Contour integral methods based on the Laplace transform integrate linear DAEs in time without the order reduction that afflicts classical Runge–Kutta schemes, and one carefully chosen contour can serve an entire parametric family.

Summary. The manuscript proposes a generalization of contour integral methods (CIM) to linear differential-algebraic equations (DAEs). The approach applies the Laplace transform to the associated Cauchy problem and reconstructs the time-domain solution by quadrature approximation of the inverse Laplace transform, claiming an efficient, accurate alternative to classical Runge–Kutta schemes that suffer order reduction on DAEs. In a second part, structured–unstructured pseudospectral computations are used to select a single integration contour that remains valid for an entire family of linear parametric DAEs, thereby facilitating model-order reduction. Numerical experiments are reported to support the methodology and theory.

Significance. If the claims hold under clearly stated and checkable hypotheses, the work would supply a practical contour-integral time integrator for linear DAEs that avoids the well-known order reduction of Runge–Kutta methods, together with a concrete contour-tuning strategy based on structured–unstructured pseudospectra for parametric families. Both contributions would be of interest to the numerical DAE and model-reduction communities. The abstract alone, however, does not yet allow verification that the analytic and stability foundations are in place, so the significance remains conditional on the full development.