This survey article traces the development of analog/hybrid and digital simulation of continuous systems since the early 1950's, describes modern aspects, and discusses future trends.

As reflected by this list of headings, a major part of the book is devoted to a description of analog/hybrid computer components and simulation techniques. In this respect, the book repeats material which is amply available in much older textbooks. The author then proceeds to extend those analog/hybrid techniques to the field of numerical algorithms. This is done specifically in chapter 3, and integrated in most of the other chapters along with the description of analog/hybrid methods of implementation.

The method of decomposition offers a means to overcome the computational difficulty associated with unstable two-point boundary value problems. The basic idea is to decompose systematically an unstable differential equation into a sequence of differential equations of a lower order, which are each stable either in the forward or in the backward direction of computer integration.

Methods of functional approximation for the computer solution of initial value partial differential equation problems provide a device by which these solutions can be approximated by those of initial value problems in sets of ordinary differential equations. A number of ways to achieve this have been suggested, some of them general, some of them utilizing specific properties of the equations at hand. What all these methods have in common is the fact that the solution u(x, t) of a partial differential equation in space x and time t is approximated by a function u*(a1(t), a2(t), . . ., aN(t)), where the dependence upon x is prescribed. In most applications, u* is linear in the ai(t), i.e., u*= =ai(t)fi(x). The ai(t) satisfy a set of ordinary differential equations obtained as the result of the approximation process. This system of ordinary differential equations is then integrated by classical analog or digital computer methods.

Error analysis in the computer simulation of dynamic systems is fundamentally a variational problem. The computing errors are small variations of the computed solutions with respect to the exact solution of the differential equations being integrated. It is not surprising, therefore, that many of the mathematical tools used to perform error analysis in the computer simulation of dynamic systems are similar to those used in the variational or perturbational analysis of those systems themselves. Fundamental papers in this direction have been published previously [8], [9], [13]. The present paper, however derives the error-propagation equations in a more basic form, which makes it easier to apply such variational mathematical tools as Liapunoff's second method to analyze error stability, and Pontryagin's maximum principle to study ``worst-case'' errors in computation.