In [P3, Maas] we develop an optimal transport approach to structure preserving discretization of nonlinear PDE. We focus on dissipative equations with a gradient flow structure, as well as on more general systems, in which dissipative and conservative effects are combined. The project builds on the development of discrete dynamical transport metrics, which have been recently introduced in independent works by A. Mielke and the PI. Our approach offers several appealing features, such as preservation of positivity, preservation of mass, existence of Lyapunov functionals, robustness and stability properties, as well as a rich interplay with discrete stochastic processes (Markov chains) and discrete geometry (Ricci curvature).
We will perform a detailed investigation of the variational structures arising in discretized PDE, and develop time-discretization methods in this context. Moreover, we will analyze the discrete-to-continuous limit using evolutionary Gamma-convergence of generalized gradient flow structures and Gromov-Hausdorff convergence for discrete dynamical transport metrics. Finally, we aim for geodesic lambda-convexity results of entropy functionals in discretized PDE, and investigate geometric and functional inequalities in this context.
This Project Part will benefit from close scientific interactions in the SFB, in particular through Joint Work Packages with [P2, Jüngel] and [P11, Stefanelli].