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