The Role of Jordan Blocks in the MOT-Scheme Time Domain EFIE Linear-in-Time Solution Instability

The marching-on-in-time electric field integral equation (MOT-EFIE) and the marching-on-in-time time differentiated electric field integral equation (MOT-TDEFIE) based on a Rao-Wilton-Glisson (RWG) spatial discretization. In both formulations we employ the Dirac-delta temporal testing functions, however they differ in temporal basis functions, i.e. hat and quadratic spline basis functions. These schemes suffer from the linear-in-time solution instability. We analyze the corresponding companion matrices using projection matrices and prove mathematically that each independent solenoidal current density corresponds to a Jordan block of size two. In combination with Lidskii-Vishik-Lyusternik perturbation theory we find that finite precision causes these Jordan block eigenvalues to split and this is the root cause of the instability of both schemes. The splitted eigenvalues cause solutions with exponentially increasing magnitudes that are initially observed as constant and/or linear-in-time, yet these become exponentially increasing at discrete time steps beyond the inverse square root of the error due to finite precision, i.e. approximately after one hundred million discrete time steps in double precision arithmetic. We provide numerical evidence to further illustrate these findings