Coordination Funds

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The numerical simulation of phenomena that can be modeled by partial differential equations (PDEs) is an essential tool in numerous scientific disciplines. While the development of accurate numerical methods for various systems of PDEs is still a very active research field, most complex applications are described by coupled systems of several PDEs, i.e. by heterogeneous models. This research unit focuses on the modeling and simulation of coupled systems arising in the description of magnetised plasmas, complex fluids and electro-chemical processes.Typically, no rigorous mathematical solution theory is available for these kind of complex, coupled, nonlinear systems. Therefore, it is desirable to develop numerical methods that can be shown to preserve certain structural properties of the underlying model. Examples of important structural properties are conservation of mass, charge, momentum and energy but also the consistency with an entropy balance equation which can be derived from the equations of interest. Other important structural properties are the preservation of asymptotic behaviour and exact approximation of steady states.In this research unit we distinguish two different situations for the appearance of heterogeneous models. In one of these situations multiple physical processes are considered in the same point or region of the domain of interest. We refer to such a situation as bulk-coupling. A typical example is the Vlasov equation of kinetic theory coupled to Maxwell's equations of electrodynamics. In another situation different mathematical models are used in different parts of the domain and glued together at common boundaries. We call this situation interface-coupling. Typical examples where interface-coupling arise are combinations of nonlinear and linearised models or the use of moment equations with different numbers of moments in different parts of the domain. For bulk- as well as interface-coupled heterogeneous models the development of structure-preserving methods is a new research direction which we address in a joint effort combining mathematical and physical modeling, numerical analysis and scientific computing. In some projects the structural elements must still be identified. In other cases we can build new numerical methods on established models. Numerical simulations will play a crucial role in all projects. In order to move from relatively simple test problems to adaptive simulations on parallel computers, the implications of coupling algorithms for high-performing computing will also be studied.