Numerical approximation of the conservative Allen–Cahn equation by operator splitting method

In this paper, a second‐order fast explicit operator splitting method is proposed to solve the mass‐conserving Allen–Cahn equation with a space–time‐dependent Lagrange multiplier. The space–time‐dependent Lagrange multiplier can preserve the volume of the system and keep small features. Moreover, we analyze the discrete maximum principle and the convergence rate of the fast explicit operator splitting method. The proposed numerical scheme is of spectral accuracy in space and of second‐order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, efficiency, mass conservation, and stability of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.     

The Allen–Cahn equation with dynamic boundary conditions and mass constraints

The Allen–Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint, which may involve either the solution inside the domain or its trace on the boundary. The system of nonlinear partial differential equations can be formulated as a variational inequality. The presence of the constraint in the evolution process leads to additional terms in the equation and the boundary condition containing a suitable Lagrange multiplier. A well‐posedness result is proved for the related initial value problem. Copyright © 2014 John Wiley & Sons, Ltd.     

On the discretisation in time of the stochastic Allen–Cahn equation

We consider the stochastic Allen–Cahn equation perturbed by smooth additive Gaussian noise in a bounded spatial domain with smooth boundary in dimension , and study the semidiscretisation in time of the equation by an Euler type split‐step method with step size . We show that the method converges strongly with a rate . By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.     

Lagrange multiplier and singular limit of double obstacle problems for the Allen–Cahn equation with constraint

We study the properties of the Lagrange multiplier for an Allen–Cahn equation with a double obstacle potential. Here, the dynamic boundary condition, including the Laplace–Beltrami operator on the boundary, is investigated. We then establish the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier of our problem. We present remarks on a trace problem as well as on the Neumann boundary condition. Moreover, we describe a numerical experiment for a problem with Neumann boundary condition using the Lagrange multiplier. Copyright © 2016 John Wiley & Sons, Ltd.     

Stabilized semi‐implicit spectral deferred correction methods for Allen–Cahn and Cahn–Hilliard equations

Stabilized semi‐implicit spectral deferred correction methods are constructed for the time discretization of Allen–Cahn and Cahn–Hilliard equations. These methods are unconditionally stable, lead to simple linear system to solve at each iteration, and can achieve high‐order time accuracy with a few iterations in each time step. Ample numerical results are presented to demonstrate the effectiveness of the stabilized semi‐implicit spectral deferred correction methods for solving the Allen–Cahn and Cahn–Hilliard equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Primal‐dual active set methods for Allen–Cahn variational inequalities with nonlocal constraints

We propose and analyze a primal‐dual active set method for discretized versions of the local and nonlocal Allen–Cahn variational inequalities. An existence result for the nonlocal variational inequality is shown in a formulation involving Lagrange multipliers for local and nonlocal constraints. Local convergence of the discrete method is shown by interpreting the approach as a semismooth Newton method. Properties of the method are discussed and several numerical simulations demonstrate its efficiency. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A mathematical analysis of a nonisothermal Allen–Cahn type system

Existence of solutions for a nonisothermal Allen–Cahn type system is proved by using a semidiscrete spectral Galerkin method together with degree theory and maximum principle. Regularity and uniqueness are obtained in a special situation for domains with dimension up to two and smooth enough data. The present system may model the evolution of solidification or melting processes occurring in certain binary alloys. Copyright © 2012 John Wiley & Sons, Ltd.     

On the convergence of operator splitting for the Rosenau–Burgers equation

We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.     

A mathematical analysis of a nonisothermal Allen–Cahn type system: error estimates

In this article, under certain conditions, we prove the regularity for the solutions of an Allen–Cahn phase‐field type system obtained as limits of approximate solutions constructed by using a semidiscrete spectral Galerkin method. With the help of this improved regularity, as one compares to previous results, we then derive error estimates for the approximate solutions in terms of the inverse of the eigenvalues of the Laplacian operator. The system under investigation may model the evolution of solidification or melting of certain binary alloys. Copyright © 2012 John Wiley & Sons, Ltd.     

Maximum norm error analysis of an unconditionally stable semi‐implicit scheme for multi‐dimensional Allen–Cahn equations

In this paper, a linearized finite difference scheme is proposed for solving the multi‐dimensional Allen–Cahn equation. In the scheme, a modified leap‐frog scheme is used for the time discretization, the nonlinear term is treated in a semi‐implicit way, and the central difference scheme is used for the discretization in space. The proposed method satisfies the discrete energy decay property and is unconditionally stable. Moreover, a maximum norm error analysis is carried out in a rigorous way to show that the method is second‐order accurate both in time and space variables. Finally, numerical tests for both two‐ and three‐dimensional problems are provided to confirm our theoretical findings.     

