Orthogonal Spline Collocation Methods for Partial Differential Equations

本项目涉及实现新的矩阵分解算法的发展，用于解决正交样条配置（即高斯点的样条配置）应用于矩形区域上可分离的二阶线性问题时产生的线性代数方程组。这些算法在解不可分边值问题、一般区域问题和时间相关问题中具有广泛的应用。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矩阵的特征值和特征向量的了解。关于一个坐标变量的特征系统的知识将原来的二维离散问题简化为求解关于另一个坐标变量的独立离散两点边值问题的集合。开发了一个软件包，实现了在各坐标方向的非均匀分区上任意阶多项式的正交样条配置算法。作为一个特例，该包包含了泊松方程在不同坐标系下分段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.