The transient time-domain simulation, of the circuit response, is a fundamental component in the Computer-Aided Design tools of all integrated circuit and systems. It is typically desirable that a method adopted in the transient circuit simulator be of high- order and numerically stable. The two requirements, however, proved to be in conflict with each other, especially in the larger class of methods that were used in traditional circuit simulators. Recent work based on utilizing the Obreshkov formula has proved that it is possible to combine the high order with the numerical stability. The objective of this thesis is to show how the present implementation of the Obreshkov- based method can be improved and generalized to handle different types of circuits. The first aspect of improvement targets the computation of the high-order derivatives re- quired by the Obreshkov formula. The second aspect of improvement, presented in the thesis, develops a generalized formulation that takes into account the presence of non- linear memory elements, whose nonlinearity is based on a capacitive or inductive-based nonlinear model.