Convergence Analysis and Numerical Implementation of a Second Order Numerical Scheme for the Three-Dimensional Phase Field Crystal Equation

Dong, Lixiu; Feng, Wenqiang; Wang, Cheng; Wise, Steven M.; Zhang, Zhengru

doi:10.48550/arxiv.1611.06288

Cited by 2 publications

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“…, where I f c is a bilinear interpolation operator (with the Nearest Neighbor Interpolation applied in Matlab, which is similar to the 2D case in [6,7] and the 3D case in [5]). This requires a relatively coarse solution, parametrized by h c , and a relatively fine solution, parametrized by h f , in particular h c = 2h f , at the same final time.…”

Section: Convergence Test and The Complexity Of The Preconditioned So...mentioning

confidence: 99%

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

Feng

Cheng²,

Wise

et al. 2018

Numerical Methods Partial

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In this article, we study a new second‐order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for higher‐order‐in‐time temporal discretizations is how to ensure an unconditional energy stability without compromising numerical efficiency or accuracy. We propose a framework for designing a second‐order numerical scheme with unconditional energy stability using the BDF method with constant coefficient stabilizing terms. Based on the unconditional energy stability property that we establish, we derive an ℓ ∞ ( 0 , T ; H h 2 ) stability for the numerical solution and provide an optimal convergence analysis. To deal with the highly nonlinear four‐Laplacian term at each time step, we apply efficient preconditioned steepest descent and preconditioned nonlinear conjugate gradient algorithms to solve the corresponding nonlinear system. Various numerical simulations are presented to demonstrate the stability and efficiency of the proposed schemes and solvers. Comparisons with other second‐order schemes are presented.

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Section: Convergence Test and The Complexity Of The Preconditioned So...mentioning

confidence: 99%

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

Feng

Cheng²,

Wise

et al. 2018

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

show abstract

A Second-Order Energy Stable Backward Differentiation Formula Method for the Epitaxial Thin Film Equation with Slope Selection

Feng¹,

Wang²,

Wise³

et al. 2017

Preprint

View full text Add to dashboard Cite

Convergence Analysis and Numerical Implementation of a Second Order Numerical Scheme for the Three-Dimensional Phase Field Crystal Equation

Cited by 2 publications

References 1 publication

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

A Second-Order Energy Stable Backward Differentiation Formula Method for the Epitaxial Thin Film Equation with Slope Selection

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