  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.

Numerical Analysis;Numerical Analysis;Optimization and Control