A proximal primal-dual approach to generalized JKO schemes for doubly nonlinear parabolic equations

Variational methods based on optimization strategies are proposed to numerically solve a large family of nonlinear partial differential equations. They are all particular instances of gradient flows with general costs, including the $ p $-Laplace equation and flux-limited equations such as the relativistic heat equation. This is achieved by deriving explicit characterizations of proximal operators for general transport costs, reducing their numerical evaluation, for the relevant $ L^p $ and relativistic costs, to independent scalar nonlinear problems at each grid point. We showcase our numerical approach via validation of the results by recovering the qualitative behavior of particular known cases of this large family of steepest descent evolutions.