Orthogonal Spline Collocation Methods for Partial Differential Equations

本项目致力于开发一种新的矩阵分解算法，用于求解矩形区域上可分的二阶线性问题的正交样条配置(即高斯点上的样条配置)时产生的线性代数方程组。这些算法在求解不可分边值问题、一般区域上的问题和含时问题上都有应用。矩阵分解算法的成功与否取决于对一维空间中相应的两点边值问题中矩阵的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的二维离散问题简化为关于另一个坐标变量的一组独立的离散两点边值问题的求解。开发了一个在每个坐标方向的非均匀划分上实现任意阶多项式正交样条配置算法的软件包。作为特例，该程序包包括了泊松方程在不同坐标系下的分段Hermite双三次正交样条配置的快速变换求解器。

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.