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

DOI10.1016/j.camwa.2017.07.012zbMath1409.82014arXiv1611.06288OpenAlexW2553175992MaRDI QIDQ1732459

Lixiu Dong, Cheng Wang, Wenqiang Feng, Steven M. Wise, Zheng-Ru Zhang

Publication date: 25 March 2019

Published in: Computers \& Mathematics with Applications (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1611.06288

zbMATH Keywords

convergence analysisfinite differenceenergy stabilitynonlinear multigrid solversecond order numerical schemethree-dimensional phase field crystal

Mathematics Subject Classification ID

Statistical mechanics of crystals (82D25) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) Dynamic and nonequilibrium phase transitions (general) in statistical mechanics (82C26)

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Uses Software

• Matlab

Cites Work

• An \(H^2\) convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation
• An unconditionally energy-stable method for the phase field crystal equation
• Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation
• High accuracy solutions to energy gradient flows from material science models
• Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations
• An adaptive time-stepping strategy for solving the phase field crystal model
• A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation
• Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations
• Global smooth solutions of the three-dimensional modified phase field crystal equation
• An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations
• Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation
• Two finite difference schemes for the phase field crystal equation
• An efficient algorithm for solving the phase field crystal model
• Advanced operator splitting-based semi-implicit spectral method to solve the binary phase-field crystal equations with variable coefficients
• A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-Cartesian grids
• Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms
• Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model
• Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models
• A multigrid finite element solver for the Cahn-Hilliard equation
• The numerical simulation of the phase field crystal (PFC) and modified phase field crystal (MPFC) models via global and local meshless methods
• Energy stable and convergent finite element schemes for the modified phase field crystal equation
• Convergence Analysis of a Second Order Convex Splitting Scheme for the Modified Phase Field Crystal Equation
• An Energy Stable and Convergent Finite-Difference Scheme for the Modified Phase Field Crystal Equation
• Local Discontinuous Galerkin Method and High Order Semi-Implicit Scheme for the Phase Field Crystal Equation
• Finite-Element Approximation of the Nonstationary Navier–Stokes Problem. Part IV: Error Analysis for Second-Order Time Discretization
• An Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation
• Spectral Methods in MATLAB
• An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System
• Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation
• Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach