  We study a Lagrangian numerical scheme for solution of a nonlinear drift
diffusion equation of the form $\partial_t u = \partial_x(u \cdot
c[\partial_x(h^\prime(u)+v)])$ on an interval. This scheme will consist of a
spatio-temporal discretization founded in the formulation of the equation in
terms of inverse distribution functions. It is based on the gradient flow
structure of the equation with respect to optimal transport distances for a
family of costs that are in some sense $p$-Wasserstein like. Additionally we
will show that, under a regularity assumption on the initial data, this also
includes a family of discontinuous, flux-limiting cost. We show that this
discretization inherits various properties from the continuous flow, like
entropy monotonicity, mass preservation, a minimum/maximum principle and
flux-limitation in the case of the corresponding cost. Convergence in the limit
of vanishing mesh size will be proven as the main result. Finally we will
present numerical experiments including a numerical convergence analysis.
