Astrophysical Fluid Dynamics and Applications to Stellar Modeling

The modeling of astrophysical objects poses a challenging multiscale multiphysics problem. Because of their large spatial extent, the description of physical processes dominating the formation, structure, and evolution of such objects is typically based on effective theories such as fluid dynamics or thermodynamics. The modeling ansatz resulting from this approach is the Euler equations in combination with appropriate source terms. In contrast to terrestrial systems, the astrophysical equations of state are usually more complex and the ranges of relevant scales in space, time, density, velocity etc., in the considered objects are orders of magnitude wider. Simulations therefore require an efficient description of physical effects, elaborate numerical techniques, and models of unresolved phenomena. We exemplify this by focusing on processes in stars. This multiphysics problem is characterized by coupling the compressible Euler equations to the simultaneous effects of gravity, nuclear reactions, hydrodynamic instabilities, and mixing processes in the stellar fluid. It implies a multis because the processes act on scales in space and time that can easily be separated by ten orders of magnitude. The traditional astrophysical approach to this challenge-one-dimensional models parametrizing the description of unresolved effects-lacks predictive power. The dramatic increase in computational power, however, enables multidimensional dynamical simulations. They pave the way to the next generation of stellar models and promise new insights into the physical processes in stars. We discuss to which degree the currently applied techniques are able to cope with the scale problems. Among other techniques, we point out the importance of finding algorithms that allow for efficient parallelization and the use of problem-adapted geometries of the discretization grids. Further progress critically depends on continuous improvement of themethods, and input from applied mathematics will play a key role in this development.