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A. Jüngel, O. Leingang:
"Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models";
in: "ASC Report 16/2017", herausgegeben von: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2017, ISBN: 978-3-902627-10-0, S. 1 - 29.

The existence of weak solutions and upper bounds for the blow-up time for time-discrete parabolic-elliptic Keller-Segel models for chemotaxis in the two-dimensional whole space are proved. For various time discretizations, including the implicit Euler, BDF, and Runge-Kutta methods, the same bounds for the blow-up time as in the contin-uous case are derived by discrete versions of the virial argument. The theoretical results are illustrated by numerical simulations using an upwind finite-element method combined with second-order time discretizations.

Time discretization, chemotaxis, finite-time blow-up of solutions, existence of weak solutions, higher-order schemes, BDF scheme, Runge-Kutta scheme.

http://www.asc.tuwien.ac.at/preprint/2017/asc16x2017.pdf