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

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.

Generalized potential games

“…More precisely, we believe that the convergence of a discrete version of this method (see e.g. [JS17]) leads to the convergence of algorithms finding equilibrium of the games defined in (21a)-(21b). This interesting fact is contained in our forthcoming work.…”

Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models

“…) be a sequence of nonnegative weak solutions to (28). Suppose that I 0 ≤ I * and τ ≤ 2τ * (see (16)).…”

“…We analyze the implicit Euler and higher-order schemes, including BDF (Backward Differentiation Formula) and Runge-Kutta schemes. Our goal is not to design efficient numerical schemes but to continue our program to "translate" mathematical techniques from continuous to discrete situations [27,28,29].…”