微分-代数方程中Obreshkov方法收敛阶的刻画
Characterizing Order of Convergence in the Obreshkov Method in Differential-Algebraic Equations
摘要：
Obreshkov方法是一种单步多导数方法，用于微分方程的数值解，近年来已被用于有效的电路仿真。结果表明，它可以在保持无条件数值稳定性的同时，具有任意高的局部收敛阶。然而，高阶收敛的理论基础仅在特殊情况下才为人所知，在这种特殊情况下，基本的微分方程系统是普通类型的，即，对于常微分方程（ODE）。另一方面，在文献中仍然缺乏对由微分和代数方程（DAE）组成的系统的更一般情况下的收敛阶的理论分析。本文给出了Obreshkov方法在求解微分方程组的数值解时局部收敛阶的理论刻画。本文的贡献表明，在DAE中，局部收敛阶是系统的微分指数的函数，在一定条件下，变得低于在常微分方程中获得的阶。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
