left for future numerical work.The stated framework requires M(x) ≥ M0 > 0, although certain positive mobilities may be accommodated.
4. Unconditional energy stability and uniform in time H1 h bound.
The first-order scheme combines convex-concave energy decomposition with implicit logarithmic and surface-diffusion terms. This yields unconditional energy stability and a uniform-in-time discrete energy bound.
- Energy stability: The proposed Cahn-Hilliard scheme is unconditionally energy stable because its convex-concave decomposition gives a discrete energy estimate.The numerical solution remains within (−1, 1) pointwise under the established solvability result.
- Uniform bound: For any Δt > 0 and h > 0, the discrete energy satisfies Eh(φm) ≤ Eh(φ0) ≤ C6.The constant C6 is independent of h, providing a uniform-in-time bound on the discrete energy.
- Relation to prior work: Existing energy-stable schemes for the logarithmic potential lacked a theoretical positivity justification for numerical-solution existence.The proposed analysis addresses both energy stability and pointwise positivity.
5. Optimal rate convergence analysis in ℓ∞(0, T; H−1) ∩ℓ2(0, T; H1).
The convergence analysis uses consistency estimates, discrete error identities, convexity of the logarithmic terms, and discrete Gronwall arguments. Under regularity and sufficiently small discretization parameters, it establishes full-order convergence for the proposed schemes.
- First-order convergence: The first-order scheme has truncation error bounded by C(Δt + h2) in the discrete H−1 norm.The estimate follows from the consistency analysis used in the error equation.
- Error estimates: The nonlinear error estimate relies on monotonicity of ln and convexity of the logarithmic terms.The pointwise bounds −1 < φn+1 < 1 and −1 < Φn+1 < 1 are used in this step.
- First-order convergence: Discrete Gronwall arguments convert the consistency and stability estimates into the desired first-order convergence estimate.The resulting constant is independent of n, Δt, and h.
- Comparison with prior analysis: Unlike earlier implicit-Euler analyses, the proposed convergence analysis requires no time-step constraint for positivity.Earlier analyses impose Δt ≤ 4ε2 to ensure positivity, whereas this scheme needs no such constraint in convergence analysis.
6. The second order numerical scheme.
The second-order BDF scheme extends the positivity, solvability, stability, and convergence framework to higher temporal accuracy. Its analysis uses convexity and artificial diffusion, while non-constant mobility is treated by analogous arguments.
- Solvability: The second-order numerical solution is characterized as a minimizer of a strictly convex discrete functional.The minimization formulation supports the existence and uniqueness proof.
- Positivity proof: The boundary-exclusion argument extends to the second-order scheme by using a restricted admissible set and directional derivatives.The logarithmic singularity prevents the minimizer from occurring at the boundary.
- Non-constant mobility: For non-constant mobility bounded below by M0 > 0, the second-order scheme has a unique solution whose pointwise magnitude remains below 1.The stated result assumes bounded previous solutions and positive mobility.
- Energy stability: With constant mobility and A ≥ 1/16, the second-order BDF scheme admits a modified energy stability estimate.The proof combines convexity-based inequalities, Cauchy estimates, and the artificial diffusion term.
- Convergence analysis: The second-order convergence proof combines temporal-stencil identities, nonlinear estimates, artificial-diffusion bounds, and discrete Gronwall arguments.Under the stated regularity and smallness assumptions, the convergence constant is independent of n, Δt, and h.
- Scheme design: The second-order method is formulated as a BDF scheme using the preceding two phase-field values.The scheme updates φn+1 and the chemical potential simultaneously.
- Computation: Numerical tests accompany a multigrid solver and demonstrate positivity of the computed Cahn-Hilliard solutions.The solver is presented as simple and efficient for the proposed schemes.
7. Numerical results.
The numerical tests examine regularization, positivity, convergence, solver complexity, and comparisons among first- and second-order schemes. Results support bounded solutions, predicted accuracy, nearly optimal multigrid complexity, and trade-offs between accuracy, efficiency, and provable stability.
- Regularization: For θ0 = 3.0, using δ = 1.0 × 10−3 or δ = 1.0 × 10−5 produced solutions and decreasing energies identical up to round-off errors.The smaller value δ = 1.0 × 10−5 was used in subsequent computations.
- Positivity: The computed solution stayed within [−0.996, 0.996], well inside (−1 + δ, 1 − δ), for the three-dimensional CS1 simulation.The initial perturbations were uniformly distributed on [−0.05, 0.05].
- Convergence: The CS1 convergence test confirmed first-order temporal and second-order spatial accuracy along the refinement path Δt = 0.4h^2.The test used θ0 = 3.0, ε = 0.2, M ≡ 1, δ = 10−5, and final time T = 0.4.
- Solver complexity: The multigrid residual decreased by nearly the same factor per iteration, providing evidence for optimal or nearly optimal complexity, although larger θ0 required more iterations.The tests varied θ0 = 3.5, 3.0, and 2.0 with two smoothing sweeps and tolerance τ = 10−9.
- Comparison results: BDF2 had the best reported accuracy and efficiency balance, while BDF2 ES was slightly more efficient but less accurate; setting A = 0 improved accuracy but lost provable stability.CS1 was worst in accuracy but second best in efficiency per step, whereas backward Euler was worst in efficiency per time step.
- Comparison results: All schemes were positivity preserving when solvable, but positivity was not proved for the fully implicit schemes in the presented analysis.The authors state that establishing this property for fully implicit schemes is possible but substantially more complicated.
8. Conclusion remarks.
The paper establishes full-order convergence estimates for both temporal schemes and demonstrates practical solver robustness and efficiency through multigrid implementation.
- Optimal-rate convergence in the discrete ℓ∞(0, T; H−1) norm is established for both first- and second-order schemes.
- An efficient multigrid solver is applied to implement the proposed numerical schemes.
- Numerical results demonstrate the robustness and efficiency of the numerical solver.