Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities∗
Laboratoire Paul Painlevé, U.M.R. CNRS 8524, Université Lille 1,
Cité Scientifique, 59655 Villeneuve,
2 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria
Received: 23 April 2014
Revised: 10 April 2015
The time decay of fully discrete finite-volume approximations of porous-medium and fast-diffusion equations with Neumann or periodic boundary conditions is proved in the entropy sense. The algebraic or exponential decay rates are computed explicitly. In particular, the numerical scheme dissipates all zeroth-order entropies which are dissipated by the continuous equation. The proofs are based on novel continuous and discrete generalized Beckner inequalities. Furthermore, the exponential decay of some first-order entropies is proved in the continuous and discrete case using systematic integration by parts. Numerical experiments in one and two space dimensions illustrate the theoretical results and indicate that some restrictions on the parameters seem to be only technical.
Mathematics Subject Classification: 65M08 / 65M12 / 76S05
Key words: Porous-medium equation / fast-diffusion equation / finite-volume method / entropy dissipation / Beckner inequality / entropy construction method
The authors have been partially supported by the Austrian-French Project Amadeé of the Austrian Exchange Service (ÖAD). The first author acknowledges support from Labex CEMPI (ANR-11-LABX-0007-01) and from the Inria-Mephysto Team. The second and last author acknowledge partial support from the Austrian Science Fund (FWF), grants P22108, P24304, I395, and W1245. Part of this work was written during the stay of the second author at the University of Technology, Munich (Germany), as a John von Neumann Professor. The second author thanks the Department of Mathematics in Munich for the hospitality and Jean Dolbeault (Paris) for very helpful discussions on Beckner inequalities.
© EDP Sciences, SMAI, 2015