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Numerical Approximations for a three components Cahn-Hilliard phase-field Model based on the Invariant Energy Quadratization method
Xiaofeng Yang, Jia Zhao, Qi Wang, Jie Shen
TL;DR
The paper addresses the challenge of designing efficient, energy-stable schemes for the nonlinearly coupled three-component Cahn–Hilliard model. It applies Invariant Energy Quadratization to build first- and second-order temporal schemes with semi-explicit nonlinear treatment, yielding well-posed symmetric positive definite linear systems and unconditional energy stability, supported by 2D and 3D simulations.
Problem
Efficient schemes for the three-component Cahn–Hilliard model must preserve energy stability despite nonlinear coupling, while existing approaches may be low-order, unstable, or highly nonlinear.
Method
The paper applies the Invariant Energy Quadratization approach to construct first- and second-order temporal schemes treating nonlinear terms semi-explicitly.
Results
The proposed schemes are unconditionally energy stable and lead to well-posed symmetric positive definite linear systems at each time step, with 2D and 3D simulations demonstrating stability and accuracy.
Takeaways & Limitations
The schemes combine temporal accuracy, unconditional energy stability, and efficient linear-system solution for the three-component phase-field model.
Takeaways & Limitations
A related variable transformation works only for partial spreading; total spreading requires the paper’s alternative variable because the transformed energy may not be bounded below.
Abstract
from arXiv · showhide
How to develop efficient numerical schemes while preserving the energy stability at the discrete level is a challenging issue for the three component Cahn-Hilliard phase-field model. In this paper, we develop first and second order temporal approximation schemes based on the "Invariant Energy Quadratization" approach, where all nonlinear terms are treated semi-explicitly. Consequently, the resulting numerical schemes lead to a well-posed linear system with the symmetric positive definite operator to be solved at each time step. We rigorously prove that the proposed schemes are unconditionally energy stable. Various 2D and 3D numerical simulations are presented to demonstrate the stability and the accuracy of the schemes.
1. Introduction.
Three-component Cahn–Hilliard modeling introduces coupled phase variables and nonlinear potentials, making efficient, energy-stable time discretization difficult. The paper uses IEQ to construct accurate, unconditionally energy-stable linear schemes.
- 1. Introduction.: Three-component models use three phase variables constrained by c1 + c2 + c3 = 1.The bulk free energy sums phase-wise double-well terms and may include a sixth-order coupling polynomial.
- 1. Introduction.: Nonlinear coupling makes it challenging to preserve energy stability while developing efficient numerical schemes.Existing methods are often first-order, energy unstable, highly nonlinear, or combinations of these properties.
- 1. Introduction.: IEQ is adopted to develop stable and more efficient linear schemes for the three-component system.The approach addresses nonlinear potentials while confronting the model’s Lagrange multiplier and sixth-order polynomial terms.
- 1. Introduction.: The proposed schemes are accurate to second order or higher in time, unconditionally energy stable, and produce symmetric positive definite linear systems.The paper also states that the schemes are efficient and easy to implement.
- 1. Introduction.: The paper includes 2D and 3D numerical simulations to validate the accuracy and efficiency of the proposed schemes.The schemes’ energy stability and symmetric positivity are also addressed analytically in the time-discrete case.
2. Model System.
The model represents three-component mixtures with linked phase variables, surface-tension-dependent free energy, and Cahn–Hilliard dynamics. Its energy dissipates over time, while the numerical section targets linear, first- and second-order schemes that remain unconditionally energy stable.
- Model formulation: The three phase variables satisfy the hyperplane link condition c1 + c2 + c3 = 1.This condition defines the admissible state space for the vector C = (c1, c2, c3).
- Model formulation: The free energy combines gradient terms, a nonlinear bulk potential, and surface-tension-dependent coefficients that may be negative.The entropic coefficients Σi can be negative in specific situations.
- Model formulation: The bulk potential is rewritten as F(c1, c2, c3) = F0(c1, c2, c3) + P(c1, c2, c3), with P containing higher-order coupling terms.The model introduces a nonnegative constant Λ in the potential construction.
- Model formulation: The phase evolution follows an H^-1 gradient flow, with a Lagrange multiplier enforcing the hyperplane constraint.The multiplier β is introduced to ensure c1 + c2 + c3 = 1.
- Model formulation: The model permits periodic or no-flux boundary conditions for the phase variables and chemical potentials.The stated no-flux conditions are ∂nci = 0 and ∇µi · n = 0 on the boundary.
- Energy properties: The bulk free energy is bounded from below on the hyperplane under the stated conditions, and its lower bound depends on Σ1, Σ2, Σ3, and Λ.For total spreading, Λ must be nonzero and sufficiently large to ensure non-negativity of F.
