We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation's gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/ maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We also present results from numerical experiments.
Mots-clés : lagrangian discretization, nonlinear Fokker-Planck equation, gradient flow, Wasserstein metric
@article{M2AN_2014__48_3_697_0,
author = {Matthes, Daniel and Osberger, Horst},
title = {Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {697--726},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
year = {2014},
doi = {10.1051/m2an/2013126},
mrnumber = {3177862},
zbl = {1293.65119},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2013126/}
}
TY - JOUR AU - Matthes, Daniel AU - Osberger, Horst TI - Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 697 EP - 726 VL - 48 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013126/ DO - 10.1051/m2an/2013126 LA - en ID - M2AN_2014__48_3_697_0 ER -
%0 Journal Article %A Matthes, Daniel %A Osberger, Horst %T Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 697-726 %V 48 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013126/ %R 10.1051/m2an/2013126 %G en %F M2AN_2014__48_3_697_0
Matthes, Daniel; Osberger, Horst. Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 697-726. doi : 10.1051/m2an/2013126. http://archive.numdam.org/articles/10.1051/m2an/2013126/
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