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

A convergent Lagrangian discretization for $p$-Wasserstein and flux-limited diffusion equations