Abstract
The Allen-Cahn model is discussed mainly in the phase field simulation. The compact difference method will be used to numerically approximate the two-dimensional nonlinear Allen-Cahn equation with initial and boundary value conditions, and then, a fully discrete compact difference scheme with second-order accuracy in time and fourth-order in space is established. And its numerical solution satisfies the discrete maximum principle under the constraints of reasonable space and time steps. On this basis, the energy stability of the scheme is investigated. Finally, numerical examples are given to illustrate the theoretical results.
Highlights
The phase field problem is a mathematical model described by partial differential equations
The numerical simulation of the phase field has always been an important field of research at home and abroad because of its important theoretical and practical significance
In 1979, the Allen-Cahn equation is considered to describe the antiphase boundary of the crystal movement by Allen and Cahn, which describes fluid dynamics problems and reaction diffusion problems in materials science, and the same model on the study of many diffusion phenomena is proposed such as the competition and repulsion of biological populations and the migration process of river beds
Summary
The phase field problem is a mathematical model described by partial differential equations. For describing the motion of the antiphase boundary in the crystal, since this type of phase field model does not have an accurate solution, different numerical methods are used to simulate. The compact difference method is applied to approximate the two-dimensional nonlinear Allen-Cahn equations with initial boundary conditions numerically. Based on the existing finite difference methods and inspired by reference [2], a compact difference scheme with second-order accuracy in time and fourth-order accuracy in space is established for the two-dimensional Allen-Cahn equation. A compact difference scheme will be establish for the following two-dimensional Allen-Cahn equation as. Ð30Þ (b) C is positive definite, such as the spatial derivative is discretized by the central finite difference scheme for the two-dimensional Allen-Cahn equation, the D2 is expressed as the corresponding discrete matrix, and UT CU > 0: ð37Þ. Substituting the matrix D2 and C into (30), we get the compact difference scheme:
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