Structure Preserving Primal Dual Methods for Gradient Flows with Nonlinear Mobility Transport Distances

Abstract

.We develop structure preserving schemes for a class of nonlinear mobility continuity equation. When the mobility is a concave function, this equation admits a form of gradient flow with respect to a Wasserstein-like transport metric. Our numerical schemes build upon such formulation and utilize modern large-scale optimization algorithms. There are two distinctive features of our approach compared to previous ones. On the one hand, the essential properties of the solution, including positivity, global bounds, mass conservation, and energy dissipation, are all guaranteed by construction. On the other hand, our approach enjoys sufficient flexibility when applied to a large variety of problems including different free energy functionals, general wetting boundary conditions, and degenerate mobilities. The performance of our methods is demonstrated through a suite of examples.Keywordsstructure preserving methodsprimal dual methodsWasserstein-like gradient flowsoptimal transport distancesminimizing movementsMSC codes35A1547J2547J3549M2965K10

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