- Energy properties: The continuous model satisfies a dissipative energy law, and the numerical section develops first- and second-order linear schemes preserving unconditional energy stability.The discretization must handle the double-well term, sixth-order polynomial, and Lagrange multiplier, including cases with Σi < 0.
3. Numerical schemes.
The paper develops IEQ-based first- and second-order schemes for the three-component Cahn–Hilliard model, targeting linear solvability, unconditional energy stability, and temporal accuracy. The transformed formulation supports discrete energy laws and symmetric positive-definite linear systems, while addressing limitations of convex-splitting and stabilization approaches.
- Motivation: Convex splitting and stabilization approaches are difficult to extend to the three-component model because of the sixth-order coupling, uncertain maximum-principle validity, and limited unconditional stability at second order.These approaches may also produce nonlinear systems or require modified potentials that can generate spurious solutions when the maximum principle does not hold.
- Transformed system: The IEQ approach rewrites the bounded-below free energy using an auxiliary variable U, producing an equivalent closed PDE system without requiring convexity splitting or a maximum principle.A constant B is introduced so that F(c1, c2, c3)+B > 0, enabling the auxiliary-variable transformation.
- Transformed system: The transformed system preserves the original continuous energy law because integrating the auxiliary-variable equation recovers the original formulation.The numerical schemes instead follow the transformed system’s discrete energy law.
- First-order scheme: The first-order backward-Euler scheme treats nonlinear coefficients explicitly and leads to a linear system whose operator is symmetric, self-adjoint, and positive definite.The explicit treatment of Hi is intended to accelerate computation while retaining a well-posed linear solve.
- First-order scheme: The first-order scheme is unconditionally energy stable and satisfies a discrete energy dissipation law.Its discrete energy includes gradient contributions from the three phase variables and a quadratic contribution from U.
- Scope and limitation: The alternative transformed energy used in one scheme is valid only for partial spreading; total spreading can make that energy unbounded below.This limitation motivates the paper’s definition of the auxiliary variable U in the main IEQ formulation.
- Second-order schemes: Second-order Crank–Nicolson and BDF schemes are also unconditionally energy stable, with discrete energy laws providing second-order approximations to the continuous dissipation law.The Crank–Nicolson and BDF formulations retain self-adjoint positive-definite linear systems, and the discrete energy law is shown to approximate the continuous law to second order.
4. Numerical Simulations.
The simulations test temporal accuracy and demonstrate the schemes’ behavior for liquid-lens equilibria, spinodal decomposition, and energy decay in two and three dimensions.
- Accuracy test: The accuracy tests use default parameters η = 0.03, B = 2, M0 = 10^-6, and Λ = 7 with 128^2 spatial grid points.The schemes are denoted LS1, LS2-CN, and LS2-BDF for the first-order and two second-order formulations.
- Accuracy test: The LS1, LS2-CN, and LS2-BDF schemes asymptotically achieve first-order, second-order, and second-order temporal accuracy, respectively.Errors are evaluated at t = 1 using adjacent time steps with L2, L1, and L∞ norms.
- Liquid lens between two stratified fluids: Changing surface tensions produces the predicted unequal contact angles, while total spreading cases yield configurations without junction points and complete spreading of selected components.For σ12 = 3, σ13 = σ23 = 1, component c3 spreads into a layer; for σ12 = 1, σ13 = 1, σ23 = 3, c1 and c3 spread while c2 remains enclosed.
- Spinodal decomposition in 2D: The free-energy curves decay over time in the simulated cases, confirming unconditional stability of the algorithms in the reported two- and three-dimensional tests.Three-dimensional simulations likewise show component accumulation and decaying free energy.
5. Concluding Remarks.
The proposed schemes combine second-order temporal accuracy, unconditional energy stability, and linear solves with symmetric positive definite systems. Their linear structure supports efficient iterative solution and extends toward broader three-phase applications.
- For the LS2-CN algorithm, the free-energy functional is tracked as log10(total energy) over time for two surface-tension parameter sets.The figure uses δt = 0.001 and compares parameter sets (1, 0.8, 1.4) and (1, 1, 1).
- The schemes achieve up to second-order temporal accuracy while remaining unconditionally energy stable and requiring only linear equations at each step.The resulting systems are symmetric positive definite and can be solved efficiently with Krylov subspace methods and suitable preconditioners.
- The resulting symmetric positive definite linear systems enable efficient Krylov-subspace solution with suitable preconditioners.
- The schemes are reported as the first linear and unconditionally energy-stable schemes for the three-component Cahn–Hilliard phase-field model.The authors also state that they can be applied to the hydrodynamically coupled three-phase model without essential difficulty.