The Project Part [P2, Jüngel] will extend entropy methods to discrete settings. Entropy methods proved to be very powerful tools for the analysis of the large-time asymptotics of diffusive and kinetic equations and to prove the well-posedness of cross-diffusion systems. The aim is to derive efficient numerical schemes with structure-preserving properties and insofar, the Project Part lies in between mathematical and numerical analysis. Besides some preliminary work, discrete entropy methods seem to be widely an open field.
The main objectives of this Project Part are the design of entropy-dissipative numerical schemes, the calculation of explicit discrete decay rates, the analysis of higher-order discretizations of gradient-type flows, and the analysis of the discrete blow-up behavior in certain multiphysics diffusion systems. These problems are of high complexity in the sense that the underlying systems are large and strongly interconnected. This may arise from numerical discretizations leading to a large number of coupled nonlinear equations or from complex phenomena inherent in the PDE models. Our key idea to tame this complexity is to translate techniques developed for continuous systems to the discrete setting, for instance Bakry-Emery-type methods, hypocoercivity techniques, and finite-time blow-up calculations.
The numerical schemes will be based on semi-discretisations in time (e.g. backward differential formulas, Runge-Kutta) and finite-difference, finite-volume, and discontinuous Galerkin approximations in space. Model equations include reaction-diffusion systems, cross-diffusion systems from thermodynamics, and Keller-Segel equations in chemotaxis. By combining tools from PDE theory, numerical analysis, and the theory of stochastic processes, we aim to develop discrete entropy methods and novel contributions to the numerical analysis of complex systems.