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

Abstract

To understand the solution of a linear, time-invariant differential-algebraic equation (DAE), 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 can enrich the conventional large-scale eigenvalue calculation used for linear stability analysis to identify 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 DAEs.