Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations
For researchers studying stability of differential-algebraic equations, particularly in fluid mechanics, this provides a tool to detect transient growth that can lead to nonlinear instability.
The paper introduces a new definition of the pseudospectrum for matrix pencils with singular E to analyze transient growth in differential-algebraic equations, and demonstrates its utility on fluid mechanics test problems.
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.