We develop in this paper a new regularized flow dynamic approach to construct efficient numerical schemes for Wasserstein gradient flows in Lagrangian coordinates. Instead of approximating the Wasserstein distance which needs to solve constrained minimization problems, we reformulate the problem using the Benamou-Brenier's flow dynamic approach, leading to algorithms which only need to solve unconstrained minimization problem in $L^2$ distance. Our schemes automatically inherit some essential properties of Wasserstein gradient systems such as positivity-preserving, mass conservative and energy dissipation. We present ample numerical simulations of Porous-Medium equations, Keller-Segel equations and Aggregation equations to validate the accuracy and stability of the proposed schemes. Compared to numerical schemes in Eulerian coordinates, our new schemes can capture sharp interfaces for various Wasserstein gradient flows using relatively smaller number of unknowns.

翻译：本文提出了一种新的正则化流动动力学方法，用于在拉格朗日坐标下构建Wasserstein梯度流的高效数值格式。该方法不通过求解约束最小化问题来逼近Wasserstein距离，而是基于Benamou-Brenier流动动力学方法重构问题，从而得到仅需在$L^2$距离下求解无约束最小化问题的算法。所提格式自动继承了Wasserstein梯度系统的若干本质特性，包括正性保持、质量守恒与能量耗散。通过对多孔介质方程、Keller-Segel方程和聚集方程的大量数值模拟，验证了所提格式的精度与稳定性。与欧拉坐标下的数值格式相比，新格式能够以较少的未知量捕捉各类Wasserstein梯度流的尖锐界面。