Orthogonal Spline Collocation Methods for Partial Differential Equations

偏微分方程的正交样条配置方法

• 批准号: 9403461
• 负责人: Graeme Fairweather
• 金额: $ 12万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-09-15 至 1996-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403461&HistoricalAwards=false
• 关键词: Orthogonal; Spline; Collocation; Methods; Partial

This project is concerned with the development of implementing new matrix decomposition algorithms for solving the systems of linear algebraic equations arising when orthogonal spline collocation (that is, spline collocation at Gauss points) is applied to separable, second order, linear problems on rectangular regions. These algorithms have application in the solution of nonseparable boundary value problems, problems on general regions, and time dependent problems. The success of matrix decomposition algorithms depends on knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two-point boundary value problems in one space dimension. Knowledge of such an eigensystem with respect to one coordinate variable reduces the original two-dimensional discrete problem to solving a collection of independent discrete two-point boundary value problems with respect to the other coordinate variable. A software package is developed implementing orthogonal spline collocation algorithms with polynomials of arbitrary order on nonuniform partitions in each coordinate direction. The package includes, as a special case, a fast transform solver for piecewise Hermite bicubic orthogonal spline collocation for Poisson's equation in various coordinate systems.

该项目涉及开发实现新的矩阵分解算法，用于求解当正交样条配置（即高斯点处的样条配置）应用于矩形区域上的可分离二阶线性问题时产生的线性代数方程组。 这些算法可应用于解决不可分离的边值问题、一般区域问题和时间相关问题。 矩阵分解算法的成功取决于对一维空间中相应两点边值问题中出现的矩阵特征值和特征向量的了解。 关于一个坐标变量的这种特征系统的知识将原始二维离散问题简化为求解关于另一个坐标变量的一组独立离散两点边值问题。 开发了一个软件包，在每个坐标方向的非均匀分区上实现具有任意阶多项式的正交样条配置算法。 作为一种特殊情况，该软件包包括一个快速变换求解器，用于各种坐标系中泊松方程的分段 Hermite 双三次正交样条配置。