Project part 2  Largetime behavior of discrete dissipative systems (Ansgar Jüngel)
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 largetime asymptotics of diffusive and kinetic equations and to prove the wellposedness of crossdiffusion systems. The aim is to derive efficient numerical schemes with structurepreserving 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 entropydissipative numerical schemes, the calculation of explicit discrete decay rates, the analysis of higherorder discretizations of gradienttype flows, and the analysis of the discrete blowup 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 BakryEmerytype methods, hypocoercivity techniques, and finitetime blowup calculations.
The numerical schemes will be based on semidiscretisations in time (e.g. backward differential formulas, RungeKutta) and finitedifference, finitevolume, and discontinuous Galerkin approximations in space. Model equations include reactiondiffusion systems, crossdiffusion systems from thermodynamics, and KellerSegel 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.
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Members of this project part
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Publications of this project part
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Only the publications of Project Part 2 are listed above. For a complete list of all the publications of the SFB 65, see this link.