  We study a discretization in space and time for a class of nonlinear
diffusion equations with flux limitation. That class contains the so-called
relativistic heat equation, as well as other gradient flows of Renyi entropies
with respect to transportation metrics with finite maximal velocity.
Discretization in time is performed with the JKO method, thus preserving the
variational structure of the gradient flow. This is combined with an entropic
regularization of the transport distance, which allows for an efficient
numerical calculation of the JKO minimizers. Solutions to the fully discrete
equations are entropy dissipating, mass conserving, and respect the finite
speed of propagation of support.
  First, we give a proof of $\Gamma$-convergence of the infinite chain of JKO
steps in the joint limit of infinitely refined spatial discretization and
vanishing entropic regularization. The singularity of the cost function makes
the construction of the recovery sequence significantly more difficult than in
the $L^p$-Wasserstein case. Second, we define a practical numerical method by
combining the JKO time discretization with a "light speed" solver for the
spatially discrete minimization problem using Dykstra's algorithm, and
demonstrate its efficiency in a series of experiments.
