Fundamental mathematical research in our field has a strategic importance for many challenges in other fields of research and development, e.g. in engineering, physics and ecology. Central topics are advanced numerical methods for nonlinear hyperbolic balance laws that are particularly important for incompressible fluid flows and related systems of equations. The numerical methods we are focused on are finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. There are still very basic and challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient , problem suited adaptive algorithms. Consequently, our main objective is to derive and analyze novel high-order accurate schemes that will reliably approximate underlying physical models and preserve important physically relevant properties. This combination remains an open and challenging problem and will be addressed in our project proposal.
Within this project we will establish a long-term cooperation between our groups, particularly among young scientists, in order to achieve a significant development in this field and to meet future demands from numerous partical applications. We will also take this project as basis to support each other to proceed research on higher level cooperation such as the framework of 973 in China, SFB in Germany and even the European framework.