Narrow Results

Parabolic Finslerian Allen–Cahn equation is a quasilinear parabolic equation on Finsler metric measure spaces. In this paper, we establish a Harnack inequality for solutions to this equation on compact Finsler metric measure spaces using geometric methods. Moreover, we give a degree 4 polynomial gradient estimate of the traveling wave solution on flat-curved spaces.

In this study, a phase-field lattice Boltzmann model based on the Allen–Cahn equation with a filtered collision operator and high-order corrections in the equilibrium distribution functions is presented. Here, we show that in addition to producing numerical results consistent with prior numerical methods, analytic solutions, and experiments with the density ratio of 1000, previous numerical deficiencies are resolved. Specifically, the new model is characterized by robustness at low viscosity, accurate prediction of shear stress at interfaces, and removal of artificial dense bubbles and rarefied droplets, etc.

Two-phase flow is important in both science and engineering. In this paper, a D2Q5 multi-relaxation-time lattice Boltzmann model is proposed for solving Allen–Cahn equation, which is a convection–diffusion equation for the order parameter. Through Chapmann–Enskog analysis, the macroscopic equations can be recovered from the D2Q5 MRT LB model. Then, a detailed numerical study on several classical problems is performed to give a comparison between the present model for the AC equation and the previous study. The results show that the present model gets better accuracy and has better stability than that of single-relaxation-time LB model, and has higher computational efficiency than that of D2Q9 MRT LB model. Furthermore, as an important application, the model is also used to study the deformation field problem because of its excellent ability to cope with complex and changeable boundaries.

In this paper, the local discontinuous Galerkin method is used to analyze numerical solutions for nonlinear Allen–Cahn equations with nonperiodic boundary conditions. To begin with, the spatial variables are discretized to generate a semidiscrete method of lines scheme. This yields an ordinary differential equation system in the temporal variable, which is then solved using the higher-order total variation diminishing Runge–Kutta method. A comparison of the generated numerical results to the exact results for various test problems using different tables and figures provides insight into the effectiveness and accuracy of the proposed method. The numerical results confirm that the proposed method is an effective numerical scheme for solving the Allen–Cahn equation since the obtained solutions are extremely close to the exact solutions while exhibiting substantially less error.

We study a model of phase segregation of the Allen–Cahn type, consisting in a system of two differential equations, one partial and the other ordinary, respectively interpreted as balances of microforces and microenergy; the two unknowns are the order parameter entering the standard A–C equation and the chemical potential. We introduce a notion of maximal solution to the o.d.e., parametrized on the order-parameter field; and, by substitution in the p.d.e. of the so-obtained chemical potential field, we give the latter equation the form of an Allen–Cahn equation for the order parameter, with a memory term. Finally, we prove the existence and uniqueness of global-in-time smooth solutions to this modified A–C equation, and we give a description of the relative ω-limit set.

This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier–Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier–Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.

A modification of the parabolic Allen–Cahn equation, determined by the substitution of Fick’s diffusion law with a relaxation relation of Cattaneo–Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution is the proof of the nonlinear stability of the wave, as a consequence of detailed spectral and linearized analyses. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front.

In this paper, for the Allen–Cahn equation, we obtain the error estimate of the temporal semi-discrete scheme, and the fully-discrete finite element numerical scheme, both of which are based on the invariant energy quadratization (IEQ) time-marching strategy. We establish the relationship between the $L^2$-error bound and the $L^{\infty }/H^2$-stabilities of the numerical solution. Then, by converting the numerical schemes to a form compatible with the original format of the Allen–Cahn equation, using mathematical induction, the superconvergence property of nonlinear terms, and the spectrum argument, the optimal error estimates that only depends on the low-order polynomial degree of ${𝜖}^{-1}$ instead of $e^{T/{𝜖}^2}$ for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis.

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakke's motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen–Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.

In this paper, we investigate numerical solution of Allen–Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals. We discretize the model equation by symmetric interior penalty Galerkin (SIPG) method in space, and by average vector field (AVF) method in time. We show that the energy stable AVF method as the time integrator for gradient systems like the Allen–Cahn equation satisfies the energy decreasing property for fully discrete scheme. Numerical results reveal that the discrete energy decreases monotonically, the phase separation and metastability phenomena can be observed, and the ripening time is detected correctly.

The aim of this paper is to prove that, for specific initial data $(u_0,u_1)$ and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen–Cahn equation on the interval $[a,b]$ shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the “energy approach” proposed by Bronsard and Kohn [On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math.43 (1990) 983–997], if $\epsilon \ll 1$ is the diffusion coefficient, we show that in a time scale of order ${\epsilon }^{-k}$ nothing happens and the solution maintains the same number of transitions of its initial datum $u_0$. The novelty consists mainly in the role of the initial velocity $u_1$, which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen–Cahn equation with relaxation.

In this paper, the second-order scalar auxiliary variable approach combined with finite difference method is employed for the Allen–Cahn equation that represents a phenomenological model for antiphase domain coarsening in a binary mixture. The second-order backward differentiation formula is used in time. The error estimation of the semi-discrete scheme is derived in the sense of $L^2$-norm. Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of the proposed scheme.

In recent years, finding precise numerical solutions to nonlinear partial differential equations (NLPDEs) has become a significant research topic. This paper addresses the two well-known nonlinear models, which are $(1+1)$-dimensional Allen–Cahn (AC) and phi-four (PF) equations, and they have applications in various real-world scenarios. These two models are solved by a numerical technique called Hermite wavelet collocation scheme (HWCS). The operational matrices of integration were developed using the Hermite wavelet basis. The proposed method is based on the collocation process. By employing appropriate grid points, this method converts the nonlinear model into a set of nonlinear algebraic equations. We obtain a solution by employing the Newton–Raphson scheme to solve these nonlinear algebraic equations. Tables and figures demonstrate that the proposed method yields superior results. The proposed examples demonstrate the efficacy of the proposed idea, and the results affirm the practicality of the proposed approach. We compared the present method with the non-polynomial spline method and trigonometric B-spline collocation technique to check the potential of the projected method.

This paper deals with the construction of solutions to autonomous semilinear elliptic equations considered in entire space. In the absence of space dependence or explicit geometries of the ambient space, the point is to unveil internal mechanisms of the equation that trigger the presence of families of solutions with interesting concentration patterns. We discuss the connection between minimal surface theory and entire solutions of the Allen-Cahn equation. In particular, for dimensions 9 or higher, we build an example that provides a negative answer to a celebrated question by De Giorgi for this problem. We will also discuss related results for the (actually more delicate) standing wave problem in nonlinear Schrödinger equations and for sign-changing solutions of the Yamabe equation.