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.

翻译：我们针对一类非线性迁移率连续性方程发展了结构保持格式。当迁移率为凹函数时，该方程可呈现为关于类Wasserstein输运度量的梯度流形式。我们的数值格式基于这种表述，并利用现代大规模优化算法。与以往方法相比，我们的方法有两个显著特征：一方面，解的基本性质（包括正性、全局有界性、质量守恒和能量耗散）均通过构造得到保证；另一方面，该方法在应用于包含不同自由能泛函、一般润湿边界条件和退化迁移率的各类问题时，展现出充分的灵活性。通过一系列算例验证了方法的性能。