The role of Jordan Blocks in the MoT-scheme Time Domain EFIE Linear-in-Time Solution Instability
article
The marching-on-in-time electric _eld integral equation (MOT-EFIE) and the marching-
on-in-time time differentiated electric _eld integral equation (MOT-TDEFIE) are based on 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 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 split 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.
on-in-time time differentiated electric _eld integral equation (MOT-TDEFIE) are based on 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 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 split 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.
TNO Identifier
970653
Source
Progress In Electromagnetics Research B, 95, pp. 123-140.
Pages
123-140
Files
To receive the publication files, please send an e-mail request to TNO Repository.