Voir la notice de l'article provenant de la source Numdam
Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these variational characterizations instead of the partial differential equations themselves, we obtain new schemes with remarkable stability properties. We show that they capture successfully the nonlinear features of the flows, such as shocks and rarefaction waves for the isentropic Euler equations. We also show how to design higher order methods for these problems in the optimal transport setting using backward differentiation formula (BDF) multi-step methods or diagonally implicit Runge-Kutta methods.
@article{M2AN_2010__44_1_133_0,
author = {Westdickenberg, Michael and Wilkening, Jon},
title = {Variational particle schemes for the porous medium equation and for the system of isentropic {Euler} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {133--166},
publisher = {EDP-Sciences},
volume = {44},
number = {1},
year = {2010},
doi = {10.1051/m2an/2009043},
mrnumber = {2647756},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2009043/}
}
TY - JOUR AU - Westdickenberg, Michael AU - Wilkening, Jon TI - Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2010 SP - 133 EP - 166 VL - 44 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2009043/ DO - 10.1051/m2an/2009043 LA - en ID - M2AN_2010__44_1_133_0 ER -
%0 Journal Article %A Westdickenberg, Michael %A Wilkening, Jon %T Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2010 %P 133-166 %V 44 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2009043/ %R 10.1051/m2an/2009043 %G en %F M2AN_2010__44_1_133_0
Westdickenberg, Michael; Wilkening, Jon. Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 44 (2010) no. 1, pp. 133-166. doi: 10.1051/m2an/2009043
Cité par Sources :