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We develop structure preserving schemes for a class of nonlinear mobilitycontinuity equation. When the mobility is a concave function, this equationadmits a form of gradient flow with respect to a Wasserstein-like transportmetric. Our numerical schemes build upon such formulation and utilize modernlarge scale optimization algorithms. There are two distinctive features of ourapproach compared to previous ones. On one hand, the essential properties ofthe solution, including positivity, global bounds, mass conservation and energydissipation are all guaranteed by construction. On the other hand, it enjoyssufficient flexibility when applies to a large variety of problems includingdifferent free energy functionals, general wetting boundary conditions anddegenerate mobilities. The performance of our methods are demonstrated througha suite of examples.

Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances