Late-time instability of the finite element time domain (FETD) method when using newmark time integration

The FETD-Newmark method [1] is typically considered unconditionally stable because its amplification matrix A has all unitary eigenvalues regardless of timestep. Herein, it is shown that the amplification matrix lacks a linearly independent set of eigenvectors, and has a nondiagonal Jordan form which permits linear growth for gradient fields [2]. Knowledge of Jordan (A) will naturally lead to a correction scheme which removes the growing gradient term and stabilizes the algorithm for long-duration simulations (multiple millions of timesteps).