Section: Scientific Foundations
Challenges related to numerical simulations of complex flows
First, we describe some typical difficulties in our fields of application which require the improvement of established and the development of new methods.
Coupling of equations and models
The general equations of fluid dynamics consist in a strongly coupled nonlinear system. Its mathematical nature depends on the precise model, but in general contains hyperbolic, parabolic, and elliptic parts. The spectrum of physical phenomena described by these equations is very large: convection, diffusion, waves... In addition, it is often necessary to couple different models in order to describe different parts of a mechanical system: chemistry, fluidfluidinteraction, fluidsolidinteraction...
Robustness with respect to physical parameters
The values of physical parameters such as diffusion coefficients and constants describing different state equations and material laws lead to different behaviour characterized for example by the Reynolds, Mach, and Weissenberg numbers. Optimized numerical methods are available in many situations, but it remains a challenging problem in some fields of applications to develop robust discretizations and solution algorithms.

The inherent nonlinearities lead to an interplay of a wide range of physical modes, wellknown for example from the study of turbulent flows. Since the resolution of all modes is often unreachable, it is a challenging task to develop numerical methods, which are still able to reproduce the essential features of the physical phenomenon under study.