Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations

TL;DR

This paper introduces a new pseudospectrum concept for matrix pencils to analyze transient growth in differential-algebraic equations, aiding stability assessment in fluid mechanics models.

Contribution

It proposes a novel pseudospectrum definition for matrix pencils, enabling better transient growth bounds and incorporating physically relevant norms.

Findings

New pseudospectrum bounds transient growth effectively.

Approximate pseudospectra derived from invariant subspaces.

Application to fluid mechanics demonstrates practical utility.

Abstract

To understand the solution of a linear, time-invariant differential-algebraic equation, one must analyze a matrix pencil (A,E) with singular E. Even when this pencil is stable (all its finite eigenvalues fall in the left-half plane), the solution can exhibit transient growth before its inevitable decay. When the equation results from the linearization of a nonlinear system, this transient growth gives a mechanism that can promote nonlinear instability. One might hope to enrich the conventional large-scale eigenvalue calculation used for linear stability analysis to signal the potential for such transient growth. Toward this end, we introduce a new definition of the pseudospectrum of a matrix pencil, use it to bound transient growth, explain how to incorporate a physically-relevant norm, and derive approximate pseudospectra using the invariant subspace computed in conventional linear stability analysis. We apply these tools to several canonical test problems in fluid mechanics, an important source of differential-algebraic equations.