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On energy dissipation theory and numerical stability for time-fractional phase field equations
Tao Tang, Haijun Yu, Tao Zhou
TL;DR
Time-fractional phase-field models lacked rigorous continuous and discrete energy-dissipation results. The paper proves integral-type laws and constructs finite-difference schemes inheriting energy stability for Allen-Cahn, Cahn-Hilliard, and molecular beam epitaxy models. Numerical studies report several coarsening stages and a -α/3 power-law stage for the time-fractional Cahn-Hilliard and molecular beam epitaxy models.
Problem
Rigorous analysis of the numerically observed energy-dissipation behavior of time-fractional phase-field models remained open.
Method
The paper establishes continuous integral-type dissipation laws and proposes finite-difference schemes satisfying discrete energy dissipation for three time-fractional phase-field models.
Results
The three time-fractional models admit the established dissipation laws, while numerical studies find several coarsening stages and a -α/3 power-law stage for Cahn-Hilliard and molecular beam epitaxy models.
Takeaways & Limitations
The results provide energy-stable continuous and discrete formulations for time-fractional Allen-Cahn, Cahn-Hilliard, and molecular beam epitaxy models within the studied settings.
Takeaways & Limitations
The -α/3 coarsening law lacks rigorous theoretical justification, and only first-order schemes are investigated.
Abstract
from arXiv · showhide
For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shall address this open issue. More precisely, we prove for the first time that the time-fractional phase field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability. Our discussion covers the time-fractional gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models. Numerical examples are presented to confirm the theoretical results. Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter $α$ reveals that there are several coarsening stages for both time-fractional Cahn-Hilliard equation and time-fractional molecular beam epitaxy model, while there exists a $-α/3$ power law coarsening stage.
1. Introduction.
The introduction frames phase-field models as energy-driven systems and identifies unresolved continuous and discrete energy dissipation laws for time-fractional variants. The paper addresses this gap by establishing dissipation results, stable finite-difference schemes, and coarsening-rate behavior.
- Phase-field models use a free energy depending on an order parameter together with a diffusive mechanism.
- Allen-Cahn, Cahn-Hilliard, and molecular beam epitaxy models are presented as representative phase-field formulations.
- Energy dissipation is important for developing stable numerical methods because it supports long-time simulations.
- Prior fractional phase-field studies included modified energies or fractional gradient flows, with qualitative behavior reported for fractional Cahn-Hilliard models.
- Numerical evidence suggested free-energy dissipation for time-fractional models, but rigorous analysis remained open.
- The paper establishes continuous and discrete dissipation laws for time-fractional phase-field models and investigates their coarsening rates.
2. Energy dissipation for time-fractional phase field equations.
The paper proves integral-type energy dissipation laws for time-fractional Allen-Cahn, Cahn-Hilliard, and molecular beam epitaxy equations under stated boundary and potential assumptions. The analysis also connects these laws to mass conservation and extends the results to discrete schemes.
- Analytical preliminaries: Fractional-calculus estimates and positivity properties of fractional-integral kernels provide the analytical tools for the dissipation proofs.
- Model assumptions: The models use potentials satisfying stated regularity and growth or derivative-sign conditions, including the quartic double-well potential as an example.
- Time-fractional Allen-Cahn equation: The time-fractional Allen-Cahn equation admits an integral-type energy dissipation law when its initial energy is finite and the boundary conditions are homogeneous.
- Time-fractional Allen-Cahn equation: For the Allen-Cahn model, the dissipated energy is bounded, but the integral dissipation law generally does not imply a pointwise time derivative inequality.
- Time-fractional Cahn-Hilliard equation: The time-fractional Cahn-Hilliard equation satisfies an energy dissipation law under periodic or no-flux boundary conditions, with mass conservation established first.
- Time-fractional molecular beam epitaxy model: The time-fractional molecular beam epitaxy model also satisfies an energy dissipation law under periodic or no-flux boundary conditions.
3. Energy stable finite difference schemes.
The paper develops finite difference schemes for time-fractional phase-field models that inherit discrete energy dissipation. The approach covers Allen–Cahn, Cahn–Hilliard, and molecular beam epitaxy models, with unconditional stability under stated conditions.
- Scheme construction: The schemes use the L1 approximation for the time-fractional derivative and treat other terms implicitly.For the Allen–Cahn equation, stabilization is applied to the nonlinear bulk force.
- Scheme construction: The L1 discretization has a special property that yields H1 stability when the scheme is paired with the new-time solution.The cross terms are bounded by one-half the sum of the corresponding squared terms.
