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三阶线性常微分方程Sinc方程组的结构预处理方法

任志茹   

  1. 中国科学院数学与系统科学研究院, 计算数学与科学工程计算研究所 科学与工程计算国家重点实验室, 北京 100190
  • 收稿日期:2013-01-31 出版日期:2013-08-15 发布日期:2013-09-07

任志茹. 三阶线性常微分方程Sinc方程组的结构预处理方法[J]. 计算数学, 2013, 35(3): 305-322.

Ren Zhiru. ON STRUCTURED PRECONDITIONING METHODS FOR SINC SYSTEMS OF LINEAR THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2013, 35(3): 305-322.

ON STRUCTURED PRECONDITIONING METHODS FOR SINC SYSTEMS OF LINEAR THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Ren Zhiru   

  1. State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing 100190, China
  • Received:2013-01-31 Online:2013-08-15 Published:2013-09-07
三阶线性常微分方程在天文学和流体力学等学科的研究中有着广泛的应用.本文介绍求解三阶线性常微分方程由Sinc方法离散所得到的线性方程组的结构预处理方法.首先, 我们利用Sinc方法对三阶线性常微分方程进行离散,证明了离散解以指数阶收敛到原问题的精确解.针对离散后线性方程组的系数矩阵的特殊结构, 提出了结构化的带状预处理子,并证明了预处理矩阵的特征值位于复平面上的一个矩形区域之内.然后, 我们引入新的变量将三阶线性常微分方程等价地转化为由两个二阶线性常微分方程构成的常微分方程组, 并利用Sinc方法对降阶后的常微分方程组进行离散.离散后线性方程组的系数矩阵是分块2×2的, 且每一块都是Toeplitz矩阵与对角矩阵的组合.为了利用Krylov子空间方法有效地求解离散后的线性方程组,我们给出了块对角预处理子, 并分析了预处理矩阵的性质.最后, 我们对降阶后二阶线性常微分方程组进行了一些比较研究.数值结果证实了Sinc方法能够有效地求解三阶线性常微分方程.
The third-order ordinary differential equations have been widely used in the studies of astronomy, fluid dynamics and other fields. In this paper, we study the structured preconditioning methods for the linear system arising from the Sinc discretizations of the linear third-order ordinary differential equations. First, we discretize the boundary value problems of linear third-order ordinary differential equations by Sinc methods and prove that the discrete solutions converge exponentially to the true solutions. According to the special structure of the coefficient matrix, we construct a banded preconditioner for the coefficient matrix of the discretized linear system and demonstrate that the eigenvalues of the preconditioned matrix are uniformly bounded within a rectangle on the complex plane. Then we introduce variable replacements to transform the linear third-order ordinary differential equations into systems of two second-order ordinary differential equations. Using Sinc methods to discretize the system of order-reduced ordinary differential equations, we get the discretized linear system with the coefficient matrix being block two-by-two structure and each block being a combination of Toeplitz and diagonal matrices. In order to solve the linear system effectively by Krylov subspace iteration methods, we propose the block-diagonal preconditioner and analyze the properties of the preconditioned matrix. Finally, we give comparative study on the system of order-reduced ordinary differential equations. Numerical examples are used to show the effectiveness of the proposed preconditioning methods.

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