The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model

The numerical simulation of the dynamics of the molecular beam epitaxy (MBE) growth is considered in this article. The governing equation is a nonlinear evolutionary equation that is of linear fourth order derivative term and nonlinear second order derivative term in space. The main purpose of this work is to construct and analyze two linearized finite difference schemes for solving the MBE model. The linearized backward Euler difference scheme and the linearized Crank‐Nicolson difference scheme are derived. The unique solvability, unconditional stability and convergence are proved. The linearized Euler scheme is convergent with the convergence order of O(τ + h²) and linearized Crank‐Nicolson scheme is convergent with the convergence order of O(τ² + h²) in discrete L²‐norm, respectively. Numerical stability with respect to the initial conditions is also obtained for both schemes. Numerical experiments are carried out to demonstrate the theoretical analysis. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A second‐order accurate difference scheme for the two‐dimensional Burgers' system

In this article, a Crank‐Nicolson‐type finite difference scheme for the two‐dimensional Burgers' system is presented. The existence of the difference solution is shown by Brouwer fixed‐point theorem. The uniqueness of the difference solution and the stability and L₂ convergence of the difference scheme are proved by energy method. An iterative algorithm for the difference scheme is given in detail. Furthermore, a linear predictor–corrector method is presented. The numerical results show that the predictor–corrector method is also convergent with the convergence order of two in both time and space. At last, some comments are provided for the backward Euler scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.     

On the numerical approach of the enthalpy method for the Stefan problem

In this article an error bound is derived for a piecewise linear finite element approximation of an enthalpy formulation of the Stefan problem; we have analyzed a semidiscrete Galerkin approximation and completely discrete scheme based on the backward Euler method and a linearized scheme is given and its convergence is also proved. A second‐order error estimates are derived for the Crank‐Nicolson Galerkin method. In the second part, a new class of finite difference schemes is proposed. Our approach is to introduce a new variable and transform the given equation into an equivalent system of equations. Then, we prove that the difference scheme is second order convergent. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Bifurcation analysis and chaos control in a host‐parasitoid model

We investigate the qualitative behavior of a host‐parasitoid model with a strong Allee effect on the host. More precisely, we discuss the boundedness, existence and uniqueness of positive equilibrium, local asymptotic stability of positive equilibrium and existence of Neimark–Sacker bifurcation for the given system by using bifurcation theory. In order to control Neimark–Sacker bifurcation, we apply pole‐placement technique that is a modification of OGY method. Moreover, the hybrid control methodology is implemented in order to control Neimark–Sacker bifurcation. Numerical simulations are provided to illustrate theoretical discussion. Copyright © 2017 John Wiley & Sons, Ltd.     

Neimark‐Sacker bifurcation of a fourth order difference equation

In this article, we study stability and bifurcation of a fourth order rational difference equation. We give condition for local stability, and we show that the equation undergoes a Neimark‐Sacker bifurcation. Moreover, we consider the direction of the Neimark‐Sacker bifurcation. Finally, we numerically validate our analytical results.     

A finite element method for the Sivashinsky equation

The Sivashinsky equation is a nonlinear evolutionary equation of fourth order in space. In this paper we have analyzed a semidiscrete finite element method and completely discrete scheme based on the backward Euler method and Crank–Nicolson–Galerkin scheme. A linearized backward Euler method have been developed and error bounds are derived for an L² projection.     

Finite difference discretization of the Benjamin‐Bona‐Mahony‐Burgers equation

Numerical solutions of the Benjamin‐Bona‐Mahony‐Burgers equation in one space dimension are considered using Crank‐Nicolson‐type finite difference method. Existence of solutions is shown by using the Brower's fixed point theorem. The stability and uniqueness of the corresponding methods are proved by the means of the discrete energy method. The convergence in L^∞‐norm of the difference solution is obtained. A conservative difference scheme is presented for the Benjamin‐Bona‐Mahony equation. Some numerical experiments have been conducted in order to validate the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On the long‐time stability of the Crank–Nicolson scheme for the 2D Navier–Stokes equations

In this article we study the stability for all positive time of the Crank–Nicolson scheme for the two‐dimensional Navier–Stokes equations. More precisely, we consider the Crank–Nicolson time discretization together with a general spatial discretization, and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the numerical scheme is stable, provided a CFL‐type condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Neimark–Sacker bifurcation of a third-order rational difference equation

In this paper, we investigate the existence and direction of the Neimark–Sacker bifurcation of a third-order rational difference equation with positive parameters. Firstly, it is found that there exists a Neimark–Sacker bifurcation when the parameter passes a critical value by analysing the characteristic equation. Secondly, the explicit algorithm for determining the direction and stability of the Neimark–Sacker bifurcations is derived by using the normal form theory. Finally, computer simulations are performed to illustrate the analytical results found.     

