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：我们为一类非线性迁移连续性方程开发了结构保持的方案。当迁移性是凹函数时，该方程具有类似于Wasserstein传输度量的梯度流形式。我们的数值方案建立在这种形式之上，利用现代大规模优化算法。与之前的方法相比，我们的方法有两个独特的特点。一方面，解的基本属性，包括正性、全局界限、质量守恒和能量耗散都是通过构造保证的。另一方面，它在应用于各种问题（包括不同的自由能泛函、一般的润湿边界条件和退化的迁移性）时具有足够的灵活性。我们的方法的性能通过一系列示例加以证明。