Geodesic
Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities
Chainais-Hillairet, Claire1  ; Jüngel, Ansgar2  ; Schuchnigg, Stefan2
1 Laboratoire Paul Painlevé, U.M.R. CNRS 8524, Université Lille 1, Cité Scientifique, 59655 Villeneuve, d’Ascq cedex, France
2 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 135-162

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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.

Reçu le :
DOI : 10.1051/m2an/2015031
Classification : 65M08, 65M12, 76S05
Keywords: Porous-medium equation, fast-diffusion equation, finite-volume method, entropy dissipation, Beckner inequality, entropy construction method

Chainais-Hillairet, Claire 1 ; Jüngel, Ansgar 2 ; Schuchnigg, Stefan 2

1 Laboratoire Paul Painlevé, U.M.R. CNRS 8524, Université Lille 1, Cité Scientifique, 59655 Villeneuve, d’Ascq cedex, France
2 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria 
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     title = {Entropy-dissipative discretization of nonlinear diffusion equations and discrete {Beckner} inequalities},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {135--162},
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Chainais-Hillairet, Claire; Jüngel, Ansgar; Schuchnigg, Stefan. Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 135-162. doi: 10.1051/m2an/2015031

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