Numerical methods play an ever more important role in astrophysics. This can be easily demonstrated through a cursory comparison of a random sample of paper abstracts from today and 20 years ago, which shows that a growing fraction of studies in astronomy is based, at least in part, on numerical work. 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. For example, cosmological inferences of large CMB experiments routinely use very large Monte-Carlo simulations as part of their Baysian parameter estimation.

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. Thanks to the rapid increase of the performance of computers, the technical limitations faced when attacking the equations numerically (in terms of calculational time, memory use, numerical resolution, etc.) become progressively smaller. But it is important to realize that they will always stay with us at some level. Computer simulations are therefore 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, which took place March 11-16, 2013, in Villars-sur-Ollon, Switzerland. As my co-lecturers Ralf Klessen und Nick Gnedin should focus on the physical processes in the interstellar medium and on galactic scales, my task was defined as covering the basics of numerically treating gravity and hydrodynamics, and on making some remarks on the use of high performance computing techniques in general. In a nutshell, my lectures hence intend to cover the basic numerical methods necessary to simulate evolving galaxies. This is still a vast field, and I necessarily had to make a selection of a subset of the relevant material. I have tried to strike a compromise between what I considered most useful for the majority of students and what I could cover in the available time.

In particular, my lectures concentrate on tech niques to compute gravitational dynamics of collisionless fluids composed of dark matter and stars in galaxies. I also spend a fair amount of time explaining basic concepts of various solvers for Eulerian gas dynamics. Due to lack of time, I am not discussing collisional N-body dynamics as applicable to star cluster, and I omit a detailed discussion of different schemes to implement adaptive mesh refinement.

The written notes presented here quite closely follow the lectures as held in Villars-sur-Ollon, apart from being expanded somewhat in detail where this seemed adequate. I note that the shear breadth of the material made it impossible to include detailed mathematical discussions and proofs of all the methods. The discussion is therefore often at an introductory level, but hopefully still useful as a general overview for students working on numerical models of galaxy evolution and star formation. Interested readers are referred to some of the references for a more detailed and mathematically sound exposition of the numerical techniques.