Operator splitting for numerical solution of the modified Burgers' equation using finite element method

The aim of this study is to obtain numerical behavior of a one‐dimensional modified Burgers' equation using cubic B‐spline collocation finite element method after splitting the equation with Strang splitting technique. Moreover, the Ext4 and Ext6 methods based on Strang splitting and derived from extrapolation have also been applied to the equation. To observe how good and effective this technique is, we have used the well‐known the error norms L₂ and L_∞ in the literature and compared them with previous studies. In addition, the von Neumann (Fourier series) method has been applied after the nonlinear term has been linearized to investigate the stability of the method.     

非定常BOLTZMANN方程的谱有限元方法

In this paper,a new numerical method,the coupling method of spherical harmonic function spectral and finite elements,for a unsteady transport equation is dlscussed,and the error analysis of this scheme is proved.     

Analysis of the operator splitting scheme for the Cahn‐Hilliard equation with a viscosity term

In this paper, we consider a second‐order fast explicit operator splitting method for the viscous Cahn‐Hilliard equation, which includes a viscosity term αΔu_t (α ∈ (0, 1)) described the influences of internal micro‐forces. The choice α = 0 corresponds to the classical Cahn‐Hilliard equation whilst the choice α = 1 recovers the nonlocal Allen‐Cahn equation. The fundamental idea of our method is to split the original problem into linear and nonlinear parts. The linear subproblem is numerically solved using a pseudo‐spectral method, and thus an ordinary differential equation is obtained. The nonlinear one is solved via TVD‐RK method. The stability and convergence are discussed in L²‐norm. Numerical experiments are performed to validate the accuracy and efficiency of the proposed method. Besides, a detailed comparison is made for the dynamics and the coarsening process of the metastable pattern for various values of α. Moreover, energy degradation and mass conservation are also verified.     

The Crank–Nicolson–Galerkin finite element method for a nonlocal parabolic equation with moving boundaries

The aim of this article is to establish the convergence and error bounds for the fully discrete solutions of a class of nonlinear equations of reaction–diffusion nonlocal type with moving boundaries, using a linearized Crank–Nicolson–Galerkin finite element method with polynomial approximations of any degree. A coordinate transformation which fixes the boundaries is used. Some numerical tests to compare our Matlab code with some existing moving finite element methods are investigated. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1515–1533, 2015     

Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B‐spline collocation method

In the present study, the operator splitting techniques based on the quintic B‐spline collocation finite element method are presented for calculating the numerical solutions of the Rosenau–KdV–RLW equation. Two test problems having exact solutions have been considered. To demonstrate the efficiency and accuracy of the present methods, the error norms L₂ and L_∞ with the discrete mass Q and energy E conservative properties have been calculated. The results obtained by the method have been compared with the exact solution of each problem and other numerical results in the literature, and also found to be in good agreement with each other. A Fourier stability analysis of each presented method is also investigated.     

二次Lagrangian有限元方程的几何多重网格法

为了构造快速求解二次Lagrangian有限元方程的几何多重网格法,在选择二次Lagrangian有限元空间和一系列线性Lagrangian有限元空间分别作为最细网格层和其余粗网格层以及构造一种新限制算子的基础上,提出了一种新的几何多重网格法,并对它的计算量进行了估计.数值实验结果,与通常的几何多重网格法和AMG01法相比,表明了新算法计算量少且稳健性强.     

DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR A FORWARD-BACKWARD HEAT EQUATION

A space-time finite element method,discontinuous in time but continuous in space, is studied to solve the nonlinear forward-backward heat equation. A linearized technique is introduced in order to obtain the error estimates of the approximate solutions. And the numerical simulations are given.     

A numerical method based on extended Raviart–Thomas (ER‐T) mixed finite element method for solving damped Boussinesq equation

In this paper, we present the approximate solution of damped Boussinesq equation using extended Raviart–Thomas mixed finite element method. In this method, the numerical solution of this equation is obtained using triangular meshes. Also, for discretization in time direction, we use an implicit finite difference scheme. In addition, error estimation and stability analysis of both methods are shown. Finally, some numerical examples are considered to confirm the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

A monotone scheme for Hamilton–Jacobi equations via the nonstandard finite difference method

A usual way of approximating Hamilton–Jacobi equations is to couple space finite element discretization with time finite difference discretization. This classical approach leads to a severe restriction on the time step size for the scheme to be monotone. In this paper, we couple the finite element method with the nonstandard finite difference method, which is based on Mickens' rule of nonlocal approximation. The scheme obtained in this way is unconditionally monotone. The convergence of the new method is discussed and numerical results that support the theory are provided. Copyright © 2009 John Wiley & Sons, Ltd.