Convergence of the fully discrete JKO scheme
Anastasiia Hraivoronska, Filippo Santambrogio
TL;DR
The paper develops a fully discrete JKO scheme by restricting minimizers to measures on a uniform grid with space step $h$ and time step $\tau$, proving convergence to continuum gradient flows as $h,\tau\to 0$ under the sharp regime $h/\tau\to 0$. It provides a rigorous De Giorgi–type variational framework with two interpolations (variational and geodesic) to pass to the limit and shows convergence to the drift-diffusion equation $\partial_t\rho + \nabla\cdot(\rho\nabla(f'(\rho)+V))=0$ with no-flux boundary, and to the Hele–Shaw-type crowd-motion system for $\mathcal{F}_{\text{CM}}$. The results hold for general convex growing $f$ and Lipschitz potentials $V$, and extend to nonlocal interaction variants with appropriate assumptions. The work clarifies the delicate balance between spatial and temporal discretization in implicit Wasserstein schemes and lays groundwork for grid-based numerical approximations of gradient-flow PDEs with diffusion and congestion constraints.
Abstract
The JKO scheme provides the discrete-in-time approximation for the solutions of evolutionary equations with Wasserstein gradient structure. We study a natural space-discretization of this scheme by restricting the minimization to the measures supported on the nodes of a regular grid. The study of the fully discrete JKO scheme is motivated by the applications to developing numerical schemes for the nonlinear diffusion equation with drift and the crowd motion model. The main result of this paper is the convergence of the scheme as both the time and space discretization parameters tend to zero in a suitable regime.