Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations
Abstract
Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps. The concavity property is shown to be related to the Bakry-Emery approach and the geodesic convexity of the entropy. The abstract conditions are verified for quasilinear parabolic equations (including the porous-medium equation), a linear diffusion system, and the fourth-order quantum diffusion equation. Numerical experiments for various Runge-Kutta finite-difference discretizations of the one-dimensional porous-medium equation show that the entropy-dissipation property is in fact global.
Cite
@article{arxiv.1506.07040,
title = {Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations},
author = {Ansgar Jüngel and Stefan Schuchnigg},
journal= {arXiv preprint arXiv:1506.07040},
year = {2015}
}