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A. Jüngel:
"Structure-preserving numerical schemes for nonlinear evolution equations";
Talk: TU Braunschweig, Braunschweig (invited); 2018-11-08.

Solutions to evolution equations often preserve a number of quantities
like positivity, mass conservation, and energy dissipation. Numerical
schemes should be designed in such a way that these structures,
including the correct large-time asymptotics, are preserved. In this
talk, we present a number of recent results on structure-preserving
schemes, including Runge-Kutta and one-leg multi-step time
approximations and finite-volume space discretizations, by combining
techniques from stochastic analysis, theory of partial differential
equations, and numerical analysis. The results are based on novel
techniques like systematic integration by parts, discrete Bakry-Emery
methods, and the boundedness-by-entropy method. Numerical
simulations for porous-medium equations and cross-diffusion systems
for ion transport illustrate the theoretical results.