"Discretizing Nonlinear Diffusion the Lagrangian Way"

Daniel Matthes (TU Muenchen),
Joint work with Horst Osberger (TU Muenchen).

One of the ground-breaking observations from the theory 
of optimal transport is that various nonlinear diffusion 
equations can be written as gradient flows with respect 
to the L2-Wasserstein metric. In this context, diffusion 
is interpreted as the motion of a particle density along 
a (gradient) vector field that sensitively depends on 
the density itself.

We use that interpretation to define spatio-temporal 
"Lagrangian" discretizations of these diffusion equations.
That is, instead of calculating the change in density at 
given points, we trace the trajectories of "mass particles".
The resulting schemes inherit various nice features of the
original diffusion equations, like conservation of mass,
preservation of positivity, energy dissipation and convexity.

Our main result concerns the rigorous analysis of the 
discrete-to-continuous-limit for a particular Lagrangian 
discretization for certain fourth order equations (like 
thin film and QDD) in one spatial dimension. The key 
estimates are obtained from the dissipation of an auxiliary 
discrete Lyapunov functional.