Symbolic-numeric methods for improving structural analysis of differential-algebraic equation systems

TL;DR

This paper enhances the reliability of structural analysis of differential-algebraic equations by developing symbolic-numeric conversion methods to address failures of the $\Sigma$-method, a popular analysis technique.

Contribution

It introduces two novel symbolic-numeric conversion methods that transform problematic DAEs into equivalent forms where the $\Sigma$-method succeeds, improving structural analysis accuracy.

Findings

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

The proposed conversion methods successfully fix these failures.

Enhanced structural analysis reliability for DAE systems.

Abstract

Systems of differential-algebraic equations (DAEs) are generated routinely by simulation and modeling environments such as Modelica and MapleSim. Before a simulation starts and a numerical solution method is applied, some kind of structural analysis is performed to determine the structure and the index of a DAE. Structural analysis methods serve as a necessary preprocessing stage, and among them, Pantelides's algorithm is widely used. Recently Pryce's $\Sigma$-method is becoming increasingly popular, owing to its straightforward approach and capability of analyzing high-order systems. Both methods are equivalent in the sense that when one succeeds, producing a nonsingular system Jacobian, the other also succeeds, and the two give the same structural index. Although provably successful on fairly many problems of interest, the structural analysis methods can fail on some simple, solvable DAEs and give incorrect structural information including the index. In this report, we focus on the $\Sigma$-method. We investigate its failures, and develop two symbolic-numeric conversion methods for converting a DAE, on which the $\Sigma$-method fails, to an equivalent problem on which this method succeeds. Aimed at making structural analysis methods more reliable, our conversion methods exploit structural information of a DAE, and require a symbolic tool for their implementation.