Contour integral methods and structured perturbations for linear differential-algebraic equations

Abstract

We generalize the contour integral methods (CIM) framework to the time integration of linear dynamical systems that are subject to algebraic constraints at all times during their evolution. The proposed approach relies on applying the Laplace transform to the Cauchy problem associated with a linear system of differential-algebraic equations (DAE), and subsequently reconstructing the time-domain solution by approximating the inverse Laplace transform via a suitable quadrature rule. This procedure yields an efficient and accurate alternative to classical Runge-Kutta schemes, which are well known to exhibit order reduction in accuracy when applied to DAE. In the second part of the paper, we address linear parametric DAE and propose an efficient strategy for tuning the integration contour in the CIM framework using suitable structured-unstructured pseudospectral computations. This allows the identification of a single integration profile capable of approximating an entire family of parametric solutions, thereby facilitating the efficient application of model order reduction techniques. Finally, numerical experiments are presented to validate the proposed methodology and support the theoretical findings.