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This paper discusses techniques employed in the discrete modeling of physical systems for digital simulation and control applications. Traditional numerical integration techniques provide accurate means of model making but prove too slow for real-time simulation of complex systems or systems with fast response. For rapid digital simulation, a simplified discrete approximation is sought for the linear integro-differential operators of a continuous system. This discrete operator, a digitized transfer function, yields difference equations hopefully permitting real-time approximation of continuous system performance on a digital computer. Determination of the discrete operator is the essential goal of each of the simulation schemes described herein, though differing initial assumptions and approximations alter the resulting forms. After a brief review of these approaches to simulation, techniques for improved approximations for linear system transforms and for discrete parameter optimization and identification are developed. The optimum discrete transfer function which minimizes the sum of error squared between a linear continuous system output and a linear discrete system output is obtained. By adjusting gain parameters in the discrete transfer function, the simulation result is shown to be improved for various inputs and system nonlinearities. Application of standard variational methods to optimize the desired parameters leads to a two-point nonlinear boundary-value problem which is resolved via the techniques of quasilinearization and differential approximation. The procedure for application of various simulation methods is summarized, and the effectiveness of the methods is shown by the simulation of a second-order, nonlinear system for various inputs and sample intervals.