A higher-order gradient flow scheme for a singular one-dimensional diffusion equation

TL;DR

This paper introduces a higher-order gradient flow scheme for a singular one-dimensional nonlinear diffusion equation, utilizing a BDF-based minimizing movement approach that ensures mass conservation and exponential decay of entropy and variance.

Contribution

It develops a novel higher-order numerical scheme based on BDF discretization for Wasserstein gradient flows, improving accuracy and preserving key properties of the diffusion process.

Findings

The scheme conserves mass and dissipates the G-norm.

Numerical decay of entropy and variance is exponential.

Decay rates are effectively computed for various grid sizes.

Abstract

A nonlinear diffusion equation, interpreted as a Wasserstein gradient flow, is numerically solved in one space dimension using a higher-order minimizing movement scheme based on the BDF (backward differentiation formula) discretization. In each time step, the approximation is obtained as the solution of a constrained quadratic minimization problem on a finite-dimensional space consisting of piecewise quadratic basis functions. The numerical scheme conserves the mass and dissipates the $G$-norm of the two-step BDF time approximation. Numerically, also the discrete entropy and variance are decaying. The decay turns out to be exponential in all cases. The corresponding decay rates are computed numerically for various grid numbers.