In the last 30 -40 years the development and application of numerical algorithms has become an essential activity in all branches of modern astronomy and astrophysics. This is, of course, a common phenomenon in many areas of scientiΓΏc research, and it was initiated by the invention and large-scale availability of digital computers. However, computer simulations play a completely di erent role in astrophysics compared to other ΓΏelds in physics or engineering. The main reason is that it is not possible to perform experiments; instead, almost all information about astrophysical objects is obtained through observations, i.e., our primary knowledge about these objects is mainly based on measuring the properties of radiation at various energies, directions, times, and polarizations. These data have to be interpreted in the framework of an appropriately designed physical model which after all consists of a set of equations, either algebraic, di erential, or integral. In general, these equations can only be solved by numerical methods. Varying the model parameters, it is then possible to generate a number of physical models, and in that sense computer simulation can be regarded as a substitute laboratory to "experiment" with astrophysical entities such as stars, galaxies, or even the universe. An early example of this procedure is the solution of the equations of stellar structure for various initial masses, and the subsequent calculation of the so-called main sequence in the famous Hertzsprung-Russell diagram (where stars entered in a luminosity vs. surface-temperature plot). More recently, detailed calculations of sound waves propagating in the sun can be compared with a multitude of observable solar oscillations, and this method of helioseismology actually has opened a window into the solar interior.
In this volume we have tried to compile a collection of contributions that, we think, is a typical representation of numerical techniques developed and used in computational astrophysics. We are aware that this collection is by no means complete, and it is certainly biased by personal preferences. For example, many contributions are concerned with problems from hydrodynamics or magneto-hydrodynamics whereas other aspects such as statistical data analysis do not appear at all. In particular, the present volume contains contributions on radiative transfer in stellar atmospheres and the calculation of atomic transitions, radiation hydrodynamics, molecular cloud dynamics, nuclear networks and neutrino transport in supernovae explosions, data inversion techniques in helioseismology, numerical methods in gas and plasma dynamics, N -body simulations for cosmic structure calculations, gravitational lensing, and modern methods in numerical relativity.