Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation

TL;DR

This paper introduces a Lagrangian numerical scheme for nonlinear drift diffusion equations that preserves key properties of the continuous model and proves its convergence under certain conditions, supported by numerical experiments.

Contribution

It presents a novel Lagrangian scheme based on the gradient flow structure and proves its convergence, which was not previously established for this class of equations.

Findings

The scheme preserves entropy monotonicity and mass.

Convergence is proven under a CFL-type condition.

Numerical experiments validate theoretical results.

Abstract

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation's gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties of the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We also present results from numerical experiments.