  Numerical methods play an ever more important role in astrophysics. This is
especially true in theoretical works, but of course, even in purely
observational projects, data analysis without massive use of computational
methods has become unthinkable. The key utility of computer simulations comes
from their ability to solve complex systems of equations that are either
intractable with analytic techniques or only amenable to highly approximative
treatments. Simulations are best viewed as a powerful complement to analytic
reasoning, and as the method of choice to model systems that feature enormous
physical complexity such as star formation in evolving galaxies, the topic of
this 43rd Saas Fee Advanced Course. The organizers asked me to lecture about
high performance computing and numerical modelling in this winter school, and
to specifically cover the basics of numerically treating gravity and
hydrodynamics in the context of galaxy evolution. This is still a vast field,
and I necessarily had to select a subset of the relevant material. The written
notes presented here quite closely follow the lectures as held in
Villars-sur-Ollon, which were meant to provide a general overview about some of
the most pertinent techniques that may be relevant for students working on
numerical models of galaxy evolution and star formation. The discussion is
hence often at an introductory level, giving precedence to a presentation of
the main numerical concepts rather than to a mathematically detailed exposition
of the techniques.
