On the numerical solution of two-point boundary value problems II
On the numerical solution of two-point boundary value problems II
复制标题
关于两点边值问题的数值解II
DOI：
10.1002/cpa.3160470806
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发表时间：
1994
影响因子：
3
通讯作者：
V. Rokhlin
中科院分区：
文献类型：
--
作者：
Page Starr;V. Rokhlin
摘要：在最近的一份文件中，格林加德和罗克林介绍了一种数值技术，快速解决积分方程的线性两点边值问题的二阶常微分方程。在本文中，我们将该方法推广到常微分方程组。在将微分方程组化为第二类积分方程组后，采用高阶Nystrom格式对第二类积分方程组进行离散。然后构造了一个有点复杂的分析装置，它允许使用O（N）求解离散系统。p的平方。n立方）运算，其中N是区间上的节点数，p是期望的收敛阶，n是系统中的方程数。因此，积分方程公式的优点（小的条件数，不敏感的边界层，不敏感的端点奇点等）被保留，同时实现了计算效率以前只提供有限差分的有限元方法。本文还提出了一种求解一阶非线性方程组边值问题的Newton方法，其中每个Newton方程都是一个第二类积分方程的解，从而得到了积分方程在求解非线性边值问题时的分析和数值优势.（kr）
Abstract : In a recent paper Greengard and Rokhlin introduce a numerical technique for the rapid solution of integral equations resulting from linear two-point boundary value problems for second order ordinary differential equations. In this paper, we extend the method to systems of ordinary differential equations. After reducing the system of differential equations to a system of second kind integral equations, we discretize the latter via a high order Nystrom scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O(N . p squared . n cubed) operations, with N the number of nodes on the interval, p the desired order of convergence, and n the number of equations in the system. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end-point singularities, etc. ) are retained, while achieving a computational efficiency previously available only to finite difference of finite element methods. We in addition present a Newton method for solving boundary value problems for nonlinear first order systems in which each Newton iterate is the solution of a second kind integral equation; the analytical and numerical advantages of integral equations are thus obtained for nonlinear boundary value problems. (kr)