二维线性双曲型方程Neumann边值问题的紧交替方向隐格式

对二维Neumann边界条件的线性双曲型方程建立了紧交替方向的隐格式.利用方程和边界条件得到在空间上的三阶与五阶导数的边界值,进而在内点、边界内点和边界角点分别建立9点、6点和4点紧差分格式;通过引进新的范数和L₂范数估计L_∞范数;借助能量估计、Gronwall不等式和Schwarz不等式等技巧,详细分析了差分格式在无穷范数下关于时间和空间分别为二阶和四阶收敛性,并给出了稳定性结果;通过数值算例,验证了理论分析结果.     

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.     

Compact difference scheme for two-dimensional fourth-order nonlinear hyperbolic equation

High-order compact finite difference method for solving the two-dimensional fourth-order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth-order equation is written as a system of two second-order equations by introducing the second-order spatial derivative as a new variable. The second-order spatial derivatives are approximated by the compact finite difference operators to obtain a fourth-order convergence. As well as, the second-order time derivative is approximated by the central difference method. Then, existence and uniqueness of numerical solution is given. The stability and convergence of the compact finite difference scheme are proved by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.     

一类半线性变系数抛物型方程的紧差分格式

对一类半线性变系数抛物型方程初边值问题建立了紧差分格式,用能量分析方法证明了差分格式解的存在唯一性、关于初值的无条件稳定性和在L_∞范数下阶数为O(τ~2+h~4)的收敛性,最后给出的数值算例验证了理论结果.     

Finite Difference Method for Reaction-Diffusion Nonlocal Boundary Conditions Equation with

In this paper, we present a numerical approach to a class of nonlinear reactiondiffusion equations with nonlocal Robin type boundary conditions by finite difference methods. A second-order accurate difference scheme is derived by the method of reduction of order. Moreover, we prove that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. A simple numerical example is given to illustrate the efficiency of the proposed method.     

Finite Difference Method for Reaction-Diffusion Equation with Nonlocal Boundary Conditions

In this paper, we present a numerical approach to a class of nonlinear reaction-diffusion equations with nonlocal Robin type boundary conditions by finite difference methods. A second-order accurate difference scheme is derived by the method of reduction of order. Moreover, we prove that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. A simple numerical example is given to illustrate the efficiency of the proposed method.     

MIXED FINITE ELEMENT METHOD FOR SOBOLEV EQUATIONS AND ITS ALTERNATING-DIRECTION ITERATIVE SCHEME

In this paper, we study the mixed element method for Sobolev equations. A time-discretization procedure is presented and analysed and the optimal order error estimates are derived.For convenience in practical computation, an alternating-direction iterative scheme of the mixed fi-nite element method is formulated and its stability and converbence are proved for the linear prob-lem. A numerical example is provided at the end of this paper.     

Convergence of a difference scheme for the heat equation in a long strip by artificial boundary conditions

The numerical solution of the heat equation on a strip in two dimensions is considered. An artificial boundary is introduced to make the computational domain finite. On the artificial boundary, an exact boundary condition is proposed to reduce the original problem to an initial‐boundary value problem in a finite computational domain. A difference scheme is constructed by the method of reduction of order to solve the problem in the finite computational domain. It is proved that the difference scheme is uniquely solvable, unconditionally stable and convergent with the convergence order 2 in space and order 3/2 in time in an energy norm. A numerical example demonstrates the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A difference equation approach to parameter estimation for differential-delay equations

A finite difference scheme for estimating parameters in linear differential-delay equations is investigated. Convergence results and rates of convergence are obtained for a simple explicit (Euler's) method. Numerical examples are given to illustrate the convergence for the Euler method. Numerical results for a “higher order” scheme are also discussed.     

对称正则长波方程的守恒差分算法

1引言SRLW方程(?)~3u/(?)x~2(?)t-(?)u/(?)t=(?)p/(?)x u((?)u/(?)x),(1.1) (?)p/(?)t (?)u/(?)x=0(1.2)是正则长波(RLW)方程的一种对称叙述,用于描述弱非线形作用下空间电荷的等离子声     

Iterative methods for a forward-backward heat equation in two-dimension

A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.     

对称正则长波方程的拟紧致守恒差分逼近

该文对称正则长波方程的初边值问题进行了数值研究, 提出了一个两层隐式拟紧差分格式,格式很好地模拟了初值问题的守恒性质. 得到了差分解的存在唯一性, 并在先验估计基础上运用能量方法分析了格式的稳定性及二阶收敛性.     

