• 论文 •

### 三阶线性常微分方程Sinc方程组的结构预处理方法

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

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

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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