A convergent Lagrangian discretization for a nonlinear fourth order equation

Abstract: A fully discrete Lagrangian scheme for numerical solution of the nonlinear fourth order DLSS equation in one space dimension is analyzed. The discretization is based on the equation's gradient flow structure in the $L^2$-Wasserstein metric. We prove that the discrete solutions are strictly positive and mass conserving. Further, they dissipate both the Fisher information and the logarithmic entropy. Numerical experiments illustrate the practicability of the scheme. Our main result is a proof of convergence of fully discrete to weak solutions in the limit of vanishing mesh size. Convergence is obtained for arbitrary non-negative initial data with finite entropy, without any CFL type condition. The key ingredient in the proof is a discretized version of the classical entropy dissipation estimate.