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

TL;DR

This paper introduces structure-preserving numerical schemes for nonlinear mobility gradient flows, ensuring key physical properties and flexibility across various problems, demonstrated through multiple examples.

Contribution

The authors develop novel schemes that preserve essential solution properties and are adaptable to diverse nonlinear mobility gradient flow problems.

Findings

Guarantee of positivity, bounds, conservation, and energy dissipation

Compatibility with various free energies and boundary conditions

Effective performance demonstrated on multiple examples

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 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, it enjoys sufficient flexibility when applies to a large variety of problems including different free energy functionals, general wetting boundary conditions and degenerate mobilities. The performance of our methods are demonstrated through a suite of examples.