Modular Algorithms for Transient Semiconductor Device Simulation, Part I: Analysis of the Outer Iteration

A general outer iteration, based upon linearization, is introduced at discrete time steps for the one-dimensional semiconductor device model. The iteration depends upon solving the semidiscrete device equations approximately, specifically, in such a way that the residual is of order ∆t in an appropriate norm. It is shown that this maintains the order of the backward Euler method. A monitoring of the constants, including time-step requirements for solvability of the semidiscrete systems, as well as boundedness, smoothness and invertibility for the maps defining the Newton approximations, is carried out. An invariant-region principle provides an important theoretical basis, and a novel proof is provided. An essential component of the theory, providing the interface with the sequel to this paper, is that the Newton iterations are small in number, typically one or two, and may be realized as approximate Newton iterations. Continuation is employed as the time-stepping bridge.