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Arbitrarily High-order Unconditionally Energy Stable Schemes for Thermodynamically Consistent Gradient Flow Models
Yuezheng Gong, Jia Zhao, Qi Wang
TL;DR
High-order numerical approximations are desired for thermodynamically consistent gradient flows that satisfy energy-dissipation laws. The paper uses energy quadratization to develop prediction-correction and Gaussian-collocation schemes, reporting improved accuracy for the former and arbitrarily high-order unconditional energy stability for the latter.
Problem
For thermodynamically consistent gradient flows, numerical approximations are sought that preserve energy dissipation while remaining computationally efficient and high order; earlier EQ and SAV schemes had been proved unconditionally energy stable only up to second order.
Source: §1 · p. 1Method
The paper applies energy quadratization to obtain an equivalent quadratic-energy system, then uses prediction-correction for linear schemes and Gaussian collocation for arbitrarily high-order schemes.
Source: §1 · p. 2Results
The HEQ schemes are arbitrarily high order and rigorously unconditionally energy stable, while prediction-correction schemes retain second-order accuracy with much smaller errors than linear schemes.
Source: §6 · pp. 16–17Takeaways & Limitations
The authors report that the schemes' high-order accuracy and unconditional energy stability allow large time steps, making them appealing for long-time computations.
Source: §1 · p. 2The schemes satisfy a discrete energy-dissipation law for the reformulated energy rather than the original energy, although that reformulated energy is a high-order approximation of the original.
Source: §3.2 · p. 9
Abstract
from arXiv · showhide
We present a systematical approach to developing arbitrarily high order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization (EQ) method, We formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic energy, arriving at an arbitrarily high-order scheme for gradient flow models. Schemes derived using both approaches are proved rigorously to be unconditionally energy stable. The proposed schemes are then implemented in four gradient flow models numerically to demonstrate their accuracy and effectiveness. Detailed numerical comparisons among these schemes are carried out as well. These numerical strategies are rather general so that they can be readily generalized to solve any thermodynamically consistent PDE models.
1. Introduction.
Thermodynamically consistent gradient flows follow energy-dissipation principles, but numerical schemes are sought that preserve dissipation rates while achieving high order. The paper develops two EQ-based strategies: prediction-correction for linear schemes and Gaussian collocation for arbitrarily high-order schemes.
1. Introduction.
- Energy-stable schemes preserve energy dissipation, and they are unconditionally energy stable when that property does not depend on the time-step size.The desired approximation is also computationally efficient and high order.Source: §1 · p. 1
- Convex splitting and stabilization can yield dissipative schemes, but neither strategy guarantees preservation of the energy-dissipation rate.Convex splitting relies on expressing free energy as a difference of convex functionals; stabilization typically adds high-order dissipative terms.Source: §1 · p. 1
- EQ and SAV had rigorous unconditional energy-stability results only up to second order in time, while a convex-splitting Runge-Kutta alternative needed many stages for third order and did not cover variable mobilities.The authors identify these limits as potentially inadequate for long simulations or sharp-transition dynamics.Source: §1 · p. 2
- The paper develops prediction-correction schemes to improve linear EQ schemes and Gaussian-collocation HEQ schemes for arbitrarily high-order energy-stable approximations.The HEQ approach transforms the model to a quadratic-energy form; classical Crank-Nicolson and backward-differentiation-formula schemes are special cases.Source: §1 · p. 2
2. Gradient Flow Models and Their EQ Reformulation.
The section formulates gradient flow models through their state variables, mobility operator, and free energy, then uses energy quadratization to obtain an equivalent quadratic-energy system. This reformulation retains an energy-dissipation law under suitable boundary conditions and provides a basis for numerical schemes.
2.1. Gradient flow models.
- A gradient flow model is specified by its state variables, a negative semi-definite mobility operator, and an effective free energy whose variational derivative is the chemical potential.The model has an energy-dissipation law for suitable boundary conditions, including periodic conditions or conditions that eliminate boundary terms after integration by parts.Source: §2.1 · p. 3
2.2. Model reformulation using the EQ approach.
- The reformulation transforms the free energy into quadratic form and expresses the resulting system so its energy-dissipation law can be preserved by numerical schemes.The section notes that the equivalent form is easier to handle than the original system.Source: §2.1 · p. 3, §2.2 · p. 4
- The EQ reformulation turns the original gradient flow system into an equivalent system whose free energy is a quadratic functional of the transformed variables.It can be applied to thermodynamically consistent, energy-dissipative gradient flows with high-order spatial derivatives.Source: §2.2 · p. 4
3. Temporal Discretization.
The section develops linear prediction-correction schemes and arbitrarily high-order schemes for the EQ reformulation of gradient flows. The linear schemes and Gaussian collocation schemes are unconditionally energy stable, whereas energy stability is not proved for the high-order BDF schemes.
