对流扩散方程的三层WENO-MMOCAA差分方法

本文把三层修正特征线法,MMOCAA 差分方法及WENO 插值相结合,提出了求解对流扩散方程的三层WENO-MMOCAA 差分格式.此格式关于时间具有二阶精度,关于空间具有二阶以上精度且可避免基于二次以上Lagrange 插值的三层MMOCAA 差分方法在解的大梯度附近所产生的振荡.本文使用新的分析方法,给出了格式的误差估计.本文的数值算例表明新格式可消除振荡.     

IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces

Based on Li’s immersed interface method (IIM), an ADI-type finite difference scheme is proposed for solving two-dimensional nonlinear convection–diffusion interface problems on a fixed cartesian grid, which is unconditionally stable and converges with two-order accuracy in both time and space in maximum norm. Correction terms are added to the right-hand side of standard ADI scheme at irregular points. The nonlinear convection terms are treated by Adams–Bashforth method, without affecting the stability of difference schemes. A new method for computing the correction terms is developed, in which the Adams–Bashforth method is employed. Thus we can get an explicit approximation for the computation of corrections, when the jump condition is solution-dependent. Three numerical experiments are displayed and analyzed. The numerical results show good agreement with the exact solutions and confirm the convergence order.     

抛物方程初边值问题连续有限元的超收敛性

研究了一类一维抛物方程初边值问题的连续有限元方法.在空间上进行任意m次有限元半离散,在时间方向上进行二次连续有限元后,获得了一个稳定的全离散计算格式.利用单元分析法校正技术的新思想进行理论分析,连续有限元解在剖分网格节点上具有超收敛性.     

对流扩散方程的经济差分格式

１．引言 对流扩散方程是一类基本的运动方程,它可描述质量、热量的输运过程以及反应扩散过程等众多物理现象．寻找稳定、快速实用的数值方法,有着重要的理论和实际意义．标准的差分方法或有限元方法对它常常失效,根本原因在于“对流项”的存在．［１］提出了解对流扩散方程的特征线修正技术,这一方法考虑沿着特征线（流动方向）的离散,利用了对流扩散问题的物理力学性质,可以有效地克服数值振荡,保证数值解的稳定,尤其对“对流占优”的问题,这一方法有突出的优越性．这方面已有大量的理论和应用研究成果［２,３,７］．对大规模…     

Convergence of Three-Step Taylor Galerkin Finite Element Scheme Based Monotone Schwarz Iterative Method for Singularly Perturbed Differential-Difference Equation

Monotone Schwarz iterative methods for parabolic partial differential equations are well known for their advantage of eliminating the search for an initial solution. In this article, we propose a monotone Schwarz iterative method for singularly perturbed parabolic retarded differential-difference equations based on a three-step Taylor Galerkin finite element scheme. The stability and ε-uniform convergence of the three-step Taylor Galerkin finite element method have been discussed. Further, by using maximum principle and induction hypothesis, the convergence of the proposed monotone Schwarz iterative method has been established.     

An Analysis of the Blended Three-Step Backward Differentiation Formula Time-Stepping Scheme for the Navier-Stokes-Type System Related to Soret Convection

In this article, we investigate the stability and convergence of a new class of blended three-step Backward Differentiation Formula (BDF) time-stepping scheme for spatially discretized Navier-Stokes-type system modeling Soret driven convective flows. A Galerkin mixed finite element spatial discretization is assumed, and the temporal discretization is by the implicit blended three-step BDF scheme. The blended BDF scheme is more accurate than the classical second order accurate two-step BDF (BDF2) scheme, yet strongly A-stable. We consider an implicit, linearly extrapolated version of the scheme to improve its efficiency. We present optimal finite element error estimates and prove the scheme is unconditionally stable and convergent. Numerical experiments are presented that compare the scheme to the classical BDF2 scheme.     

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.     

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.     

A scheme for constructing algorithms for correcting a local perturbation in a finite semimetric

A three-step scheme for constructing algorithms for transforming metric information in data mining is proposed and investigated. The correction problem of a local perturbation of a semimetric on a finite set of objects is considered. In the framework of the proposed scheme, algorithms correcting the changes of the distance between a pair of objects by a given quantity that preserve the metric properties are examined. Sufficient conditions under which the correction of semimetrics using the proposed three-step scheme actually completes in two steps and in some special cases even after the first step are obtained. Semimetric similarity functionals are considered, and the correction algorithms are matched to those functionals.     

