精细积分方法研究综述

A survey of the precise integration method

对于线性常微分方程初值和两点边值问题,精细积分方法可给出计算机上的精确解.本文总结了精细积分方法的基本思想和算法的进一步发展.在初值问题精细积分方法方面,详细综述了精细积分方法的基本思想、对非齐次项的处理技术、大规模问题求解技术以及时变、非线性微分方程的求解.在两点边值问题精细积分方法方面,介绍了处理边值问题的基本思想和求解过程,总结了两点边值问题精细积分方法在各个领域的应用.最后,讨论了初值和边值问题精细积分方法的联系和区别,从而为精细积分方法的理解和应用提供了新的视角.

For the initial value and two-point boundary value problems of linear ordinary differential equation, the Precise Integration Method（PIM） gives exact solutions in the computer accuracy sense. In this paper, the basic idea and the further development of PIM are surveyed. For the initial value problems, the basic idea of PIM, the methods for integrating the nonhomogeneous term, the methods for large scale problems and the application of PIM in time-varying and nonlinear system are surveyed. For two-point boundary value problems, the basic idea and the fundamental formula of PIM are given and the application of the PIM for two-point boundary value problems in many fields are surveyed. Finally, the relationship between the PIM for the initial value and two-point boundary value problems is discussed, which gives a new angle for understanding and application of PIM.