  We establish sharp energy decay rates for a large class of nonlinearly
first-order damped systems, and we design discretization schemes that inherit
of the same energy decay rates, uniformly with respect to the space and/or time
discretization parameters, by adding appropriate numerical viscosity terms. Our
main arguments use the optimal-weight convexity method and uniform
observability inequalities with respect to the discretization parameters. We
establish our results, first in the continuous setting, then for space
semi-discrete models, and then for time semi-discrete models. The full
discretization is inferred from the previous results. Our results cover, for
instance, the Schr\"odinger equation with nonlinear damping, the nonlinear wave
equation, the nonlinear plate equation, as well as certain classes of equations
with nonlocal terms.
