The general convergence problems encountered when applying Newton's method to the circuit simulation domain are discussed. The authors identify and explore one source of difficulties for these methods and discuss a solution. The basic transconductance element, widely used to construct FET and bipolar transistor models, results in a pathological failure case for L/sub 2/-based norm-reducing methods due to the unidirectional nature between its input and output mode. Their particular solution retains the generic nature of norm-reducing methods but replaces the L/sub 2/-norm with a nonconsistent point of view. This norm determines which equations should converge first, prioritizes them, and guides the damping of the Newton updates accordingly. From a mathematical point of view, the steepest-descent direction in the Nu-norm is parallel to each Newton update at the iterate point and, therefore, allows more effective damping of the updates. The result has been an order-of-magnitude reduction in the number of Newton iterations. The performance of this norm on a series of high-electron-mobility transistor (HEMT) circuits is presented. The nonconsistency of the Nu-norm and its impact on global convergence properties for norm-reducing methods are discussed.>

A methodology for handling a class of stiff multiparticle parabolic PDE's in one and two dimensions is presented. The particular example considered in this work is the interaction and diffusion of two point defects in silicon, interstitials and vacancies. Newton's method, latency techniques, and second-order time-stepping approaches all contribute in significantly reduced computation times. A general class of diffusion-reaction problems is defined and conditions under which the corresponding Newton matrix is invertible and Newton's method converges to a globally unique solution are derived. The convergence properties of the purely reactive system are also derived and compared to those given by a Picard iteration. Application of basic iterative matrix techniques for the general diffusion-reaction system is discussed and specific numerical examples of point defect kinetics are given.