Two finite difference schemes for the phase field crystal equation

Cao, Hai-Yan; Sun, Zhi‐zhong

doi:10.1007/s11425-015-5025-1

Cited by 11 publications

(4 citation statements)

References 15 publications

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“…Another strategy is to discretize directly the fourth-order derivatives using the second-order accurate center-difference formula. Numerical tests imply that the derived schemes will be satisfactory as long as the spatial mesh size is sufficiently small, see, e.g., [11,12] for the Cahn-Hilliard dynamics, [28,32] for the molecular beam epitaxy models and [2,38] for the phase field crystal equation. In addition, discretizing the fourth-order derivatives using a fourth-order compact difference scheme could be found in a recent work [20].…”

Section: Introductionmentioning

confidence: 99%

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

2016

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In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first-and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.

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Section: Introductionmentioning

confidence: 99%

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

2016

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“…Tegze et al [60] developed a semi-implicit spectral scheme for the binary PFC equations that is not expected to unconditionally stable. Also see other related numerical works [5,13,42,45] in recent years.…”

Section: Introductionmentioning

confidence: 84%

Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation

Dong

Feng

Cheng

et al. 2018

Computers & Mathematics with Applications

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In this paper we analyze and implement a second-order-in-time numerical scheme for the three-dimensional phase field crystal (PFC) equation. The numerical scheme was proposed in [46], with the unique solvability and unconditional energy stability established. However, its convergence analysis remains open. We present a detailed convergence analysis in this article, in which the maximum norm estimate of the numerical solution over grid points plays an essential role. Moreover, we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain, and various three-dimensional numerical results are presented, including the numerical convergence test, complexity test of the multigrid solver and the polycrystal growth simulation.

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“…The convex splitting paradigm, popularized by Eyre's work [16], is a well-known approach to achieve unconditional numerical energy stability and unconditional unique solvability, at the cost of extra numerical dissipation. The approach has successfully applied to the original PFC equation (2.2) and the modified version; see the related references [5,6,10,19,21,25,42,45,48], et cetera. Meanwhile, for the PFC amplitude equations (2.12) -(2.15), there has been no theoretical justification of the unique solvability and energy stability in the existing literature.…”

Section: Energy Dissipation and Convexity Analysis Of The Amplitude Ementioning

confidence: 99%

An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations

Guan

Heinonen

Lowengrub

et al. 2016

Journal of Computational Physics

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Please cite this article in press as: Z. Guan et al., An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations, J. Comput. Phys. (2016), http://dx. AbstractIn this paper we construct an energy stable finite difference scheme for the amplitude expansion equations for the two-dimensional phase field crystal (PFC) model. The equations are formulated in a periodic hexagonal domain with respect to the reciprocal lattice vectors to achieve a provably unconditionally energy stable and solvable scheme. To our knowledge, this is the first such energy stable scheme for the PFC amplitude equations. The convexity of each part in the amplitude equations is analyzed, in both the semi-discrete and fully-discrete cases. Energy stability is based on a careful convexity analysis for the energy (in both the spatially continuous and discrete cases). As a result, unique solvability and unconditional energy stability are available for the resulting scheme. Moreover, we show that the scheme is point-wise stable for any time and space step sizes. An efficient multigrid solver is devised to solve the scheme, and a few numerical experiments are presented, including grain rotation and shrinkage and grain growth studies, as examples of the strength and robustness of the proposed scheme and solver.

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Two finite difference schemes for the phase field crystal equation

Cited by 11 publications

References 15 publications

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation

An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations

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