  We introduce a novel variant of the JKO scheme to approximate Darcy's law
with a pressure dependent source term. By introducing a new variable that
implicitly controls the source term, our scheme is still able to use the
standard Wasserstein-2-metric even though the total mass changes over time.
Leveraging the dual formulation of our scheme, we show that the
discrete-in-time approximations satisfy many useful properties expected for the
continuum solutions, such as a comparison principle and uniform
$L^1$-equicontinuity. Many of these properties are new even in the
well-understood case where the growth term is absent. Finally, we show that our
discrete approximations converge to a solution of the corresponding PDE system,
including a tumor growth model with a general nonlinear source term.
