关于Ut=aUxx差分格式的进一步研究

对于热传导方程构造了两个高阶精度的差分格式，一个是三层七点显格式．另一个是三层九点隐格式．证明了差分格式的收敛性和稳定性，最后给出数值计算结果。     

高阶Schr(o)dinger方程双参数高精度恒稳差分格式

对高阶Schrodinger方程эu/эt=i(-1)^mэ^2mu/эx^2m构造一族含双参数的三层高精度隐式差分格式．当参数α=1／2，β=0时得到一个两层格式．并证明了：对任意非负参数α≥0，β≥0该格式都是绝对稳定的，并且其截断误差阶达到O((△t)^2 (△x)^6)．数值例子表明：本文所建立的差分格式是有效的，理论分析与实际计算相吻合．     

关于色散方程u_t=au_(xxx)一类显式差分格式的讨论

关于色散方程u_t=au_(xxx)差分格式的讨论,在[1]和[2]中,分别提出了中层为五点和六点的显式差分格式,其稳定区域分别为 0≤r≤0.7016和-0.0625 ≤r≤1.1851.本文针对这一问题,讨论中层为七点的一类差分格式的稳定性.[1]中格式是本文的特例,并且这类格式的最佳稳定区域为0≤r≤2.394,大约是[2]中稳定范围的二倍,[1]中稳定范围的三倍.     

二维抛物型偏微分方程的绝对稳定显格式

提出了一个解二维抛物型偏微分方程初边值问题的绝对稳定分支三层显式差分格式，格式的局部截断误差阶为Ｏ（△ｔ２＋△ｘ２＋△ｙ２）．实算表明，格式的稳定性能与理论分析是一致的．     

一个 油藏模型的二阶差分格式

本文应用伸缩变换和阶降法在动网格上对石油科学研究中提出的一个漂移聚集油藏模型建立了一个三层线性化弱耦合差分格式，证明了所构造的差分格式是唯一可解的且在ｌ２范数下以Ｏ（ｒ＾２＋ｈ＾２）阶收敛。所建立的差分方程组均是三对角的，可用追赶法求解。     

解弥散问题的高精度差分格式

用待定系数法 ,对弥散方程构造了一个二层六阶精度的差分格式 ,给出了稳定条件 .用该格式可以直接从初始条件出发逐层求解 ,也可以在使用三层差分格式时 ,用来求第一层的数值解 u1j.     

RLW-KdV方程的紧致有限差分格式

本文研究了RLW-KdV方程的一个三层线性紧致有限差分格式.该格式是质量守恒和能量守恒的,用离散能量法证明了差分格式的收敛性和稳定性.所建格式的收敛阶为O(τ~2+h~4).数值实验验证了该格式的有效性和可靠性.     

色散方程两层绝对稳定隐格式

对色散方程ut=auxxx的初边值问题,构造了两组带参数绝对稳定两层四点去心隐式差分格式,其截断误差为0（τ＋h^2）．若适当选取参数,格式的精确度可高达0（τ＋h^3）．若特殊的令某个节点前的系数为0,则得到二阶的半显格式．最后的数例验证了理论分析的正确性．这是两组灵活、实用的差分格式．     

一维单个守恒律的一类二阶、几乎熵耗散的格式

本文考虑一维单个守恒律方程，对其设计了一个基于熵耗散的非线性守恒型差分格式．本格式的数值流函数是Lax-Freidrichs格式和Lax-Wendroff格式数值流函数的凸组合，凸组合中的系数是由考虑耗散熵来决定的．这样在解的光滑区域内，格式几乎、甚至完全是Lax-Wendroff格式，而在解的间断处，格式几乎、甚至完全是Lax—Freidrichs格式．从而消除了间断附近的非物理振荡，实现了计算的非线性稳定性．理论分析表明本格式在解的非极值点处是二阶精度的，而在解的极值点处至少有一阶精度．数值试验表明格式是有效的．     

对称正则长波方程的一个新的守恒差分格式

本文考虑了具有齐次边界条件的对称正则长波方程的有限差分格式,提出了一个三层守恒的有限差分格式,证明了格式的收敛性和稳定性,从理论上得到了收敛阶为O(h2+τ2).通过数值试验表明,所提的方法是可靠有效的.     

A linearized and second‐order unconditionally convergent scheme for coupled time fractional Klein‐Gordon‐Schrödinger equation

In this work, we study finite difference scheme for coupled time fractional Klein‐Gordon‐Schrödinger (KGS) equation. We proposed a linearized finite difference scheme to solve the coupled system, in which the fractional derivatives are approximated by some recently established discretization formulas. These formulas approximate the solution with second‐order accuracy at points different form the grid points in time direction. Taking advantage of this property, our proposed linearized scheme evaluates the nonlinear terms on the previous time level. As a result, iterative method is dispensable. The coupled terms in the scheme bring difficulties in analysis. By carefully studying these effects, we proved that the proposed scheme is unconditionally convergent and stable in discrete norm with energy method. Numerical results are included to justify the theoretical statements.     