Numerical time-delayed feedback control in a neural network with delay

In this article, by a nonstandard finite-difference method we obtain the general time delayed feedback control numerical discrete scheme for a delayed neural network model. Firstly, the local stability of the equilibria point is discussed according to the Neimark–Sacker bifurcation theory. Then, from the point of view of control, for any step-size, a general time delayed feedback control numerical algorithm is introduced to delay the onset of the Neimark–Sacker bifurcation at a desired point by choosing appropriate control parameters. This controller can deal with the general system that the natural equilibrium cannot be given by analytic expression. Finally, numerical examples are provided to illustrate the theoretical results. The results show that the time delayed feedback numerical scheme is better than a polynomial function time delayed feedback method.     

Finite difference approximate solutions for the Cahn‐Hilliard equation

In this article, we analyze a Crank‐Nicolson‐type finite difference scheme for the nonlinear evolutionary Cahn‐Hilliard equation. We prove existence, uniqueness and convergence of the difference solution. An iterative algorithm for the difference scheme is given and its convergence is proved. A linearized difference scheme is presented, which is also second‐order convergent. Finally a new difference method possess a Lyapunov function is presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 437–455, 2007     

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

In this paper, we consider the Crank‐Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank‐Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two‐grid Crank‐Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two‐grid Crank‐Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H¹‐norm and for pressure in L²‐norm. However, the two‐grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes.     

Stability of the Crank–Nicolson–Adams–Bashforth scheme for the 2D Leray‐alpha model

We consider the stability of an efficient Crank–Nicolson–Adams–Bashforth method in time, finite element in space, discretization of the Leray‐α model. We prove finite‐time stability of the scheme in L², H¹, and H², as well as the long‐time L‐stability of the scheme under a Courant‐Freidrichs‐Lewy (CFL)‐type condition. Numerical experiments are given that are in agreement with the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1155–1183, 2016     

Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

In this paper, stability and bifurcation of a two‐dimensional ratio‐dependence predator–prey model has been studied in the close first quadrant . It is proved that the model undergoes a period‐doubling bifurcation in a small neighborhood of a boundary equilibrium and moreover, Neimark–Sacker bifurcation occurs at a unique positive equilibrium. We study the Neimark–Sacker bifurcation at unique positive equilibrium by choosing b as a bifurcation parameter. Some numerical simulations are presented to illustrate theocratical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Galerkin finite element methods for two-dimensional RLW and SRLW equations

In this paper, Galerkin finite methods for two-dimensional regularized long wave and symmetric regularized long wave equation are studied. The discretization in space is by Galerkin finite element method and in time is based on linearized backward Euler formula and extrapolated Crank–Nicolson scheme. Existence and uniqueness of the numerical solutions have been shown by Brouwer fixed point theorem. The error estimates of linearlized Crank–Nicolson method for RLW and SRLW equations are also presented. Numerical experiments, including the error norms and conservation variables, verify the efficiency and accuracy of the proposed numerical schemes.     

A high‐order compact boundary value method for solving one‐dimensional heat equations

We combine fourth‐order boundary value methods (BVMs) for discretizing the temporal variable with fourth‐order compact difference scheme for discretizing the spatial variable to solve one‐dimensional heat equations. This class of new compact difference schemes achieve fourth‐order accuracy in both temporal and spatial variables and are unconditionally stable due to the favorable stability property of BVMs. Numerical results are presented to demonstrate the accuracy and efficiency of the new compact difference scheme, compared to the standard second‐order Crank‐Nicolson scheme. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 846–857, 2003.     

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

In this work we construct and analyze some finite difference schemes used to solve a class of time‐dependent one‐dimensional convection‐diffusion problems, which present only regular layers in their solution. We use the implicit Euler or the Crank‐Nicolson method to discretize the time variable and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize the spatial variable. In both cases we prove that the numerical method is uniformly convergent with respect to the diffusion parameter, having order near two in space and order one or 3/2, depending on the method used, in time. We show some numerical examples which illustrate the theoretical results, in the case of using the Euler implicit method, and give better numerical behaviour than that predicted theoretically, showing order two in time and order N^?2log²N in space, if the Crank‐Nicolson scheme is used to discretize the time variable. Finally, we construct a numerical algorithm by combining a third order A‐stable SDIRK with two stages and a third‐order HODIE difference scheme, showing its uniformly convergent behavior, reaching order three, up to a logarithmic factor. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Analysis and computational method based on quadratic B‐spline FEM for the Rosenau–Burgers equation

L_∞‐error estimates for B‐spline Galerkin finite element solution of the Rosenau–Burgers equation are considered. The semidiscrete B‐spline Galerkin scheme is studied using appropriate projections. For fully discrete B‐spline Galerkin scheme, we consider the Crank–Nicolson method and analyze the corresponding error estimates in time. Numerical experiments are given to demonstrate validity and order of accuracy of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 877–895, 2016     

Finite element approximation of a forward and backward anisotropic diffusion model in image denoising and form generalization

A new forward–backward anisotropic diffusion model is introduced. The two limit cases are the Perona‐Malik equation and the Total Variation flow model. A fully discrete finite element scheme is studied using C⁰‐piecewise linear elements in space and the backward Euler difference scheme in time. A priori estimates are proven. Numerical results in image denoising and form generalization are presented.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008