Numerical Diagnostics for Systems of Differential Algebraic Equations

TL;DR

This paper introduces two dynamical systems-based methods to detect and localize convergence issues in numerical solutions of differential algebraic equations, applicable to various solvers and systems.

Contribution

The paper presents novel techniques rooted in bifurcation analysis for diagnosing convergence problems in homotopy and DAE systems, including localization of anomalies.

Findings

Methods successfully detect convergence anomalies in nonlinear circuits.

Localization of issues to specific components demonstrated.

Applicable to a range of numerical solvers and systems.

Abstract

In many commercial and academic settings, numerical solvers fail to achieve their theoretical performance levels due to issues in the system definition, parameterization, and even implementation. We propose a pair of methods for detecting and localizing these convergence rate issues in applications that can be treated as homotopy problems including numerical continuation and the evolution of differential algebraic equations. Both approaches are rooted in dynamical systems theory, in particular, the numerical techniques used to perform bifurcation studies on "black-box" systems, and can be applied across a range of numerical solvers and systems without significant modification. In a general problem, these methods will detect certain classes of convergence anomalies, and indicate the states that are affected by their existence. However in applications like circuit simulation, certain classes of anomalies can be localized to an individual component through the use of a small number of additional computational experiments. To demonstrate the efficacy of the approaches for detection and localization, we apply them to a pair of nonlinear circuits: a diode bridge and a simplified model of a power channel