Conversion Methods for Improving Structural Analysis of Differential-Algebraic Equation Systems

TL;DR

This paper examines the limitations of existing structural analysis methods for differential-algebraic equations and introduces two conversion techniques to improve their success rate in identifying correct structural information.

Contribution

It identifies failure modes of the $\Sigma$-method and proposes two novel conversion methods to enhance structural analysis accuracy for DAEs.

Findings

The $\Sigma$-method can fail on simple, solvable DAEs.

Two conversion methods improve the success rate of structural analysis.

Converted DAEs are more likely to yield correct structural information.

Abstract

Differential-algebraic equation systems (DAEs) are generated routinely by simulation and modeling environments. Before a simulation starts and a numerical method is applied, some kind of structural analysis (SA) is used to determine which equations to be differentiated, and how many times. Both Pantelides's algorithm and Pryce's $\Sigma$-method are equivalent: if one of them finds correct structural information, the other does also. Nonsingularity of the Jacobian produced by SA indicates a success, which occurs on many problems of interest. However, these methods can fail on simple, solvable DAEs and give incorrect structural information including the index. This article investigates $\Sigma$-method's failures and presents two conversion methods for fixing them. Both methods convert a DAE on which the $\Sigma$-method fails to an equivalent problem on which this SA is more likely to succeed.