## Abstract

The evolution of both scientiﬁc and engineering developments results in an ever growing demand to
address problems of increasing complexity in a swift and precise manner. Numerical simulations using
digital computing devices are a valuable asset in meeting these demands. The resulting evolution of
digital computers has lead to the development of several programming techniques and paradigms, which
are shortly outlined. While it is critically important to be aware of the capabilities as well as the
limitations of the deployed digital tools, which are also brieﬂy touched upon, it is not suﬃcient for the
setting of scientiﬁc computing, since at least a rudimentary understanding of the problem domain to be
treated is also required.

As both purely scientiﬁc as well as applied engineering make extensive use of mathematical formalisms,
it is only natural to use the mathematical structures as a guide. This is especially important in order to
transcend the still comparatively primitive abstractions, consequently also the implemented
structures. Therefore a two pronged approach has been chosen, presenting on the one hand the
theoretical outline, including the mathematical backbone as well as a physical application of the
outlined mathematical structures, while on the other hand showing realizations of the described
structures.

Thus, the two parts mirror each other as both progress from essential base components, such as sets and
algebraic structures, to form abstract constructs of increasing complexity and sophistication, including
entities from diﬀerential geometry, integration, and probability.

The components are then put to use in order to model what is perceived as physical reality with a special
attention to the ﬁeld of dynamics, including even a small foray into the ﬁeld of quantum mechanics.
Again, the corresponding sections of theory and treatment in the environment of application attempt to
mirror each other.