  In this work, we use the JKO scheme to approximate a general class of
diffusion problems generated by Darcy's law. Although the scheme is now
classical, if the energy density is spatially inhomogeneous or irregular, many
standard methods fail to apply to establish convergence in the continuum limit.
To overcome these difficulties, we analyze the scheme through its dual problem
and establish a novel $L^1$-contraction principle for the density variable.
Notably, the contraction principle relies only on the existence of an optimal
transport map and the convexity structure of the energy. As a result, the
principle holds in a very general setting, and opens the door to using
optimal-transport-based variational schemes to study a larger class of
non-linear inhomogeneous parabolic equations.
