| 求解常微分方程边值问题新的数值方法 |
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| 引用本文： | 刘秋生,沈孟育,刘晔.求解常微分方程边值问题新的数值方法[J].清华大学学报(自然科学版),1996(4). |
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| 作者姓名： | 刘秋生 沈孟育 刘晔 |
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| 作者单位： | 清华大学工程力学系!北京100084(刘秋生，沈孟育)，应用数学系!北京100084(刘晔) |
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| 摘 要： | 提出求解常微分方程的一种新的数值方法——解析离散法。与差分法不同，该法在计算域离散化后，直接解出离散点处的函数值和任意阶导数值，并通过Ｔａｙｌｏｒ展开获得任意点处的函数及其导数值。文中给出了求解常微分方程的、格式点数任意、任意高阶精度的通用方法。该法可用大步长，精度可达到步长的任意次幂，各类边界条件的处理都很方便。具有十阶精度的算例证实本文方法是成功的。
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| 关 键 词： | 解析离散法 高精度格式 常微分方程 边值 |
New numerical method for general boundary value problems of ordinary differential equations |
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| Liu Qiusheng, Shen Mengyu, Liu Ye.New numerical method for general boundary value problems of ordinary differential equations[J].Journal of Tsinghua University(Science and Technology),1996(4). |
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| Authors: | Liu Qiusheng Shen Mengyu Liu Ye |
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| Institution: | Liu Qiusheng, Shen Mengyu, Liu Ye Department of Engineering Mechanics,Tsinghua University,Department of Applied Mathematics,Tsinghua University |
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| Abstract: | A new numerical method, analytical discrete method, for solving ordinary differential equations is proposed. Different from any existing numerical method, the present method directly works out the function value and its some-order derivertives at grids after the discretization of the solution domain. Then the function value and its derivertives at any point are Obtained by use of Taylor's expansion. In this method, a large step length and schemes with any number of grids can be used and any-order accuracy can be obtained. Besides , any type boundary conditions can be treated easily and conveniently. The numerical experiments with ten-order accuracy show that the present method is successful. |
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| Keywords: | analytical discrete method high accuracy scheme ordinary differential equation boundary value |
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