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.

中文翻译：

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具有非线性迁移传输距离的梯度流的结构保持原始对偶方法

SIAM 数值分析杂志，第 62 卷，第 1 期，第 376-399 页，2024 年 2 月。
摘要。我们为一类非线性迁移连续性方程开发了结构保持方案。当迁移率是凹函数时，该方程允许相对于类似 Wasserstein 的传输度量的梯度流形式。我们的数值方案建立在这种公式的基础上，并利用现代大规模优化算法。与以前的方法相比，我们的方法有两个显着特征。一方面，解的基本性质，包括正性、全局界限、质量守恒和能量耗散，都是由构造保证的。另一方面，我们的方法在应用于各种问题时具有足够的灵活性，包括不同的自由能泛函、一般润湿边界条件和简并迁移率。我们的方法的性能通过一系列示例来证明。