Explicit, linear, energy stable, and maximum principle preserving method for Allen–Cahn equation

In this thesis, we study systems of nonlinear partial differential equations that can be used to describe phase separation. This process can be modeled from both a macroscopic and microscopic point of view. Typically, the macroscopic model contains a local differential operator, which accounts for short-range interactions. In the microscopic model, however, this operator is then replaced by a nonlocal operator, which describes long-range interactions between the particles. This is done by convolution integrals weighted by suitable interaction kernels. In the first part, we analyze the asymptotic behavior of the nonlocal operator in the case, when the corresponding interaction kernel concentrates around the origin suitably. We prove that the nonlocal operator converges to a local differential operator as the parameter in the interaction kernel is sent to zero. More precisely, we even provide concrete rates of convergence depending on this parameter. The analysis is done in the case of sufficiently smooth bounded domains and a large class of interaction kernels. Indeed, the kernel can be either anisotropic or isotropic as well as of $W^{1,1}$-regularity or even more singular. The convergence is shown with respect to the $L^p$-norm, where $p\in[1,\infty)$. The proof is based on localization and perturbation arguments. In the case of regular kernels, we also prove convergence on the torus with respect to the $L^2$-norm. This proof is based on methods from Fourier analysis. In the second part, we focus on models, where two different phases are separated by a diffuse interface. We intend to apply the results from the first part to show that solutions to the nonlocal model converge to a solution to the local model as the parameter in the nonlocal operator is sent to zero. More precisely, we prove strong convergence for the sequence of solutions along with certain rates of convergence. This is done using an energy method. We consider the following models: - Cahn-Hilliard equation - Allen-Cahn equation - Cahn-Hilliard model for tumor growth - Cahn-Hilliard/Navier-Stokes system Finally, we also investigate the case, where both the parameter in the nonlocal operator and the thickness of the diffuse interface are sent to zero suitably. The analysis is done for the nonlocal Allen--Cahn equation on the torus. We first prove a well-posedness result and suitable error estimates for the solutions. Then, we combine the convergence result from the first part together with existing results for the sharp interface limit of the local Allen--Cahn equation.

The nonlocal Allen--Cahn equation, a generalization of the classic Allen--Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, satisfies the maximum principle as its local counterpart. In this paper, we develop and analyze first and second order exponential time differencing schemes for solving the nonlocal Allen--Cahn equation, which preserve the discrete maximum principle unconditionally. The fully discrete numerical schemes are obtained by applying the stabilized exponential time differencing approximations for time integration with quadrature-based finite difference discretization in space. We derive their respective optimal maximum-norm error estimates and further show that the proposed schemes are asymptotically compatible, i.e., the approximating solutions always converge to the classic Allen--Cahn solution when the horizon, the spatial mesh size, and the time step size go to zero. We also prove that the schemes are energy stable in the discrete sense. Various experiments are performed to verify these theoretical results and to investigate numerically the relation between the discontinuities and the nonlocal parameters.

Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation

In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:=\tau_k/\tau_{k-1}<(3+\sqrt{17})/2\approx3.561.$ Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum principle of BDF2 scheme under the time-step ratio restriction $r_k<1+\sqrt{2}\approx 2.414$ and a practical time step constraint. The second-order rate of convergence in the maximum norm is also presented. Numerical experiments are provided to support the theoretical findings.

We investigate a family of axisymmetric solutions to a coupling of Navier–Stokes and Allen–Cahn equations in $\mathbb{R}^3$. First, a one-dimensional system of equations is derived from the method of separation of variables, which approximates the three-dimensional system along its symmetry axis. Then based on them, by adding perturbation terms, we construct finite energy solutions to the three-dimensional system. We prove the global regularity of the constructed solutions in both large viscosity and small initial data cases. These solutions can be considered as perturbations near infinite-energy solutions.

Second order convex splitting schemes for periodic nonlocal Cahn–Hilliard and Allen–Cahn equations

The fractional Allen–Cahn equation with the sextic potential

Application of the maximal entropy production principle to rapid solidification: A multi-phase-field model

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

We consider numerical methods for solving the fractional-in-space Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. A standard fully discretized scheme for this equation is considered, namely, using the conventional second-order Crank–Nicolson scheme in time and the second-order central difference approach in space. For the resulting nonlinear scheme, we propose a nonlinear iteration algorithm, whose unique solvability and convergence can be proved. The nonlinear iteration can avoid inverting a dense matrix with only \(\mathcal {O}(N\log N)\) computation complexity. One major contribution of this work is to show that the numerical solutions satisfy discrete maximum principle under reasonable time step constraint. Based on the maximum stability, the nonlinear energy stability for the fully discrete scheme is established, and the corresponding error estimates are investigated. Numerical experiments are performed to verify the theoretical results.

A structure-preserving explicit numerical scheme for the Allen–Cahn equation with a logarithmic potential

Phase-field models such as the Allen–Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen–Cahn equation with a potential with N\geq 3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow. In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: as long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen–Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter \varepsilon\searrow 0 . We even establish the rate of convergence O(\varepsilon^{1/2}) . Our approach is based on the gradient-flow structure of the Allen–Cahn equation and its limiting motion: building on the recent concept of “gradient-flow calibrations” for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen–Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the Allen–Cahn operator or additional convergence hypotheses for the energy at positive times.

Deep learning-based algorithms have recently been utilized to solve several nonlinear partial differential equations (PDEs) on curved surfaces. The mesh-free nature of these algorithms reduces computational complexity, especially for curved surfaces, where generating meshes is significantly challenging. The purpose of this paper is to generate an accurate solution of the highly nonlinear Allen–Cahn equation on various curved surfaces using a deep learning-based algorithm. However, generating a convergent solution of the Allen–Cahn equation for sharp interphase is significantly challenging, especially for curved surfaces, where the curvature of the geometry plays a crucial role in the solution accuracy. Due to the sharp interphase between two layers, the equation becomes very stiff; as a result, instability is a common practice. To mitigate this, we employ a transfer learning-based approach with PINNs to solve the Allen–Cahn equation on curved surfaces. A thorough numerical study is carried out to show the effectiveness of the numerical results presented for various complex closed curved surfaces with several initial conditions. Benchmark examples are provided for some cases to compare against the previous literature, along with a comparison with the analytic solution for the sphere case to test the convergence and accuracy of the present solution.

This paper presents an energy stable scheme for the numerical solution of the Allen–Cahn equation. The scheme combines the Crank–Nicolson/Adams–Bashforth technique with the Galerkin method, complementing a thoughtfully designed stabilization term. While the Allen–Cahn equation is inherently nonlinear, our proposed algorithm effectively addresses this nonlinearity by utilizing the implicit nature of the resulting system of equations. Additionally, the L 2 error estimate for the fully discretized scheme is carried out in a rigorous way with the optimal error bound O ( τ 2 + h r + 1 ) . This analysis demonstrates the precision and accuracy of our approach. In tandem with the theoretical framework, we conduct comprehensive numerical experiments to assess the practical validity of our method. Our findings not only validate the theoretical results but also reveal an intriguing observation: the judiciously chosen stabilization term significantly enhances the performance of the model. This finding underscores the practical effectiveness of our proposed scheme, which consistently produces precise and reliable solutions, reaffirming existing evidence.

Numerical solutions to low and high-dimensional Allen–Cahn equations using stochastic differential equations and neural networks