Efficient numerical scheme for solving the Allen‐Cahn equation

This article presents an efficient and robust algorithm for the numerical solution of the Allen‐Cahn equation, which represents the motion of antiphase boundaries. The proposed algorithm is based on the diagonally implicit fractional‐step θ − scheme for time discretization and the conforming finite...

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Bibliographic Details
Published in:Numerical methods for partial differential equations 2018-09, Vol.34 (5), p.1820-1833
Main Authors: Shah, Abdullah, Sabir, Muhammad, Qasim, Muhammad, Bastian, Peter
Format: Article
Language:English
Subjects:
Algorithms
Allen‐Cahn equation
Antiphase boundaries
diagonally implicit fractional‐step θ − scheme
Discretization
DUNE‐PDELab
Finite element method
interfacial dynamics
Nonlinear differential equations
Nonlinear equations
Nonlinear programming
Partial differential equations
Robustness (mathematics)
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Summary:This article presents an efficient and robust algorithm for the numerical solution of the Allen‐Cahn equation, which represents the motion of antiphase boundaries. The proposed algorithm is based on the diagonally implicit fractional‐step θ − scheme for time discretization and the conforming finite element method for space discretization. For the steady‐state solution, both uniform and adaptive grids are used to illustrate the effectiveness of adaptive grids over uniform grids. For the unsteady solution, the diagonally implicit fractional‐step θ − scheme is compared with other time discretization schemes in terms of computational cost and temporal error estimation accuracy. Numerical examples are presented to illustrate the capabilities of the proposed algorithm in solving nonlinear partial differential equations.
ISSN:0749-159X
1098-2426
DOI:10.1002/num.22255