Over the past decades, Scientific Computing and Mathematical Modeling successfully derived equations and developed numerical schemes to simulate numerous variants of processes in any conceivable area of application. The success came from the ability to identify the mathematical essence to be found in the applications and construct models and algorithms sometimes highly specialized to the specific process and scenario. Consequently, one of the next challenges is the coupling of different mathematical models and different computational methods in order to utilize the achievements for complex simulations involving many different processes.
For many mathematical models of physical processes, no rigorous mathematical solution theory is available. While it is not possible to show existence and uniqueness of solutions or convergence of numerical schemes, it is possible to identify basic structural properties of the mathematical models. Hence, modern numerical methods rely on structure-preservation. Following this paradigm, our goal is to construct new numerical schemes for complex coupled systems, which obey the same structural properties on a discrete level. This imposes conditions on the numerical methods, which in return will improve stability, accuracy and robustness in simulations. Additionally, the structural properties investigated in this research unit may also generate ideas to eventually construct a convergence theory in subsequent research proposals.
In the real world, coupled systems of PDEs are the norm and not the exception. Mathematical models of physical systems are inherently heterogeneous. Relevant processes might be heat conduction, fluid flow, electromagnetic fields, multi-component chemistry and others. Additionally, these processes may come in different descriptions like compressible/incompressible fluids, electrostatics and -dynamics, as well as fluid mechanics and kinetic, i.e., mesoscopic, models. This research unit is motivated by three coupled scenarios: magnetized plasmas, complex fluids, and electro-chemical processes. These can be found in relevant fields like astrophysics and nuclear fusion, polymers and multi-phase flows, and batteries and biological systems. Their joint feature is the coupling pattern between fluid flow, electrodynamics and kinetic processes.
Achieving structure-preservation across bulk- and interface-coupling requires a concerted research effort combining mathematical modeling, numerical analysis and scientific computing. In some cases, the structural elements must still be clarified in the context of model derivation, while in other cases numerical algorithms are lacking. An important aspect are the algorithmic implications for the coupling, in particular when aiming for high-performance computing. Here, we will identify fundamental requirements and challenges and provide a flexible code library that can be adopted to a wide collection of different coupling scenarios.
The results of this research unit will provide both mathematical insights into coupling mechanisms and significant progress in the ability to simulate complex, real-world, multi-scale, multi-physics processes.