【作者】 王海霞；

【导师】 黄乘明；

【作者基本信息】 华中科技大学 ， 计算数学， 2006， 硕士

【摘要】 在电路分析、计算机辅助设计、多体力学系统的实时仿真、化学反应模拟以及管理系统等科学与工程应用领域中的许多问题,都可以抽象为一个微分代数系统。常用于解微分代数方程的方法主要是求解常微分方程的数值方法的推广。20世纪80年代,E.Lelarasmee等人在对超大规模集成电路进行数值模拟时,提出了波形松弛方法,用该方法处理模拟电路中相应的微分代数方程系统,取得了很好的数值效果。波形松弛方法的提出,为微分代数方程的数值求解提供了一种新的途径。随后,波形松弛方法的并行性及解耦性引起了研究者们的注意。波形松弛方法又称动态迭代方法,是一种连续时间的迭代过程,对迭代后的方程进行时间上的离散称为离散时间的动态迭代过程。波形松弛算法的基本思想就是把原来的大规模系统分解成多个松散耦合的规模较小的子系统,然后分别求解每个子系统。本文首先回顾了波形松弛方法的产生背景,针对连续时间和离散时间波形松弛迭代格式,总结了这一方法的发展历史及研究概况。接着在吸收前人的优秀成果基础上,研究了微分代数方程的离散波形松弛。第二章涉及隐式微分代数方程,对应用向后微分公式导致的离散时间波形松弛方法进行了进一步的研究,得到了收敛性结果。第三章涉及显式微分代数方程,针对单支方法和线性多步法导致的离散时间波形松弛法开展收敛性研究。通过运用矩阵分裂的相关性质,我们在一定的假设条件下证明了离散波形松弛方法的收敛性,并获得了明确的时间步长收敛区间。第四章针对一类积分微分代数方程,研究了基于单支θ?方法和线性θ?方法的离散时间波形松弛方法,给出了相应的收敛性结论。最后第五章针对本文所得的主要结论,给出了具体的数值试验,验证结论的正确性和方法的有效性。更多还原

【Abstract】 Many problems in areas of science and engineering can be modeled by differential algebraic systems. Methods commonly used in the solution of differential-algebraic equation are an extension of numerical methods for solving ordinary differential equations. In 1980’s, E.Lelarasmee et.al introduced waveform relaxation approach to solve the dynamical systems which are described by a system of mixed implicit algebraic-differential equations. Theoretical and computational studies show the method to be efficient and reliable. For the numerical solution of differential-algebraic equation waveform relaxation method provides a new way. After that, Parallel and decoupling properties of waveform relaxation approach begin to get more attentions.Waveform relaxation (WR) is a decoupling technique of large systems, which is first applied in circuit simulation. It is sometimes called dynamic iteration. The iterative equation for the discrete time is called as discrete-time dynamic iterative process. The key idea of waveform relaxation algorithm is to decompose original large systems into several subsystems, and then solve for each subsystem.In this paper we first review the background to the selection of waveform relaxation algorithm and introduce the history and development of this approach research. We mainly study the convergence conditions of the discrete-time waveform relaxation algorithm. In section two, we first discuss the discrete-time waveform relaxation method for an implicit system of nonlinear differential-algebraic equations which is derived by the backward-differentiation formulas (BDFs) and give some conclusions for the convergence of the methods. Then in section three, by discretising continuous-time waveforms using one-leg methods and linear multistep methods, we study the discrete-time waveform relaxation solutions, and some convergence conclusions are achieved. The proof is based on the properties of matrices. Under some assumptions, we get the convergence interval of the time step. By use of the one legθ-method and the linearθ-method, we obtain some conclusions for the convergence of the discrete waveforms of nonlinear integral-differential-algebraic equations in section four. Numerical experiments are provided to illustrate the theoretical results in section five.更多还原