The Tustin Method for approximating eigenvalues of delay systems

Numerical approximation of eigenvalues of Delay Differential Equations (DDEs) is an active field of research due to its impact in the modeling of many processes of industrial interest. In this work we introduce the Tustin Method, a new numerical method for Linear Time-Invariant (LTI) systems based on the Cayley transform. We introduce two alternatives, one using a trigonometric basis, and other using splines with maximum smoothness. For splines, we exploit the so called $h$-$p$-$k$ refinement, achieving very high accuracy. We prove that the Tustin method recovers all the eigenvalues, and we give estimates for the rate of convergence. We make numerical experiments for generic systems, for systems with large delay, and for systems close to the stability boundary.