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.     

一类多维非线性反应扩散方程有限差分格式的稳定性问题(英文)

本文研究了一类多维线性反应扩散方程差分格式的稳定性.利用量未知元方法,建立了具有增量未知元的有限差分格式;然后利用非线性Galerkin方法,得到该差分格式的稳定性条件.通过对该格式的稳定性分析,说明和经典的差分格式的稳定性相比较,带有增量未知元的有限差分格式的稳定性得到了提高.     

Finite Difference Method for an Optimal Control Problem for a Nonlinear Time-dependent Schrödinger Equation

The finite difference method is applied to an optimal control problem for a system governed by a nonlinear Schrödinger equation with a complex coe?cient. The optimal control problem is discretized by the finite difference method, the error estimate for the finite difference scheme is established and the convergence of difference approximations of the optimal control according to the functional is proved.     

双根树上二阶非齐次马氏链的强大数定律和Shannon-McMillan定理

本文首先研究了双根树上转移矩阵为逐点转移的二阶非齐次马氏链的强极限定理,同时得到双根树上二阶非齐次马氏链的强大数定律.最后,给出了双根树上二阶非齐次马氏链几乎处处收敛意义下的Shannon-McMillan定理.     

Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces

In this paper, we introduce a general iteration scheme for a finite family of asymptotically quasi-nonexpansive mappings. The new iterative scheme includes the modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor and Khan and Takahashi scheme as special cases. Our results are generalizations as well as refinement of several known results in the current literature.     

On finite difference approximation of a matrix-vector product in the Jacobian-free Newton-Krylov method

The Jacobian-free Newton-Krylov (JFNK) method is a special kind of Newton-Krylov algorithm, in which the matrix-vector product is approximated by a finite difference scheme. Consequently, it is not necessary to form and store the Jacobian matrix. This can greatly improve the efficiency and enlarge the application area of the Newton-Krylov method. The finite difference scheme has a strong influence on the accuracy and robustness of the JFNK method. In this paper, several methods for approximating the Jacobian-vector product, including the finite difference scheme and the finite difference step size, are analyzed and compared. Numerical results are given to verify the effectiveness of different finite difference methods.     

On finite difference approximation of a matrix-vector product in the Jacobian-free Newton–Krylov method

The Jacobian-free Newton–Krylov (JFNK) method is a special kind of Newton–Krylov algorithm, in which the matrix-vector product is approximated by a finite difference scheme. Consequently, it is not necessary to form and store the Jacobian matrix. This can greatly improve the efficiency and enlarge the application area of the Newton–Krylov method. The finite difference scheme has a strong influence on the accuracy and robustness of the JFNK method. In this paper, several methods for approximating the Jacobian-vector product, including the finite difference scheme and the finite difference step size, are analyzed and compared. Numerical results are given to verify the effectiveness of different finite difference methods.     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes

In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared.     

A marching-on in time meshless kernel based solver for full-wave electromagnetic simulation

A meshless particle method based on an unconditionally stable time domain numerical scheme, oriented to electromagnetic transient simulations, is presented. The proposed scheme improves the smoothed particle electromagnetics method, already developed by the authors. The time stepping is approached by using the alternating directions implicit finite difference scheme, in a leapfrog way. The proposed formulation is used in order to efficiently overcome the stability relation constraint of explicit schemes. In fact, due to this constraint, large time steps cannot be used with small space steps and vice-versa. The same stability relation holds when the meshless formulation is applied together with an explicit finite difference scheme accounted for the time stepping. The computational tool is assessed and first simulation results are compared and discussed in order to validate the proposed approach.     

自然对流换热问题基于混合元法的差分格式及其数值模拟

研究自然对流换热问题,通过对于空间变量采用有限元离散而对于时间变量用差分离散,导出一种基于混合有限元法的最低阶的差分格式,这种格式可以同时求出流体的速度、温度和压力的数值解,并给出了模拟方腔流的自然换热的数值例子。