【作者】 朱婷；

【导师】 肖爱国；

【作者基本信息】 湘潭大学 ， 计算数学， 2009， 硕士

【摘要】 刚性振荡问题常出现在现代科学技术的许多领域,其数值方法的研究具有广泛的应用前景.由于刚性振荡问题具有刚性和振荡性双重特性,其高效数值求解具有一定的挑战性.多年来,许多学者一直在关注并努力获得数值求解的高效算法。本文主要是在前人工作的基础上对求解刚性振荡问题的对角隐式Runge-Kutta方法进行研究,通过提高方法的代数阶、稳定性条件,以及对相误差和耗散误差的控制来实现方法的有效性.全文由五章组成.第一章阐述了研究背景和本文的主要工作。第二章介绍刚性振荡问题及其数值方法,包括2级3阶对角隐式Runge-Kutta方法、第一级为显式的3级3阶对角隐式Runge-Kutta方法、3级4阶对称对角隐式Runge-Kutta方法和第一级为显式的4级4阶对称对角隐式Runge-Kutta方法.第三章给出了方法A-稳定时方法系数应该满足的范围.第四章对所构造的满足阶条件和稳定性要求的方法进行相误差和耗散误差分析.第五章通过求解实际的刚性振荡问题,来验证所构造的方法的有效性.更多还原

【Abstract】 Stiff oscillatory problems axe involved in various fields of modern science and technology. The research on their numerical methods has wide prospect in applications. Due to its two-sided characteristics, namely, stiffness and oscillation, it is rather difficult and challenging to obtain highly-efficient numerical methods, which are attracting many scholars’ attention for a long time.This paper mainly consider diagonally implicit Runge-Kutta methods for solving stiff problems with oscillating solutions based on other scholars’ research.We make our methods more effective by improving the algebraic order, stability conditions of the methods, and controlling the phase error and amplification error. This paper contains five parts.In chapter 1, we introduce the background of research and main work of this paper.In chapter 2, we introduce the considered problem and the RK methods for solving it, such as two-stage diagonally-implicit Runge-Kutta methods of order three, three-stage diagonally-implicit Runge-Kutta methods with an explicit first stage of order three, three-stage symmetric diagonally-implicit Runge-Kutta methods of order four and four-stage symmetric diagonally-implicit Runge-Kutta methods with an explicit first stage of order four.In chapter 3, we analyze the scope which the coefficients of the methods should satisfy when the methods are A-stable.In chapter 4, we study the phase error and amplification error of the methods to make the order of them as high as possible.In chapter 5, we take our methods to numerical experiments. It is shown that the constructed methods are efficient.更多还原