常系数扩散方程的三层九点差分格式的稳定性（英文）

In this paper,we study a numerical solution of diffusion equation.We propose a three level-nine-point implicit difference scheme and prove the difference scheme is compatible with diffusion equation,second order convergent,unconditionally stable.A numerical experiments show,the difference scheme works well inside domain,but not near the discontinuous initial-boundary points,there are still has a vibration even though it was proved unconditionally stable theoretically.We take an action to solve the disturbance,give an Algorithm,Algorithm says,we must do some primal work at the discontinuous-initial-boundary points,then starting numerical solution according the three level-nine-point implicit difference scheme we proposed in this paper.The numerical example is done once again,and there is no disturbance or vibration,our Algorithm performed well all in domain and on the boundary points with small error and good accuracy,so the Algorithm we recommended is feasible and effective.     

An O(k2 + kh2 + h4) arithmetic average discretization for the solution of 1‐D nonlinear parabolic equations

This article develops a new two‐level three‐point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation ε u_xx = f(x, t, u, u_x, u_t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where ε > 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (?u/?x). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A quadrature finite element Galerkin scheme for a biharmonic problem on a rectangular polygon

A quadrature Galerkin scheme with the Bogner–Fox–Schmit element for a biharmonic problem on a rectangular polygon is analyzed for existence, uniqueness, and convergence of the discrete solution. It is known that a product Gaussian quadrature with at least three‐points is required to guarantee optimal order convergence in Sobolev norms. In this article, optimal order error estimates are proved for a scheme based on the product two‐point Gaussian quadrature by establishing a relation with an underdetermined orthogonal spline collocation scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A hybrid approximation method for equilibrium,variational inequality and fixed point problems

The purpose of this paper is to present an iterative scheme by a hybrid method for finding a common element of the fixed points of $?$-asymptotically nonexpansive mapping, the set of solutions of the equilibrium problem and the set of solutions of the variational inequality for an inverse strongly monotone operator in the framework of Banach spaces. We show that the iterative scheme converges strongly to a common element of the above three sets under appropriate conditions.     

关于三个非扩张映射收敛于其公共不动点的具误差的三重迭代法

在一致凸Banach空间中,研究了关于三个不同非扩张映射的具误差的三步迭代算法,并在一定条件下证明了此迭代序列收敛于其公共不动点的弱强收敛定理.     

Solving second order initial value problems by a hybrid multistep method without predictors

A three-step seventh order hybrid linear multistep method (HLMM) with three non-step points is proposed for the direct solution of the special second order initial value problems (IVPs) of the form y″ = f(x, y) with an extension to y″ = f(x, y, y′). The main method and additional methods are obtained from the same continuous scheme derived via interpolation and collocation procedures.The stability properties of the methods are discussed by expressing them as a one-step method in higher dimension. The methods are then applied in block form as simultaneous numerical integrators over non-overlapping intervals. Numerical results obtained using the proposed block form reveal that it is highly competitive with existing methods in the literature.     

Gaussian spectral rules for second order finite-difference schemes

Earlier the authors suggested an algorithm of grid optimization for a second order finite-difference approximation of a two-point problem. The algorithm yields exponential superconvergence of the Neumann-to-Dirichlet map (or the boundary impedance). Here we extend that approach to PDEs with piecewise-constant coefficients and rectangular homogeneous subdomains. Examples of the numerical solution of the 2-dimensional oscillatory Helmholtz equation exhibit exponential convergence at prescribed points, where the cost per grid node is close to that of the standard five-point finite-difference scheme. Our scheme demonstrates high accuracy with slightly more than two grid points per wavelength and reduces the grid size by more than three orders as compared with the standard scheme.     

Smooth interpolation of a mesh of curves

The interpolation of a mesh of curves by a smooth regularly parametrized surface with one polynomial piece per facet is studied. Not every mesh with a well-defined tangent plane at the mesh points has such an interpolant: the curvature of mesh curves emanating from mesh points with an even number of neighbors must satisfy an additional vertex enclosure constraint. The constraint is weaker than previous analyses in the literature suggest and thus leads to more efficient constructions. This is illustrated by an implemented algorithm for the local interpolation of a cubic curve mesh by a piecewise [bi]quarticC ¹ surface. The scheme is based on an alternative sufficient constraint that forces the mesh curves to interpolate second-order data at the mesh points. Rational patches, singular parametrizations, and the splitting of patches are interpreted as techniques to enforce the vertex enclosure constraint.Communicated by Wolfgang Dahmen.     

A high-order compact difference scheme for 2D Laplace and Poisson equations in non-uniform grid systems

In this study, a high-order compact scheme for 2D Laplace and Poisson equations under a non-uniform grid setting is developed. Based on the optimal difference method, a nine-point compact difference scheme is generated. Difference coefficients at each grid point and source term are derived. This is accomplished through the consideration of compatibility between the partial differential equation and its difference discretization. Theoretically, the proposed scheme has third- to fourth-order accuracy; its fourth-order accuracy is achieved under uniform grid settings. Two examples are provided to examine performance of the proposed scheme. Compared with the traditional five-point difference scheme, the proposed scheme can produce more accurate results with faster convergence. Another reference scheme with the same nine-point grid stencil is derived based on the five-point scheme. The two nine-point schemes have the same coefficients for each grid points; however, their coefficients for the source term are different. The overall accuracy level of the solution resulting from the proposed scheme is higher than that of the nine-point reference scheme. It is also indicated that the smoothness of grids has significant effects on accuracy and convergence of the solutions; efforts in optimizing the grid configuration and allocation can improve solution accuracy and efficiency. Consequently, with the proposed method, solution under the non-uniform grid setting with appropriate grid allocation would be more accurate than that under the uniform-grid manipulation, with the same number of grid points.