The Obreshkov method is a single-step multi-derivative method used in the numerical solution of differential equations and has been used in recent years in efficient circuit simulation. It has been shown that it can be made of arbitrary high local order of convergence while maintaining unconditional numerical stability. Nevertheless, the theoretical basis for the high order of convergence has been known only for the special case where the underlying system of differential equations is of the ordinary type, i.e., for ordinary differential equations (ODE). On the other hand, theoretical analysis of the order of convergence for the more general case of a system consisting of differential and algebraic equations (DAE) is still lacking in the literature. This paper presents the theoretical characterization for the local order of convergence of the Obreshkov method when used in the numerical solution of a system of DAE. The contribution presented in this paper demonstrates that, in DAE, the local order of convergence is a function of the differentiation index of the system and, under certain conditions, becomes lower than the order obtained in ODE.
翻译:Obreshkov方法是用于差别方程数字解决办法的单步多分流方法,近年来一直用于高效电路模拟,已经表明,它可以在保持无条件数字稳定性的同时,以任意的高度当地趋同顺序进行,但是,高度趋同的理论基础只对差异方程基础系统属于普通类型的特殊情况,即普通差别方程(ODE)是已知的。另一方面,文献中仍然缺乏对由差别和代数方程构成的系统(DAE)这一更为一般性情况的趋同顺序的理论分析,本文介绍了在DAE系统数字解决办法中使用的Obreshkov方法对当地趋同顺序的理论定性。本文中提供的材料表明,在DAE中,当地趋同顺序是系统差别指数的函数,在某些条件下,与Obreshkov方法的相趋同顺序的理论特征比在数字解决办法中使用的Obreshkov方法。