# A discrete Bakry-Emery method and its application to the porous-medium equation

## TL;DR

This paper introduces a discrete Bakry-Emery method to prove exponential decay of entropy in a fully discrete porous-medium equation, supported by a new nonlinear summation-by-parts formula and numerical simulations.

## Contribution

It develops an abstract discrete Bakry-Emery framework and a novel nonlinear summation-by-parts formula for analyzing entropy decay in discrete porous-medium equations.

## Key findings

- Proves exponential decay of entropy in the discrete setting.
- Introduces a new nonlinear summation-by-parts formula.
- Numerical simulations confirm theoretical results.

## Abstract

The exponential decay of the relative entropy associated to a fully discrete porous-medium equation in one space dimension is shown by means of a discrete Bakry-Emery approach. The first ingredient of the proof is an abstract discrete Bakry-Emery method, which states conditions on a sequence under which the exponential decay of the discrete entropy follows. The second ingredient is a new nonlinear summation-by-parts formula which is inspired by systematic integration by parts developed by Matthes and the first author. Numerical simulations illustrate the exponential decay of the entropy for various time and space step sizes.