We design and compute first-order implicit-in-time variational schemes with high-order spatial discretization for initial value gradient flows in generalized optimal transport metric spaces. We first review some examples of gradient flows in generalized optimal transport spaces from the Onsager principle. We then use a one-step time relaxation optimization problem for time-implicit schemes, namely generalized Jordan-Kinderlehrer-Otto schemes. Their minimizing systems satisfy implicit-in-time schemes for initial value gradient flows with first-order time accuracy. We adopt the first-order optimization scheme ALG2 (Augmented Lagrangian method) and high-order finite element methods in spatial discretization to compute the one-step optimization problem. This allows us to derive the implicit-in-time update of initial value gradient flows iteratively. We remark that the iteration in ALG2 has a simple-to-implement point-wise update based on optimal transport and Onsager's activation functions. The proposed method is unconditionally stable for convex cases. Numerical examples are presented to demonstrate the effectiveness of the methods in two-dimensional PDEs, including Wasserstein gradient flows, Fisher--Kolmogorov-Petrovskii-Piskunov equation, and two and four species reversible reaction-diffusion systems.

翻译：我们设计并计算了广义最优输运度量空间中初值梯度流问题的一阶时间隐式变分格式，并采用高阶空间离散方法。首先基于Onsager原理回顾广义最优输运空间中的若干梯度流实例。随后利用单步时间松弛优化问题构建时间隐式格式，即广义Jordan-Kinderlehrer-Otto格式。其最小化系统满足初值梯度流的一阶时间精度隐式时间格式。我们采用一阶优化算法ALG2（增广拉格朗日方法）结合空间离散中的高阶有限元方法求解该单步优化问题，从而迭代推导初值梯度流的隐式时间更新。值得注意的是，ALG2中的迭代基于最优输运和Onsager激活函数实现了易于实现的逐点更新。对于凸情形，所提方法具有无条件稳定性。数值算例验证了该方法在二维偏微分方程中的有效性，涵盖Wasserstein梯度流、Fisher-Kolmogorov-Petrovskii-Piskunov方程以及二物种与四物种可逆反应-扩散系统。