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     

A fourth-order finite difference scheme for two-dimensional nonlinear elliptic partial differential equations

A finite difference scheme for the two-dimensional, second-order, nonlinear elliptic equation is developed. The difference scheme is derived using the local solution of the differential equation. A 13-point stencil on a uniform mesh of size h is used to derive the finite difference scheme, which has a truncation error of order h⁴. Well-known iterative methods can be employed to solve the resulting system of equations. Numerical results are presented to demonstrate the fourth-order convergence of the scheme. © 1995 John Wiley & Sons, Inc.     

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     

A Legendre spectral element method (SEM) based on the modified bases for solving neutral delay distributed‐order fractional damped diffusion‐wave equation

The main purpose of the current paper is to propose a new numerical scheme based on the spectral element procedure for simulating the neutral delay distributed‐order fractional damped diffusion‐wave equation. To this end, the temporal direction has been discretized by a finite difference formula with convergence order where 1<α<2. In the next, to obtain a full‐discrete scheme, we apply the spectral finite element method on the spatial direction. Furthermore, the unconditional stability of semidiscrete scheme and convergence order of full‐discrete scheme of new technique are discussed. Finally, 2 test problems have been considered to demonstrate the ability and efficiency of the proposed numerical technique.     

计算高维带弱奇异核发展型方程的交替方向隐式欧拉方法

本文主要研究高维带弱奇异核的发展型方程的交替方向隐式(ADI)差分方法.向后欧拉(Euler)方法联立一阶卷积求积公式处理时间方向的离散,有限差分方法处理空间方向的离散,并进一步构造了ADI全离散差分格式.然后将二维问题延伸到三维问题,构造三维空间问题的ADI差分格式.基于离散能量法,详细证明了全离散格式的稳定性和误差分析.随后给出了2个数值算例,数值结果进一步验证了时间方向的收敛阶为一阶,空间方向的收敛阶为二阶,和理论分析结果一致.     

A HYBRID EXPLICIT-IMPLICIT SCHEME FOR THE TIME-DEPENDENT WIGNER EQUATION

This paper designs a hybrid scheme based on finite difference methods and a spectral method for the time-dependent Wigner equation,and gives the error analysis for the full discret ization of its initial value problem.An explicit-implicit time-splitting scheme is used for time integration and the second-order upwind finite difference scheme is used to dis-cretize the advection term.The consistence error and the stability of the full discretization are analyzed.A Fourier spectral method is used to approximate the pseudo-differential operator term and the corresponding error is studied in detail.The final convergence result shows clearly how the regularity of the solution affects the convergence order of the pro-posed scheme.N umerical results are presented for confirming the sharpness of the analysis.The scattering effects of a Gaussian wave packet tunneling through a Gaussian potential barrier are investigated.The evolution of the density function shows that a larger portion of the wave is reflected when the height and the width of the barrier increase.Mathematics subject classification:65M06,65M70.     

Convergence analysis of some high–order time accurate schemes for a finite volume method for second order hyperbolic equations on general nonconforming multidimensional spatial meshes

The present work is an extension of our previous work (Bradji, Numer Methods Partial Differ Equations, to appear) which dealt with error analysis of a finite volume scheme of a first convergence order (both in time and space) for second‐order hyperbolic equations on general nonconforming multidimensional spatial meshes introduced recently in (Eymard et al. IMAJ Numer Anal 30(2010), 1009–1043). We aim in this article to get some higher‐order time accurate schemes for a finite volume method for second‐order hyperbolic equations using the same class of spatial generic meshes stated above. We derive a family of finite volume schemes approximating the wave equation, as a model for second‐order hyperbolic equations, in which the discretization in time is performed using a one‐parameter scheme of the Newmark's method. We prove that the error estimate of these finite volume schemes is of order two (or four) in time and it is of optimal order in space. These error estimates are analyzed in several norms which allow us to derive approximations for the exact solution and its first derivatives whose the convergence order is two (or four) in time and it is optimal in space. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, that the convergence order is \begin{align*}k^2+h_\mathcal{D}\end{align*} or \begin{align*}k^4+h_\mathcal{D}\end{align*}, where \begin{align*}h_\mathcal{D}\end{align*} (resp. k) is the mesh size of the spatial (resp. time) discretization. These estimates are valid under the regularity assumption \begin{align*}u\in C^4(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*}, when the schemes are second‐order accurate in time, and \begin{align*}u\in C^6(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*}, when the schemes are four‐order accurate in time for the exact solution u. The proof of these error estimates is based essentially on a comparison between the finite volume approximate solution and an auxiliary finite volume approximation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A high order difference method for fractional sub-diffusion equations with the spatially variable coefficients under periodic boundary conditions

In this paper, we propose a difference scheme with global convergence order $O(\tau^{2}+h^4)$ for a class of the Caputo fractional equation. The difficulty caused by the spatially variable coefficients is successfully handled. The unique solvability, stability and convergence of the finite difference scheme are proved by use of the Fourier method. The obtained theoretical results are supported by numerical experiments.