An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

An implicit backward differential formula (BDF) scheme is constructed for nonlinear reaction‐diffusion equation, and superconvergence results are studied with the Galerkin finite element method (FEM). The existence and uniqueness of the numerical solution are given by using of the function's monotonicity. Splitting technique is utilized to get rid of the ratio between the time step τ and the subdivision parameter h. Temporal error estimates in H2‐norm are derived, which implies the boundedness of the solutions about the time‐discrete equations in H2‐norm. Unconditional spatial error estimates in L2‐norm are deduced, which help bound the numerical solutions in L∞‐norm. The unconditional superconvergent property of un in H1‐norm with order Ofalse(h2+τ2false) is obtained by the relationship between the classic Ritz projection operator and the the corresponding interpolation operator. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.

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“…Comparing with the method in previous studies, [33][34][35][36][37] we derive the energy stability as follows:…”

Section: Theorem 1 Problem Of System (23)-(25) Is Uniquely Solvablementioning

confidence: 99%

Superconvergence analysis for nonlinear reaction‐diffusion equation with BDF‐FEM

Wang

Shi

2020

Math Meth Appl Sci

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“…In this section, we mainly discuss an efficient algorithm for solving eq. (11). Firstly, we note that the scheme (11) can be reformulated as a closed equation for u n+1 :…”

Section: Linear Iteration Algorithmmentioning

confidence: 99%

“…More extensive applications of energy-stable or energy conservative method to a wide class of physical models also are available. See the related works for wave equations [7,8], the phase field crystal equation [9] and the Cahn-Hilliard equation [10][11][12], etc.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulations of the energy-stable scheme for Swift-Hohenberg equation

Jun

2019

THERM SCI

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A collocation Fourier scheme for Swift-Hohenberg equation based on the convex splitting idea is implemented. To ensure an efficient numerical computation, we propose a general framework with linear iteration algorithm to solve the non-linear coupled equations which arise with the semi-implicit scheme. Following the contraction mapping theorem, we present a detailed convergence analysis for the linear iteration algorithm. Various numerical simulations, including verification of accuracy, dissipative property of discrete energy and pattern formation, are presented to demonstrate the efficiency and the robustness of proposed method.

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“…Therefore, a second-order-accurate in time, energy stable numerical algorithm is highly desired. Instead of the modified Crank-Nicolson approach for the gradient structure, which has been successfully applied to the Cahn-Hilliard [8,11,14,15,16,18,31,32,33,34] and epitaxial thin film equation [44], we make use of a modified backward differentiation formula (BDF) approach. In more details, we apply the second order BDF concept to derive second order temporal accuracy, but modified so that the concave diffusion term is treated by an explicit extrapolation.…”

Section: Introductionmentioning

confidence: 99%

An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

Cheng¹,

Wang²,

Wise³

2019

CICP

Self Cite

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In this paper we propose and analyze an energy stable numerical scheme for the square phase field crystal (SPFC) equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. In particular, a modification of the free energy potential to the standard phase field crystal model leads to a composition of the 4-Laplacian and the regular Laplacian operators. To overcome the difficulties associated with this highly nonlinear operator, we design numerical algorithms based on the structures of the individual energy terms. A Fourier pseudo-spectral approximation is taken in space, in such a way that the energy structure is respected, and summation-by-parts formulae enable us to study the discrete energy stability for such a high-order spatial discretization. In the temporal approximation, a second order BDF stencil is applied, combined with an appropriate extrapolation for the concave diffusion term(s). A second order artificial Douglas-Dupont-type regularization term is added to ensure energy stability, and a careful analysis leads to the artificial linear diffusion coming at an order lower that that of surface diffusion term. Such a choice leads to reduced numerical dissipation. At a theoretical level, the unique solvability, energy stability are established, and an optimal rate convergence analysis is derived in the ∞ (0, T ; 2 ) ∩ 2 (0, T ; H 3 N ) norm. In the numerical implementation, the preconditioned steepest descent (PSD) iteration is applied to solve for the composition of the highly nonlinear 4-Laplacian term and the standard Laplacian term, and a geometric convergence is assured for such an iteration. Finally, a few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

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An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

Cited by 152 publications

References 66 publications

Superconvergence analysis for nonlinear reaction‐diffusion equation with BDF‐FEM

Superconvergence analysis for nonlinear reaction‐diffusion equation with BDF‐FEM

Numerical simulations of the energy-stable scheme for Swift-Hohenberg equation

An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

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