Asymptotic analysis of the Allen–Cahn equation in expanding cylindrical domains

We are concerned with a special form of the (stochastic) Allen-Cahn equation, which can be seen as a model of hybrid zones in population genetics. Individuals in the population can be of one of three types; aaare fitter than AA, and both are fitter than the aA heterozygotes. The hybrid zone is the region separating a subpopulation consisting entirely of aa individuals from one consisting of AA individuals. We investigate the interplay between the motion of the hybrid zone and the shape of the habitat, both with and without genetic drift (corresponding to stochastic and deterministic models respectively). In the deterministic model, we investigate the effect of a wide opening and provide some explicit sufficient conditions under which the spread of the advantageous type is halted, and complementary conditions under which it sweeps through the whole population. As a standing example, we are interested in the outcome of the advantageous population passing through an isthmus. We also identify rather precise conditions under which genetic drift breaks down the structure of the hybrid zone, complementing previous work that identified conditions on the strength of genetic drift under which the structure of the hybrid zone is preserved. Our results demonstrate that, even in cylindrical domains, it can be misleading to caricature allele frequencies by one-dimensional travelling waves, and that the strength of genetic drift plays an important role in determining the fate of a favoured allele.

A reduced order method for Allen–Cahn equations

An unconditionally stable scheme for the Allen–Cahn equation with high-order polynomial free energy

Allen Cahn (AC) equation is highly nonlinear due to the presence of cubic term and also very stiff; therefore, it is not easy to find its exact analytical solution in the closed form. In the present work, an approximate analytical solution of the AC equation has been investigated. Here, we used the variational iteration method (VIM) to find approximate analytical solution for AC equation. The obtained results are compared with the hyperbolic function solution and traveling wave solution. Results are also compared with the numerical solution obtained by using the finite difference method (FDM). Absolute error analysis tables are used to validate the series solution. A convergent series solution obtained by VIM is found to be in a good agreement with the analytical and numerical solutions.

ABSTRACTIn this paper, we propose a two‐grid finite element method for solving the time‐fractional Allen–Cahn equation with the logarithmic potential. Firstly, with the L1 method to approximate Caputo fractional derivative, we solve the fully discrete time‐fractional Allen–Cahn equation on a coarse grid with mesh size and time step size . Then, we solve the linearized system with the nonlinear term replaced by the value of the first step on a fine grid with mesh size and the same time step size . We obtain the energy stability of the two‐grid finite element method and the optimal order of convergence of the two‐grid finite element method in the L2 norm when the mesh size satisfies . The theoretical results are confirmed by arithmetic examples, which indicate that the two‐grid finite element method can keep the same convergence rate and save the CPU time.

In this paper, a wavelet-based approximation method is introduced for solving the Newell-Whitehead (NW) and Allen-Cahn (AC) equations. To the best of our knowledge, until now there is no rigorous Legendre wavelets solution has been reported for the NW and AC equations. The highest derivative in the differential equation is expanded into Legendre series, this approximation is integrated while the boundary conditions are applied using integration constants. With the help of Legendre wavelets operational matrices, the aforesaid equations are converted into an algebraic system. Block pulse functions are used to investigate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence of the proposed methods is proved. Finally, we have given some numerical examples to demonstrate the validity and applicability of the method.

