  We consider a discretization of Caputo derivatives resulted from deconvolving
a scheme for the corresponding Volterra integral. Properties of this
discretization, including signs of the coefficients, comparison principles, and
stability of the corresponding implicit schemes, are proved by its linkage to
Volterra integrals with completely monotone kernels. We then apply the backward
scheme corresponding to this discretization to two time fractional dissipative
problems, and these implicit schemes are helpful for the analysis of the
corresponding problems. In particular, we show that the overdamped generalized
Langevin equation with fractional noise has a unique limiting measure for
strongly convex potentials and establish the convergence of numerical solutions
to the strong solutions of time fractional gradient flows. The proposed scheme
and schemes derived using the same philosophy can be useful for many other
applications as well.
