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.
DOI : 10.1051/m2an/2015031
Mots clés : Porous-medium equation, fast-diffusion equation, finite-volume method, entropy dissipation, Beckner inequality, entropy construction method
@article{M2AN_2016__50_1_135_0, author = {Chainais-Hillairet, Claire and J\"ungel, Ansgar and Schuchnigg, Stefan}, title = {Entropy-dissipative discretization of nonlinear diffusion equations and discrete {Beckner} inequalities}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {135--162}, publisher = {EDP-Sciences}, volume = {50}, number = {1}, year = {2016}, doi = {10.1051/m2an/2015031}, zbl = {1341.65034}, mrnumber = {3460104}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015031/} }
TY - JOUR AU - Chainais-Hillairet, Claire AU - Jüngel, Ansgar AU - Schuchnigg, Stefan TI - Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 135 EP - 162 VL - 50 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015031/ DO - 10.1051/m2an/2015031 LA - en ID - M2AN_2016__50_1_135_0 ER -
%0 Journal Article %A Chainais-Hillairet, Claire %A Jüngel, Ansgar %A Schuchnigg, Stefan %T Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 135-162 %V 50 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015031/ %R 10.1051/m2an/2015031 %G en %F M2AN_2016__50_1_135_0
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. http://www.numdam.org/articles/10.1051/m2an/2015031/
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