Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation

Daniel Matthes; Horst Osberger

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 3, page 697-726
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Matthes, Daniel, and Osberger, Horst. "Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.3 (2014): 697-726. <http://eudml.org/doc/273272>.

@article{Matthes2014,
abstract = {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.},
author = {Matthes, Daniel, Osberger, Horst},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {lagrangian discretization; nonlinear Fokker-Planck equation; gradient flow; Wasserstein metric; Lagrangian discretization; nonlinear drift diffusion equation; entropy monotonicity; mass preservation; metric contraction; minimum/maximum principles; convergence; Courant-Friedrichs-Lewy condition; numerical results},
language = {eng},
number = {3},
pages = {697-726},
publisher = {EDP-Sciences},
title = {Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation},
url = {http://eudml.org/doc/273272},
volume = {48},
year = {2014},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 3
SP - 697
EP - 726
AB - 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.
LA - eng
KW - lagrangian discretization; nonlinear Fokker-Planck equation; gradient flow; Wasserstein metric; Lagrangian discretization; nonlinear drift diffusion equation; entropy monotonicity; mass preservation; metric contraction; minimum/maximum principles; convergence; Courant-Friedrichs-Lewy condition; numerical results
UR - http://eudml.org/doc/273272
ER -

References

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