- Assumptions: The modified bulk potential assumes F ∈ C2(R) with a finite global Lipschitz constant for its derivative.The paper notes that modifying the far-end nonlinearity can produce a quadratic-growth double-well potential satisfying this condition.
- Allen–Cahn equation: The Allen–Cahn scheme satisfies a discrete energy dissipation law and is unconditionally energy stable when S ≥ γL/2ε.The estimate follows by multiplying the scheme by the time difference, integrating in space, summing over time steps, and applying the discrete lemma.
- Cahn–Hilliard and MBE equations: The same construction extends to the time-fractional Cahn–Hilliard and molecular beam epitaxy models, which also satisfy discrete energy dissipation properties.For the molecular beam epitaxy scheme without slope selection, unconditional stability holds for any time step when the stated stabilization bound is met.
- Alternative schemes: Convex-splitting schemes are another energy-stable option, but they require solving a nonlinear system at each time step.The paper contrasts this computational requirement with the stabilization-based schemes.
4. Maximum principle for the Allen-Cahn equation.
The section establishes maximum-principle results for the time-fractional Allen–Cahn equation and its semi-discretized scheme. These bounds support use of the standard double-well potential instead of a modified potential under appropriate conditions.
- Discrete maximum principle: A discrete maximum principle can remove the global Lipschitz requirement on f because numerical solutions remain bounded by the initial data.Consequently, the modified bulk potential may be unnecessary and the standard double-well potential can be used.
- Discrete maximum principle: The semi-discretized scheme preserves the initial upper and lower bounds by induction when its stated assumptions and time-step condition hold.The proof excludes violations at successive maximum points and establishes the lower bound similarly.
- Regularity: The time-fractional Allen–Cahn solution has limited temporal regularity, with ∂tφ(t) ∈ L2(Ω) and ∥∂tφ(t)∥L2(Ω) ≤ ct^α−1.This behavior reflects an initial singularity known for linear time-fractional parabolic equations.
- Continuous maximum principle: The weak-solution maximum principle follows from convergence of the piecewise linear extension in C([0,T]; L2(Ω)) and uniform boundedness for sufficiently small τ.A corresponding strong-solution result is discussed under additional regularity assumptions.
5. Numerical experiments.
Numerical experiments examine phase-field evolution, energy dissipation, and coarsening for time-fractional Allen–Cahn, Cahn–Hilliard, and molecular beam epitaxy models. Smaller fractional orders delay equilibration, while the Cahn–Hilliard and MBE models exhibit multi-stage dissipation with a -α/3 power-law stage.
- Numerical setup: Fourier-Galerkin simulations evaluate phase-field evolution and energy behavior, using a fast sum-of-exponentials algorithm for fractional-derivative history terms.The computations use periodic boundary conditions for the coarsening studies.
- 5.1. Numerical results for the time-fractional AC equation: Smaller α values delay equilibration in the time-fractional Allen–Cahn equation and produce a long-tail energy dissipation effect.The comparison uses α = 1, 0.5, and 0.3.
- 5.2. Numerical results for the time-fractional CH equation: The time-fractional Cahn–Hilliard dissipation process has three stages: small-scale separation, interacting structures with power-law decay, and minimum-energy equilibrium.The intermediate regime follows E[φ(t)] ≈ C_α t^p_α, with fitted p_α ≈ -α/3.
- 5.2. Numerical results for the time-fractional CH equation: p_α ≈ -α/3 describes the fitted intermediate power-law stage of time-fractional Cahn–Hilliard coarsening.This is reported as consistent with the -1/3 law for the classical case α = 1.
- 5.3. Numerical results for the time-fractional MBE model: The time-fractional molecular beam epitaxy model also shows three dissipation stages, including an intermediate power-law with asymptotic power -α/3.The numerical study compares α = 1, 0.7, and 0.4.
6. Concluding remarks.
The work establishes integral-type energy dissipation for time-fractional phase-field equations and develops finite-difference schemes that inherit discrete energy dissipation. Numerical experiments support the theory while identifying unresolved questions about rigorous coarsening-law justification and higher-order schemes.
- The continuous theory establishes an integral-type energy dissipation law for time-fractional phase-field equations.
- The proposed finite-difference schemes inherit the discrete energy dissipation property for time-fractional AC, CH, and MBE models.
- Numerical experiments verify the theoretical predictions across the time-fractional AC, CH, and MBE equations.
- The energy dissipation rate shows an asymptotic power law of −α/3 during coarsening for the time-fractional CH equation and MBE model.A rigorous theoretical justification remains open.
- Only first-order schemes are investigated, leaving high-order energy-stable schemes for future study.