Efficient and linear schemes for anisotropic Cahn–Hilliard model using the Stabilized-Invariant Energy Quadratization (S-IEQ) approach

Xu, Zhen; Yang, Xiaofeng; Zhang, Hui; Xie, Zhongliang

doi:10.1016/j.cpc.2018.12.019

Cited by 49 publications

(11 citation statements)

References 36 publications

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“…The second-order backward differentiation formula (BDF) finite difference schemes in [8-10, 17, 52] are also based on a convex splitting and stabilisation. For the anisotropic Cahn-Hilliard equation, the stabilisation terms used in IEQ approach [50] and SAV approach [3] aim to suppress non-physical spatial oscillations caused by the strong anisotropy. Besides, the stabilised-IEQ (S-IEQ) approach is considered in [62,63] and stabilised-SAV (S-SAV) approach in [64,65].…”

Section: Introductionmentioning

confidence: 99%

Modeling and Simulation of a Ternary System for Macromolecular Microsphere Composite Hydrogels

Ji¹

2021

EAJAM

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In this paper, we study a ternary system for macromolecular microsphere composite (MMC) hydrogels. Assuming that the graft chains are distributed randomly around the macromolecular microspheres, a phase transition model was constructed. The stabilised-scalar auxiliary variable (S-SAV) approach is used to present a first-order energy stable scheme for solving the nonlinear system. Some numerical experiments are carried out to show the accuracy of the scheme, including the mass conservation of the volume fractions, the decrease in the modified energy, and the influence of different parameters.

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Section: Introductionmentioning

confidence: 99%

Modeling and Simulation of a Ternary System for Macromolecular Microsphere Composite Hydrogels

Ji¹

2021

EAJAM

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“…In Figure 18, we plot the time evolution of the original energy (9), the modified energy (11), the discrete energy (55) and the mass, which shows that these three energy functions match very well and they all decay with time while the mass conserves with time. We set the parameters 𝛼 = 1, 𝛽 = 1, 𝜀 = 0.9, M = 1, h vac = 3000, S = 3000, the final time T = 200.…”

Section: Crystal Growth Without Vacancies In 2dmentioning

confidence: 95%

“…Using the IEQ approach and the stabilized strategies (S-IEQ approach), we obtain three time discretization schemes based on the first-order Euler method, the BDF2 and the second-order Crank-Nicolson method, respectively. The S-IEQ approach has been applied to many phase field models [55,57,66,67,[69][70][71]. Thanks to the IEQ approach, the nonlinear cubic term and the complicated nonlinear potential are transformed into a simple quadratic form about a new variable.…”

Section: Introductionmentioning

confidence: 99%

Efficient unconditionally stable numerical schemes for a modified phase field crystal model with a strong nonlinear vacancy potential

Pei

Hou

Yan

2021

Numerical Methods Partial

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We consider numerical approximations for a modified phase field crystal model with a strong nonlinear vacancy potential. Based on the invariant energy quadratization approach and stabilized strategies, we develop linear, unconditionally energy stable numerical schemes using the first-order Euler method, the second-order backward differentiation formulas and the second-order Crank-Nicolson method, respectively. We rigorously prove the unconditional energy stability, the mass conservation of these three numerical schemes and carry out error estimates in time for the first-order numerical scheme. Various numerical experiments in 2D and 3D are carried out to validate the accuracy, energy stability, mass conservation, and efficiency of the proposed schemes.

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“…Although the Lagrange multiplier approach makes a great contribution for the energy stability, it will causes the lack of accuracy at large time steps because the new auxiliary variable will affects the coefficient matrix. To overcome this disadvantage, some researchers (Xu et al 2019;Zhang and Yang 2019) added an extra stabilized term to suppress the bad influence of auxiliary variable and explicit nonlinear term. By combing the Lagrange multiplier approach and the idea of stabilized method, we propose a stabilized Lagrange multiplier method for the SH equation, where the Crank-Nicolson (CN) temporal discretization is used to construct second-order scheme.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation and analysis of the Swift–Hohenberg equation by the stabilized Lagrange multiplier approach

Yang

Kim

2021

Comp. Appl. Math.

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In this work, we develop linear, first-and second-order accurate, and energy-stable numerical scheme for the Swift-Hohenberg (SH) equation. An auxiliary variable (Lagrange multiplier) is used to control the nonlinear term so that the linear temporal scheme can be easily constructed. To further achieve the accuracy with large time steps, a proper stabilized term is adopted. For the first-order time-accurate scheme, the backward Euler approximation is adopted. The Crank-Nicolson (CN) and explicit Adams-Bashforth (AB) approximations are applied to achieve temporally second-order accuracy. We analytically perform the estimations of the semi-discrete solvability, the energy stability with respect to the original and pseudo-energy functionals, and the convergence error. To numerically solve the resulting discrete system of equations, we use an efficient linear multigrid method. We present various two-(2D) and three-dimensional (3D) computational examples to demonstrate the accuracy and energy stability of the proposed scheme.

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Efficient and linear schemes for anisotropic Cahn–Hilliard model using the Stabilized-Invariant Energy Quadratization (S-IEQ) approach

Cited by 49 publications

References 36 publications

Modeling and Simulation of a Ternary System for Macromolecular Microsphere Composite Hydrogels

Modeling and Simulation of a Ternary System for Macromolecular Microsphere Composite Hydrogels

Efficient unconditionally stable numerical schemes for a modified phase field crystal model with a strong nonlinear vacancy potential

Numerical simulation and analysis of the Swift–Hohenberg equation by the stabilized Lagrange multiplier approach

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