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This chapter explores a new way of approximating differential equations, replacing the time discretization by a quantization of the state variables. We shall see that this idea will lead us to discrete event systems in terms of the DEVS formalism instead of difference equations, as in the previous approximations.

Thus, before formulating the numerical methods derived from this approach, we shall introduce the basic definitions of DEVS. This methodology, as a general discrete event systems modeling and simulation formalism, will provide us the tools to describe and translate into computer programs the routines that implement a new family of methods for the numerical integration of continuous systems.

Further, the chapter explores the principles of quantization-based approximations of ordinary differential equations and their representation as DEVS simulation models.

Finally, we shall briefly introduce the QSS method in preparation for the next chapter, where we shall study this numerical method in more detail.

This chapter focuses on the method and its extensions. After a brief explanation concerning the connections between this discrete event method and perturbation theory, the main theoretical properties of the method, i.e., convergence, stability, and error control properties, are presented.

The reader is then introduced to some practical aspects of the method related to the choice of quantum and hysteresis, the incorporation of input signals, as well as output interpolation.

In spite of the theoretical and practical advantages that the QSS method offers, the method has a serious drawback, as it is only first-order accurate. For this reason, a second-order accurate quantization-based method is subsequently presented that conserves the main theoretical properties that characterize the QSS method.

Further, we shall focus on the use of both quantization-based methods in the simulation of DAEs and discontinuous systems, where we shall observe some interesting advantages that these methods have over the classical discrete-time methods.

Finally, and following the discussion of a real-time implementation of these methods, some drawbacks and open problems of the proposed methodology shall be discussed with particular emphasis given to the simulation of stiff system.