Fisher型方程的Jacobi谱配点法（英文）

主要讨论了以Jacobi-Gauss-Lobatto点为配置点的谱配点法数值求解具有初边值条件的Fisher型方程.借助于插值和由此产生的微分矩阵,将Fisher型方程转化为常微分方程组,再利用四阶Runge-Kutta法求解该常微分方程组.文中以一维Fisher型方程为例证明了该方法具有谱精度,并给出了四个Fisher型方程算例.数值例子验证了Jacobi谱配点法具有高精度和快速收敛性.     

多维区域中非线性偏微分方程的修正Laguerre谱与拟谱方法

研究多维区域中非线性偏微分方程的谱与拟谱方法．建立了修正Laguerre正交逼近与插值结果,这些结果对于建立和分析无界区域中的数值方法起着重要的作用．作为结果的一个应用,研究了二维无界区域中的Logistic方程的修正Laguerre谱格式,证明了它的稳定性和收敛性．数值试验结果表明所提出方法具有很高的精度,与理论分析结果完全吻合．     

偏微分方程的区间小波自适应精细积分法

利用插值小波理论构造了拟Shannon区间小波,并结合外推法给出了一种求解非线性常微分方程组的时间步长自适应精细积分法,在此基础上构造了求解非线性偏微分方程的区间小波自适应精细积分法（AIWPIM）．数值结果表明,该方法在计算精度上优于将小波和四阶Runge-Kutta法组合得到的偏微分方程的数值求解方法,而计算量则相差不大．该文方法通过Burgers方程给出,但适用于一般情形．     

求解二维Poisson方程的重心有理插值配点法

首先介绍了重心Lagrange插值法,然后通过改变重心Lagrange插值法的插值权函数,重点给出了重心有理插值的具体形式.基于等距节点和Chebyshev节点这两类插值节点,利用重心有理插值配点法求解了二维Poisson方程,并比较了采用上述两种插值节点时的计算精度.数值算例表明,重心有理插值配点法具有稳定性好,计算精度高和程序编写简单的特点.     

第二类Volterra型积分方程的Chebyshev-Legendre谱配置方法

提出了一种新的求解第二类线性Volterra型积分方程的Chebyshev谱配置方法.该方法分别对方程中积分部分的核函数和未知函数在Chebyshev-Gauss-Lobatto点上进行插值,通过Chebyshev-Legendre变换,把插值多项式表示成Legendre级数形式,从而将积分转换为内积的形式,再利用Legendre多项式的正交性进行计算.利用Chebyshev插值算子在不带权范数意义下的逼近结果,对该方法在理论上给出了L∞范数意义下的误差估计,并通过数值算例验证了算法的有效性和理论分析的正确性.     

任意四边形杂交Mindlin板单元

本文把基于虚功原理的杂交有限元模型用于板弯曲问题，构造了一个考虑横向剪应变的任意四边形新型板单元，本单元的突出优点是采用了一种比较合理的位移插值函数，使之能较真实地模拟各类板的变形，且用的自由度最少。文中对此单元作了比较广泛的数值试验，计算结果表明它对板厚有相当宽的适用范围，对于各种例题均能在较粗的网格下得到满意的精度．     

Thiele重心型矩阵值混合有理插值算法

利用Samelson型矩阵广义逆,构造了一种基于Thiele型连分式插值与重心有理插值的相结合的二元矩阵值混合有理插值格式,这种新的混合矩阵值有理插值函数继承了连分式插值和重心插值的优点,它的表达式简单,计算方便,数值稳定性好.该算法满足有理插值问题所给的插值条件,同时给出了误差估计分析.最后用数值算例验证了插值算法的有效性.     

计算半线性椭圆问题多解的一类谱Galerkin型搜索延拓法的收敛性分析

本文提出计算半线性椭圆边值问题多解的一类高效的谱Galerkin型搜索延拓法(SGSEM).该方法基于模型方程相应线性特征值问题的若干特征函数的线性组合构造多解初值,充分利用了传统搜索延拓法构造多解初值方面的优势.同时,采用插值系数Legendre-Galerkin谱方法离散模型问题,具有计算成本低、计算精度高的优点.运用Schauder不动点定理和其他技巧,本文严格证明了对应于每个特定真解的数值解的存在性以及限制在该真解一个充分小的邻域内的数值解的唯一性,并证明了其谱收敛性.数值结果验证了算法的可行性与高效性,并展示了不同类型的多解.     

非定常 N-S 方程的非线性 Galerkin 算法及其误差估计

本文给出了二维非定常N-S方程的三种数值格式,其中空间变量用谱非线性Galerkin算法进行离散,时间变量用有限差分离散,并研究了这些格式数值解的逼近精度．最后,给出了部分数值计算结果．     

无网格Chebyshev配点法求解二维位势问题

基于Chebyshev正交多项式插值理论和无网格配点技术,提出一种新型的无网格数值离散方法,称之为Chebyshev配点法.所提方法采用Chebyshev多项式的零点(Gauss-Lobatto节点)为插值节点,可最大限度地降低龙格现象,并且提供插值多项式的最佳一致逼近.数值算例表明,本文算法稳定,效率高,并可达到很高的计算精度.     

