A discrete Bakry-Emery method and its application to the porous-medium equation
Provides a theoretical framework for entropy decay in discrete settings, relevant for numerical analysis of nonlinear diffusion equations.
The paper proves exponential decay of relative entropy for a fully discrete porous-medium equation using a discrete Bakry-Emery method, supported by numerical simulations.
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.