It is demonstrated that the source of late time instabilities in the non-orthogonal FDTD (NFDTD) and discrete integral equation/generalized Yee (DSI/GY) methods is due to the ill-posed nature of the original formulations. Specifically, for general problems the explicit operators are not well posed. This leads to solutions that are unstable in the late time independent of the time step. Secondly, methods by which to repose the problem in a manner that is well posed, accurate and stable are presented.