Various degenerate diffusion equations exhibit a waiting time phenomenon: Dependening on the flatness of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount of time. We show that this phenomenon is captured by particular Lagrangian discretizations of the porous medium and the thin-film equations, and we obtain suffcient criteria for the occurrence of waiting times that are consistent with the known ones for the original PDEs. Our proof is based on estimates on the fluid velocity in Lagrangian coordinates. Combining weighted entropy estimates with an iteration technique `a la Stampacchia leads to upper bounds on free boundary propagation. Numerical simulations show that the phenomenon is already clearly visible for relatively coarse discretizations.
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