Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation

Li, Qi; Mei, Liquan; Yang, Xiaofeng; Li, Yibao

doi:10.1007/s10444-019-09678-w

Cited by 54 publications

(17 citation statements)

References 27 publications

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“…For simplicity, we take periodic boundary conditions for the system (2)-(3). The periodic boundary conditions have been used in lots of numerical analysis for the PFC model [6,10,24,27,48,51,53,58] and the MPFC model [1,2,9,23,25,29,30,49,50,52]. The initial condition of the model ( 2…”

Section: Governing System and Its Energy Lawmentioning

confidence: 99%

“…We first simulate the phase transition without vacancies (i.e., h vac = 0) in 2D. It is a benchmark simulation for the PFC model or MPFC model and similar numerical experiments could also be found in Reference [29,30,68]. We set the computation domain Ω = [0, 32] 2 , the parameters M = 1, ϵ = 0.8, the final time T = 1000.…”

Section: 21mentioning

confidence: 99%

“…The authors in Reference [35] developed a stabilized SAV scheme for a new Allen-Cahn type SPFC model. For the MPFC model, many numerical strategies have also been applied to construct energy stable numerical schemes, for instance, the convex splitting methods [1,2,29,52], the IEQ approach [30], the SAV approach [31], and many others [9,23,25]. In contrast, there are few numerical approximations for the VMPFC model.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Governing System and Its Energy Lawmentioning

confidence: 99%

Section: 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…Stability of different numerical approaches including backward Euler, high-order BDF, Fourier methods, and semi-implicit convex-splitting are carefully considered in works. 34,35…”

Section: Numerical Solution Of Mpfc Equationmentioning

confidence: 99%

About one unified description of the first‐ and second‐order phase transitions in the phase‐field crystal model

Ankudinov

Starodumov

Galenko

2020

Math Methods in App Sciences

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Modeling of crystal micro-structures and their dynamics during fast phase transitions can be performed by the phase-field crystal (PFC) model in the hyperbolic formulation (Modified Phase Field Crystal [MPFC] model). This method is suitable for a continual modeling of the atomic density field at diffusion time intervals (slow diffusion dynamics) and short intervals of atomic flux relaxation (fast structural relaxation). Since the PFC model describes transitions of the first and second order, we present a description of both transitions in a unified manner. We show how phase transitions of each order can be treated using specific analytical transformations. To justify the unified approach to description of the first-and second-order transformations, we provide results of numerical simulation of phase changes between homogeneous structure and Body Centered Cubic (BCC) crystal lattice. The set of benchmarks for different domains shows coincidence of instantaneous atomic distributions and free energies in both forms.

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“…Many efforts have been done in order to resolve this problem (cf. [4,7,10,13,15,14,22,26,28,34,27]).…”

mentioning

confidence: 99%

Accurate and efficient algorithms with unconditional energy stability for the time fractional Cahn–Hilliard and Allen–Cahn equations

Liu

Huang

2021

Numerical Methods Partial

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Comparing with the classic phase filed models, the fractional models such as time fractional Allen-Cahn and Cahn-Hilliard equations are equipped with Caputo fractional derivative and can describe more practical phenomena for modeling phase transitions. In this paper, we construct two accurate and efficient linear algorithms for the time fractional Cahn-Hilliard and Allen-Cahn equations with general nonlinear bulk potential. The main contribution is that we have proved the unconditional energy stability for the time fractional Cahn-Hilliard and Allen-Cahn models and their semi-discrete schemes carefully and rigorously. Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes.

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Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation

Cited by 54 publications

References 27 publications

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

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

About one unified description of the first‐ and second‐order phase transitions in the phase‐field crystal model

Accurate and efficient algorithms with unconditional energy stability for the time fractional Cahn–Hilliard and Allen–Cahn equations

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