The Linear Rational Pseudospectral Method for Boundary Value Problems

We consider the version of the pseudospectral method for solving boundary value problems which replaces the differential operator with a matrix constructed from the elementary differentiation matrices whose elements are the derivatives of the Lagrange fundamental polynomials at the collocation points. The iterative solution of the resulting system of equations then requires the recurrent application of that differentiation matrix. Since global polynomial interpolation on the interval only gives useful approximants for points which accumulate in the vicinity of the extremities, the matrix is ill-conditioned. To reduce this drawback, we use Kosloff and Tal-Ezer's suggestion to shift the collocation points closer to equidistant by a conformal map. However, instead of applying their change of variable setting, we extend to stationary equations the linear rational collocation method introduced in former work on partial differential equations. Numerically about as efficient, this does not require any new coding if one starts from an efficient program for the polynomial differentiation matrices.This revised version was published online in October 2005 with corrections to the Cover Date.     

Collocation points distributions for optimal spacecraft trajectories

The method of direct collocation with nonlinear programming (DCNLP) is a powerful tool to solve optimal control problems (OCP). In this method the solution time history is approximated with piecewise polynomials, which are constructed using interpolation points deriving from the Jacobi polynomials. Among the Jacobi polynomials family, Legendre and Chebyshev polynomials are the most used, but there is no evidence that they offer the best performance with respect to other family members. By solving different OCPs with interpolation points not only taken within the Jacoby family, the behavior of the Jacobi polynomials in the optimization problems is discussed. This paper focuses on spacecraft trajectories optimization problems. In particular orbit transfers, interplanetary transfers and station keepings are considered.     

Callocation methods for volterra integrodifferential equations with singular kernels

A collocation method which uses Hermite cubic elements is proposed for the solution of Volterra integrodifferential equations with singular kernels. Optimum error estimates in the uniform norm are obtained by means of interpolation operators. We also report on results of numerical comparisons with one well established method and another new “modified collocation” scheme.     

On the stability of meshless symmetric collocation for boundary value problems

In this paper, we study the stability of symmetric collocation methods for boundary value problems using certain positive definite kernels. We derive lower bounds on the smallest eigenvalue of the associated collocation matrix in terms of the separation distance. Comparing these bounds to the well-known error estimates shows that another trade-off appears, which is significantly worse than the one known from classical interpolation. Finally, we show how this new trade-off can be overcome as well as how the collocation matrix can be stabilized by smoothing. AMS subject classification (2000)  65N12, 65N15, 65N35     

Linear barycentric rational collocation method for solving heat conduction equation

The linear barycentric rational collocation method for solving heat conduction equation is presented. The matrix form of discrete heat conduction equation by collocation method is also obtained. With the help of convergence rate of the barycentric interpolation, the convergence rate of linear barycentric rational collocation method for solving heat conduction equation is proved. At last, several numerical examples are provided to validate the theoretical analysis.     

Least Squares Solution of Bivariate Surface Fitting Problems Using Tensor Product Splines

The paper treats bivariate surface fitting problems, where the data points lie on lines parallel to one of the axes. The associated bivariate collocation matrix is investigated as a block Kronecker product of univariate collocation matrices. Based on various properties of this block Kronecker product, such scattered data are characterized where the associated interpolation problem using tensor product splines admits a unique solution.     

瞬态热传导的奇异边界法及其MATLAB实现

基于动力学问题时间依赖基本解的奇异边界法是一种无网格边界配点法.该方法引入源点强度因子的概念从而避免了基本解的源点奇异性,具有数学简单、编程容易、精度高等优点.将该方法用于瞬态热传导问题的数值模拟,运用MATLAB实现该问题的数值研究,并创建相应的MATLAB工具箱.针对二维和三维瞬态热传导问题,进行了基于反插值技术和经验公式的奇异边界法MATLAB算例实现.针对支撑圆坯低温瞬态温度场的模拟结果表明,瞬态热传导奇异边界法的MATLAB工具箱具有简单、方便、精确可靠的优点.研究成果有助于发展瞬态热传导的奇异边界法,并为瞬态热传导问题的数值分析和仿真提供了一种简单高效的模拟工具.     

On spline collocation methods for boundary integral equations in the plane

Nowadays boundary elemen; methods belong to the most popular numerical methods for solving elliptic boundary value problems. They consist in the reduction of the problem to equivalent integral equations (or certain generalizations) on the boundary Γ of the given domain and the approximate solution of these boundary equations. For the numerical treatment the boundary surface is decomposed into a finite number of segments and the unknown functions are approximated by corresponding finite elements and usually determined by collocation and Galerkin procedures. One finds the least difficulties in the theoretical foundation of the convergence of Galerkin methods for certain classes of equations, whereas the convergence of collocation methods, which are mostly used in numerical computations, has yet been proved only for special equations and methods. In the present paper we analyse spline collocation methods on uniform meshes with variable collocation points for one-dimensional pseudodifferential equations on a closed curve with convolutional principal parts, which encompass many classes of boundary integral equations in the plane. We give necessary and sufficient conditions for convergence and prove asymptotic error estimates. In particular we generalize some results on nodal and midpoint collocation obtained in [2], [7] and [8]. The paper is organized as follows. In Section 1 we formulate the problems and the results, Section 2 deals with spline interpolation in periodic Sobolev spaces, and in Section 3 we prove the convergence theorems for the considered collocation methods.     

积分算子方程的连续型配置方法

1．引言由于对积分算子方程来说，配置法比Galerkin法具计算量小的优点（少算一重积分），故配置法更受人们重视．但已有的文献几乎都是将配置空间取作非连续的分片多项式样条空间，以得到某种超收敛结果（如［1，2］）．这种方法存在下列不足：（a）光滑核Volterra积分方程与光滑核Fredholm积分方程具完全不同的收敛性质［1］，且需用不同的方法获得其加速收敛结果（比较［31与［4］），尽管Volterra积分方程在理论上被看作是Fredholm积分方程的特殊情形；（b）光滑核Volterra积分方程的配置解不具任何超收敛性，其迭代配置解也只在结点…