  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 continuous 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.