3.1. Linear energy stable schemes.
- Linear prediction-correction schemes use predicted values to correct explicit terms and are unconditionally energy stable.The schemes include CN and BDF2 variants; several tested iteration steps N ≤5 noticeably improve accuracy.Source: §3 · p. 4, §3.1 · pp. 4–6
3.2. Arbitrarily high-order, unconditionally energy stable schemes.
- High-order BDF schemes combine backward differentiation with matched extrapolation, but their energy stability is unproved and decay is reported only for reasonable time steps.The authors state they cannot prove energy stability for these schemes.Source: §3.2 · p. 6, §3.1 · p. 6
- Gaussian collocation schemes have order 2s and are proved unconditionally energy stable for the reformulated EQ system.The proof uses Gaussian quadrature nodes and weights.Source: §3.2 · pp. 6–7
- The discrete dissipation law applies to the reformulated energy, which approximates the original energy at high order rather than enforcing its exact discrete decay.For the Ginzburg-Landau example, the stabilized modified energy implies H1 boundedness of the phase variable.Source: §3.2 · p. 9
- For singular energy potentials, the proposed methods are not proved to preserve positivity.The stated example is the Cahn-Hilliard equation with Flory-Huggins free energy.Source: §3.2 · p. 9
4. Spatial discretization.
The spatial discretization uses Fourier pseudospectral methods to preserve the energy-dissipation structure of the quadratized gradient-flow system. The resulting ODE system can be advanced with the paper’s time-discretization methods to obtain fully discrete energy-stable schemes.
4. Spatial discretization
- The Fourier pseudospectral discretization preserves the negative semi-definite mobility operator and positive definite self-adjoint energy operator, yielding a spatially semi-discrete energy-dissipation law.Applying the paper’s time methods to the resulting ODE system gives fully discrete energy-stable schemes.Source: §4 · pp. 9–10
- Trigonometric interpolation on a uniform periodic grid provides the spatial representation, with derivatives evaluated at the collocation points.The domain is partitioned uniformly, and the interpolation space is spanned by trigonometric polynomials.Source: §4 · pp. 9–10
- The spatial method is applied to Cahn–Hilliard and molecular beam epitaxy models as two examples.The examples are introduced as applications of the general spatial discretization.Source: §4 · pp. 9–11
5. Numerical Results.
The numerical tests compare prediction-correction, BDF, and HEQ schemes across gradient-flow examples, assessing temporal accuracy and behavior at large time steps.
Time-step refinement
- HEQ-RK reaches fourth- and sixth-order accuracy, while ICN reaches second order at larger time steps and has lower same-step errors than LCN.The experiments assume periodic boundaries; the schemes also apply to physical boundaries when those boundaries annihilate terms in the energy-dissipation-rate formula.Source: §5 · p. 12, Figure 5.1 · p. 13
Coarsening dynamics
- At Δt=0.2, fourth- and sixth-order HEQ Gaussian collocation give very accurate energy predictions; BDF4 and BDF6 violate energy dissipation at large steps.ICN gives a very accurate energy prediction at Δt=0.1, while LCN requires Δt<0.0125 for its calculated energy to converge to the accurate energy.Source: §5 · p. 13, Figure 5.2 · p. 14
- A sixth-order Gaussian-collocation run with Δt=0.01 predicts coarsening dynamics fairly accurately across the reported times.The profiles show φ at t=10, 20, 50, 100, 200, and 380.Source: §5 · p. 13, Figure 5.3 · p. 15
Phase field crystal growth model
- The fourth-order Gaussian-collocation phase-field-crystal simulation gives qualitatively similar predictions to.Source: §5 · p. 15
Molecular beam epitaxy model
- For MBE, ICN and HEQ collocation give robust large-step dynamics, whereas LCN is erratic and BDF-6 needs relatively small steps for accurate energy curves.Source: §5 · p. 15
Allen–Cahn equation
- With Δt=5, HEQ Gaussian collocation predicts the Allen–Cahn disk’s volume-decay rate −2π accurately, versus ICN at 0.5 and BDF4/BDF6 at about 0.025.Source: §5 · p. 14, Figure 5.4 · p. 16
6. Conclusion.
The conclusion reports improved accuracy from prediction-correction schemes and arbitrary-order, unconditionally energy-stable HEQ schemes. Numerical comparisons and examples illustrate their performance and broad potential applicability.
6. Conclusion.
- Prediction-correction schemes retain second-order time accuracy with much smaller errors than linear schemes, and attain second-order convergence even with a large time step.The conclusion describes this as the first of the two strategies.Source: §6 · p. 16
- Fig. 5.6 depicts phase-field-crystal growth dynamics at t = 10, 20, 30, 50, 100, and 1000.The figure presents the dynamics at six times.Source: Figure 5.6 · p. 17
- In MBE, ICN and HEQ Gaussian collocation predict fairly accurate free-energy profiles at relatively large time steps, while LCN and BDF6 require finer steps for similar accuracy.The comparison is shown in Fig. 5.7.Source: Figure 5.7 · p. 17
- The HEQ schemes use quadratic-invariant-preserving Runge–Kutta multistage discretization to obtain arbitrary-order, unconditionally energy-stable approximations, with stability established rigorously.The conclusion presents this as the second strategy.Source: §6 · p. 17
- Numerical experiments compare the proposed schemes on several gradient-flow problems to illustrate their accuracy and efficiency.The conclusion lists Cahn–Hilliard, Allen–Cahn, crystal-growth, and molecular-beam-epitaxy models.Source: §6 · p. 17
- The authors state that prediction-correction and Runge–Kutta-based high-order schemes can be applied to a large class of thermodynamically consistent models, and that the strategies can also be used with SAV reformulations.Source: §6 · p. 18