This study focuses on solving the Allen-Cahn (AC) equation with consideration of varying phase boundary thickness using Physics-Informed Neural Networks (PINN). Traditional numerical methods often exhibit declining computational efficiency and accuracy as problem complexity increases when handling complex phase boundary conditions; PINNs offer a novel approach to addressing such challenges. The study first conducts a detailed derivation of the AC equation and constructs a PINN-based solution model, including three key steps: designing a specific neural network architecture; building a loss function composed of partial differential equation (PDE) residual terms, boundary condition residual terms, and initial condition residual terms; and performing model training. The computational results demonstrate that the established PINN model can effectively solve the AC equation with varying phase boundary thicknesses, accurately revealing the distribution patterns of the order parameter under different phase boundary thicknesses, and adapt to the challenges brought by changes in phase boundary thickness by adjusting the number of iterations to achieve stable and efficient numerical solutions. The phase boundary thickness parameter has a significant impact on order parameter distribution, computational stability, and convergence. Additionally, a discussion is conducted between PINN and the finite difference method. This research provides support for in-depth understanding of the physical connotation of the AC equation and optimization of numerical calculation methods, and also lays a foundation for the application of PINN in solving similar complex problems in fields such as materials science and condensed matter physics. Future work can further expand the application of PINN in high-dimensional phase-field models and multi-physics coupling problems, combine other technologies to improve the accuracy and efficiency of phase-field simulations, and carry out more studies on practical application cases. 本研究聚焦于物理信息神经网络（PINN）求解不同相界厚度变化的Allen-Cahn（AC）方程。传统数值方法在处理复杂相边界条件时，其计算效率与精确度常随问题复杂度而下降；PINN为解决此类问题提供了新途径。本研究首先详细推导了AC方程，进而构建了基于PINN的求解模型，包括设计特定的神经网络架构、构建由偏微分方程残差项、边界条件残差项和初始条件残差项所组成的损失函数，并进行模型训练。计算结果表明，所建立的PINN模型能有效求解不同相界厚度变化的AC方程，准确揭示序参数在不同相界厚度下的分布规律，且能通过改变迭代次数适应相界厚度变化带来的挑战，实现稳定高效的数值求解。此外，将PINN计算结果与有限差分法进行了对比。本研究为深入理解AC方程物理内涵、优化数值计算方法提供支持，也为PINN在材料科学、凝聚态物理等领域解决类似复杂问题奠定基础。未来可进一步拓展PINN在高维相场模型、多物理场耦合问题中的应用，结合其他技术提高相场模拟精确度和效率，并开展更多实际应用案例研究。

Operator splitting based structure-preserving numerical schemes for the mass-conserving convective Allen-Cahn equation

Morse index of layered solutions to the heterogeneous Allen–Cahn equation

This paper develops an a posteriori error estimate of residual type for finite element approximations of the Allen--Cahn equation ut ? ?u+ ??2 f(u)=0. It is shown that the error depends on ??1 only in some low polynomial order, instead of exponential order. Based on the proposed a posteriori error estimator, we construct an adaptive algorithm for computing the Allen--Cahn equation and its sharp interface limit, the mean curvature flow. Numerical experiments are also presented to show the robustness and effectiveness of the proposed error estimator and the adaptive algorithm.

The Allen--Cahn equation is one of fundamental equations of phase-field models, while the logarithmic Flory--Huggins potential is one of the most useful energy potentials in various phase-field models. In this paper, we consider numerical schemes for solving the Allen--Cahn equation with logarithmic Flory--Huggins potential. The main challenge is how to design efficient numerical schemes that preserve the maximum principle and energy dissipation law due to the strong nonlinearity of the energy potential function. We propose a novel energy factorization approach with the stability technique, which is called stabilized energy factorization approach, to deal with the Flory--Huggins potential. One advantage of the proposed approach is that all nonlinear terms can be treated semi-implicitly and the resultant numerical scheme is purely linear and easy to implement. Moreover, the discrete maximum principle and unconditional energy stability of the proposed scheme are rigorously proved using the discrete variational principle. Numerical results are presented to demonstrate the stability and effectiveness of the proposed scheme.

Numerical solutions of the Allen–Cahn equation with the [formula omitted]-Laplacian

Stable second-order schemes for the space-fractional Cahn–Hilliard and Allen–Cahn equations

We present the image segmentation model using the modified Allen–Cahn equation with a fractional Laplacian. The motion of the interface for the classical Allen–Cahn equation is known as the mean curvature flows, whereas its dynamics is changed to the macroscopic limit of Lévy process by replacing the Laplacian operator with the fractional one. To numerical implementation, we prove the unconditionally unique solvability and energy stability of the numerical scheme for the proposed model. The effect of a fractional Laplacian operator in our own and in the Allen–Cahn equation is checked by numerical simulations. Finally, we give some image segmentation results with different fractional order, including the standard Laplacian operator.

Numerical study of two operator splitting localized radial basis function method for Allen–Cahn problem