A numerical multiscale approach called equation-free analysis is in the framework of slow-fast dynamical systems further improved and applied to particle models. The method allows to perform numerical investigations of the macroscopic behavior of microscopically defined systems including continuation and bifurcation analysis on the coarse or macroscopic level where no explicit equations are available. This approach fills a gap in the analysis of several complex real-world applications including particle models with intermediate number of particles where the microscopic system is too large for direct investigations of the full system and too small to justify large-particle limits. An implicit equation-free method is presented which reduced numerical errors of the analysis considerably. It can be shown, that the implicitly defined coarse-level time stepper converges to the true dynamics on the slow manifold. The method is applied to perform a coarse bifurcation analysis of microscopic particle models describing first car traffic on single lane highways and second pedestrian flow in two space dimensions. The results include an equation-free continuation of traveling wave solutions, identification of saddle-node and Hopf-bifurcations as well as two-parameter continuations of bifurcation points.
报告人简介:  Jens Starke studied both, Mathematics and Physics at the University of Stuttgart and received his PhD in 1997 from the same University. After that he worked at the University of Heidelberg first as postdoc, then as head of a junior research group (financed by the Heidelberg Academy of Science and Toyota) and Assistant Professor at the Institute of Applied Mathematics and the Interdisciplinary Centre of Scientific Computing. Since 2006 he is Associate Professor at the Technical University of Denmark. His main research interest focuses on problems concerning dynamics (differential equations, dynamical systems, stochastic systems) in modeling, analysis, numerics, simulation and optimization of complex scientific, medical and engineering systems. Besides mathematical theory, real-world applications of nonlinear mathematics are an important part of his work. His research contains in particular the interplay between discrete and continuous as well as deterministic and stochastic models and corresponding methods. Jens Starke is one of the PIs of the VKR Centre of Excellence “Life in a Changing Ocean”. He has in total more than 70 publications and is the current chair of the Advisory Board of the “Dynamics Days Europe”.