A robust and low frequency stable time domain PMCHWT equation

The time domain PMCHWT equation models transient scattering by piecewise homogeneous dielectrics. After discretization, it can be solved using the marching-on-in-time algorithm. Unfortunately, the PMCHWT equation suffers from DC instability: it supports constant in time regime solutions. Upon discretization, the corresponding poles of the system response function shift into the unstable region of the complex plane, rendering the MOT algorithm unstable. Furthermore, the discrete system becomes ill-conditioned when a large time step is used. This phenomenon is termed low frequency breakdown. In this contribution, the quasi-Helmholtz components of the PMCHWT equation are separated using projector operators. Judicially integrating or differentiating these components of the basis and testing functions leads to an algorithm that (i) does not suffer from unstable modes even in the presence of moderate numerical errors, (ii) remains well-conditioned for large time steps, and (iii) can be applied effectively to both simply and multiply connected geometries.