Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations

In this paper, we study the classical Allen-Cahn equations and investigate the maximum- principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen- Cahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its flexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to dem-onstrate the energy decay property. Moreover, due to the stiff source given in the equation, we use the conservative modified exponential Runge-Kutta methods and thus can use rela-tively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

Semi-Implicit Scheme to Solve Allen-Cahn Equation with Different Boundary Conditions

The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part of the operator and an implicit Euler discretization of the linear part. Finite difference schemes are used for the spatial part. This finally leads to the numerical solution of a sparse linear system that can be solved efficiently.

A Finite Volume Unstructured Mesh Method for Fractional-in-space Allen-Cahn Equation

Fractional-in-space Allen-Cahn equation containing a very strong nonlinear source term and small perturbation shows metastability and a quartic double well potential.Using a finite volume unstructured triangular mesh method, the present paper solves the twodimensional fractional-in-space Allen-Cahn equation with homogeneous Neumann boundary condition on different irregular domains. The efficiency of the method is presented through numerical computation of the two-dimensional fractional-in-space Allen-Cahn equation on different domains.

Nondegeneracy of Solution to the Allen-Cahn Equation with Regular Triangle Symmetry

The Allen-Cahn equation on the plane has a 6-end solution U with regular triangle symmetry. The angle between consecutive nodal lines of U is . We prove in this paper that U is non-degenerated in the class of functions possessing regular triangle symmetry. As an application, we show the existence of a family of solutions close to U.

VARIABLE STEP-SIZE BDF3 METHOD FOR ALLEN-CAHN EQUATION

In this work,we analyze the three-step backward differentiation formula(BDF3)method for solving the Allen-Cahn equation on variable grids.For BDF2 method,the discrete orthogonal convolution(DOC)kernels are positive,the stability and convergence analysis are well established in[Liao and Zhang,Math.Comp.,90(2021),1207–1226]and[Chen,Yu,and Zhang,arXiv:2108.02910,2021].However,the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial due to the additional degrees of freedom and the non-positivity of DOC kernels.By developing a novel spectral norm inequality,the unconditional stability and convergence are rigorously proved under the updated step ratio restriction rk:=τk/τk−1≤1.405 for BDF3 method.Finally,numerical experiments are performed to illustrate the theoretical results.To the best of our knowledge,this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.

IMPLICIT-EXPLICIT SCHEME FOR THE ALLEN-CAHN EQUATION PRESERVES THE MAXIMUM PRINCIPLE

It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes？ To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.

Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation

This paper is concerned with the multidimensional asymptotic stability of V-shaped traveling fronts in the Allen-Cahn equation under spatial decaying initial values. We first show that V-shaped traveling fronts are asymptotically stable under the perturbations that decay at infinity. Then we further show that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which indicates that V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations. Our main technique is the supersolutions and subsolutions method coupled with the comparison principle.

A Regularity Criterion for the 3D Incompressible Density-Dependent Navier-Stokes-Allen-Cahn Equations

In this paper, we study the initial boundary problem for 3D incompressible density-dependent Navier-Stokes-Allen-Cahn equations, and give a regularity criterion for local strong solutions. Our result refines the blow-up criterion in [1].

HIGH ORDER LOCAL DISCONTINUOUS GALERKIN METHODS FOR THE ALLEN-CAHN EQUATION： ANALYSIS AND SIMULATION

In this paper, we present a local discontinuous Galerkin （LDG） method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k ＋ 1 in L2 norm and present the （2k＋1）-th order negative-norm estimate of the semi- discrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction （SDC） method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of （O（hk＋1） in L2 norm and improve the LDG solution from （O（hk＋1） to （O（h2k＋1） with the accuracy enhancement post-processing technique.

Catenoidal Layers for the Allen-Cahn Equation in Bounded Domains

This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α~2Δu + u(1- u^2) = 0 in a smooth bounded domain Ω R^3, with Neumann boundary condition and α > 0 a small parameter. These solutions have the property that as α→ 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ?Ω orthogonally and that is non-degenerate respect to ?Ω. The authors provide explicit examples of surfaces to which the result applies.

