Local Discontinuous Galerkin Methods for the Functionalized Cahn–Hilliard Equation

In this paper, we develop a local discontinuous Galerkin (LDG) method for the sixth order nonlinear functionalized Cahn–Hilliard (FCH) equation. We address the accuracy and stability issues from simulating high order stiff equations in phase-field modeling. Within the LDG framework, various boundary...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 63; no. 3; pp. 913 - 937
Main Authors: Guo, Ruihan, Xu, Yan, Xu, Zhengfu
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2015
Springer Nature B.V
Subjects:
Accuracy
Algorithms
Boundary conditions
Computational Mathematics and Numerical Analysis
Computer simulation
Efficiency
Energy
Finite element method
Galerkin method
Galerkin methods
Linear equations
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Methods
Morphology
Nonlinearity
Numerical analysis
Ordinary differential equations
Partial differential equations
Polymers
Simulation
Solvers
Stability
Temporal logic
Theoretical
Time marching
Functionalized Cahn–Hilliard equation
Stability
Multigrid algorithm
Local discontinuous Galerkin method
ISSN:0885-7474, 1573-7691
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Summary:In this paper, we develop a local discontinuous Galerkin (LDG) method for the sixth order nonlinear functionalized Cahn–Hilliard (FCH) equation. We address the accuracy and stability issues from simulating high order stiff equations in phase-field modeling. Within the LDG framework, various boundary conditions associated with the background physics can be naturally implemented. We prove the energy stability of the LDG method for the general nonlinear case. A semi-implicit time marching method is applied to remove the severe time step restriction ( Δ t ∼ O ( Δ x 6 ) ) for explicit methods. The h - p adaptive capability of the LDG method allows for capturing the interfacial layers and the complicated geometric structures of the solution with high resolution. To enhance the efficiency of the proposed approach, the multigrid (MG) method is used to solve the system of linear equations resulting from the semi-implicit temporal integration at each time step. We show numerically that the MG solver has mesh-independent convergence rates. Numerical simulation results for the FCH equation in two and three dimensions are provided to illustrate that the combination of the LDG method for spatial approximation, semi-implicit temporal integration with the MG solver provides a practical and efficient approach when solving this family of problems.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-014-9920-3