Overview
In the second half of the 20th century, a new machine was invented and established, which by now has influenced and fundamentally changed all parts of human life: the computer. The crucial characteristic of this machine is its capacity to process large am
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1.1 Introduction In the second half of the 20th century, a new machine was invented and established, which by now has influenced and fundamentally changed all parts of human life: the computer. The crucial characteristic of this machine is its capacity to process large amounts of data at high velocity according to precisely formulated conditions. Basically, there are two possibilities: the numerical computation in the narrow sense, and the organization, search, manipulation, and description of symbols. Initially, computation was the more important aspect, a fact still conveyed by the word "computer". In the meantime, however, particularly the combination of numerics and symbol processing is the motor for the enormous progress concerning the application of computer technology (data bases, graphics, internet, virtual reality, ... ). The capacities of computers opened up a wide range of new possibilities for mathematical sciences and engineering. In the previous centuries, many fundamental natural phenomena were formulated by equations that, in principle, described the relative processes exactly and thoroughly. Most of these equations, however, were so complicated that their solution for arbitrary initial and boundary conditions was absolutely impossible. Whole generations of scientists tried to find mainly analytic solutions for specifically simplified situations, in order to understand the structures behind the equations. These methods led to deep and beautiful insights into the system of the laws of nature, and their application to engineering techniques was the basis for considerable progress. However, if the real problem could not be solved by the existing simplifying approaches (disturbance methods, use of symmetries, separation approaches, ... ), the limits quickly became clear. This was particularly the case for most nonlinear problems. The computer radically changed this situation. Today, a numerical solution is possible for the entire complex structure of the relevant equations of many real problems. The number of problems that can now be solved is continuously increasing. Due to these technical developments, enormous progress was also made with respect to the search for algorithms, a necessary and worthwhile field of research because the new technical applications require efficient algorithms for large amounts of data and complicated numerical operations.
H. Loch et al. (eds.), Mathematical Simulation in Glass Technology © Springer-Verlag Berlin Heidelberg 2002
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Overview
The computer and the implemented algorithms together form a kind of "mathematical machine". It is an attractive and very popular vision today that we might be able to simulate the behavior of interesting real systems virtually (i.e., without referring to the real world) on such mathematical machines in order to understand and optimize them. However, before we can start analyzing a technically relevant system in a "virtual laboratory", it must first be formulated by equations, i.e., apart from the usual conservation laws, material
