Jordan Block Eigenvalue Shift in the Marching-on-in-Time Electric Field Integral Equation

The Marching-on-in-Time Electric Field Integral Equation (MOT-EFIE) and its time differentiated variant (MOT-TDEFIE) both suffer from the linear-in-time instability caused by dimension-two Jordan blocks in the companion-matrix representation. We prove that these Jordan blocks are centered around a specific eigenvalue that depends on the recurrence relation between the interaction matrices. By adapting the recurrence relation, we can move the pertaining eigenvalues of the Jordan blocks to the interior of the unit circle and therefore remove the linear-in-time instability from the scheme. We provide numerical evidence to further illustrate these findings.