  The exponential decay of the relative entropy associated to a fully discrete
porous-medium equation in one space dimension is shown by means of a discrete
Bakry-Emery approach. The first ingredient of the proof is an abstract discrete
Bakry-Emery method, which states conditions on a sequence under which the
exponential decay of the discrete entropy follows. The second ingredient is a
new nonlinear summation-by-parts formula which is inspired by systematic
integration by parts developed by Matthes and the first author. Numerical
simulations illustrate the exponential decay of the entropy for various time
and space step sizes.
