Abstract:
We derive a new particle method for linear and nonlinear diffusive gradient flows. This method is essentially based on the regularization of the Wasserstein internal energy through the convolution with a specific class of mollifiers. This regularized energy being of a new type, we characterize its subdifferential and we prove basic properties such as lower semicontinuity, differentiability and convexity. This enables us to prove that the regularized gradient flow is well-posed, and that the regularized energy Gamma-converges to the original internal energy. When the diffusion is at least quadratic, we show that the regularized gradient flow converges to the original diffusion equation using Serfaty's approach on the convergence of gradient flows. The final numerical scheme is obtained by solving an ODE system whose solutions are the solutions to the regularized gradient flow with Dirac masses as initial data. We provide error plots and numerical simulations illustrating the behavior of this particle method in test cases such as the heat and porous medium equations, and we recover critical properties of the Keller--Segel equation in one and two space dimensions by adding an interaction term to the energy. This is joint work with J. A. Carrillo and K. Craig. |