Characterizing Order of Convergence in the Obreshkov Method in Differential-Algebraic Equations

TL;DR

This paper analyzes the convergence order of the Obreshkov method for differential-algebraic equations, revealing its dependence on the system's differentiation index and extending understanding beyond ordinary differential equations.

Contribution

It provides the first theoretical characterization of the Obreshkov method's convergence order for DAE systems, linking it to the differentiation index.

Findings

Convergence order depends on the differentiation index.

Under certain conditions, the order is lower than in ODE.

Theoretical basis for high-order convergence in DAE is established.

Abstract

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.