Two-Phase Image Segmentation by the Allen-Cahn Equation and a Nonlocal Edge Detection Operator

Based on a nonlocal Laplacian operator,a novel edge detection method of the grayscale image is proposed in this paper.This operator utilizes the information of neighbor pixels for a given pixel to obtain effective and delicate edge detection.The nonlocal edge detection method is used as an initialization for solving the Allen-Cahn equation to achieve two-phase segmentation of the grayscale image.Efficient exponential time differencing(ETD)solvers are employed in the time integration,and finite difference method is adopted in space discretization.The maximum bound principle and energy stability of the proposed numerical schemes are proved.The capability of our segmentation method has been verified in numerical experiments for different types of grayscale images.

NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION

We consider numerical methods to solve the Allen-Cahn equation using the second-order Crank-Nicolson scheme in time and the second-order central difference approach in space.The existence of the finite difference solution is proved with the help of Browder fixed point theorem.The difference scheme is showed to be unconditionally convergent in Loo norm by constructing an auxiliary Lipschitz continuous function.Based on this result,it is demonstrated that the difference scheme preserves the maximum principle without any restrictions on spatial step size and temporal step size.The numerical experiments also verify the reliability of the method.

The Variable-Step L1 Scheme Preserving a Compatible Energy Law for Time-Fractional Allen-Cahn Equation

In this work,we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo’s form.The L1 implicit scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools,i.e.,the discrete orthogonal convolution kernels and discrete complementary convolution kernels.Then the discrete embedding techniques and the fractional Gronwall inequality are applied to establish an L^(2)norm error estimate on nonuniform time meshes.An adaptive time-stepping strategy according to the dynamical feature of the system is presented to capture the multi-scale behaviors and to improve the computational performance.

One-Dimensional Symmetry and Liouville Type Results for the Fourth Order Allen-Cahn Equation in R^N

In this paper, the authors prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R^N, as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. The authors also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.

A Formulation of the Porous Medium Equation with Time-Dependent Porosity: A Priori Estimates and Regularity Results

We consider a generalized form of the porous medium equation where the porosity ϕis a function of time t: ϕ=ϕ(x,t): ∂(ϕS)∂t−∇⋅(k(S)∇S)=Q(S).In many works, the porosity ϕis either assumed to be independent of (or to depend very little of) the time variable t. In this work, we want to study the case where it does depend on t(and xas well). For this purpose, we make a change of unknown function V=ϕSin order to obtain a saturation-like (advection-diffusion) equation. A priori estimates and regularity results are established for the new equation based in part on what is known from the saturation equation, when ϕis independent of the time t. These results are then extended to the full saturation equation with time-dependent porosity ϕ=ϕ(x,t). In this analysis, we make explicitly the dependence of the various constants in the estimates on the porosity ϕby the introduced transport vector w, through the change of unknown function. Also we do not assume zero-flux boundary, but we carry the analysis for the case Q≡0.

Some Modified Equations of the Sine-Hilbert Type

Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based on these bilinear equations, some exact solutions to the three modified equations are derived.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

Theoretical study of particle and energy balance equations in locally bounded plasmas

In this study,new particle and energy balance equations have been developed to predict the electron temperature and density in locally bounded plasmas.Classical particle and energy balance equations assume that all plasma within a reactor is completely confined only by the reactor walls.However,in industrial plasma reactors for semiconductor manufacturing,the plasma is partially confined by internal reactor structures.We predict the effect of the open boundary area(A′_(L,eff))and ion escape velocity(u_(i))on electron temperature and density by developing new particle and energy balance equations.Theoretically,we found a low ion escape velocity(u_(i)/u_(B)≈0.2)and high open boundary area(A′_(L,eff)/A_(T,eff)≈0.6)to result in an approximately 38%increase in electron density and an 8%decrease in electron temperature compared to values in a fully bounded reactor.Additionally,we suggest that the velocity of ions passing through the open boundary should exceedω_(pi)λ_(De)under the condition E^(2)_(0)?(Φ/λ_(De))^(2).

Data-Driven Ai-and Bi-Soliton of the Cylindrical Korteweg-de Vries Equation via Prior-Information Physics-Informed Neural Networks

By the modifying loss function MSE and training area of physics-informed neural networks(PINNs),we propose a neural networks model,namely prior-information PINNs(PIPINNs).We demonstrate the advantages of PIPINNs by simulating Ai-and Bi-soliton solutions of the cylindrical Korteweg-de Vries(cKdV)equation.

Matrix Riccati Equations in Optimal Control

In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied thoroughly, matrix Riccati equation of which scalar Riccati equations is a particular case, is much less investigated. This article proposes a change of variable that allows to find explicit solution of the Matrix Riccati equation. We then apply this solution to Optimal Control.