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

SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 376-399, February 2024.
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.