A convergent Lagrangian discretization for a nonlinear fourth order equation

TL;DR

This paper introduces a fully discrete Lagrangian scheme for solving the nonlinear fourth order DLSS equation, demonstrating convergence, positivity, mass conservation, and entropy dissipation without CFL restrictions.

Contribution

It presents a novel convergent discretization based on the gradient flow structure in Wasserstein metric for the DLSS equation.

Findings

Discrete solutions are strictly positive and conserve mass.

Solutions dissipate Fisher information and logarithmic entropy.

Convergence to weak solutions is proven without CFL condition.

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.