# Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations

Michael Westdickenberg; Jon Wilkening

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 1, page 133-166
- ISSN: 0764-583X

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topWestdickenberg, Michael, and Wilkening, Jon. "Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations." ESAIM: Mathematical Modelling and Numerical Analysis 44.1 (2010): 133-166. <http://eudml.org/doc/250812>.

@article{Westdickenberg2010,

abstract = {
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.
},

author = {Westdickenberg, Michael, Wilkening, Jon},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Optimal transport; Wasserstein metric;
isentropic Euler equations; porous medium equation; numerical methods; optimal transport; isentropic Euler equations},

language = {eng},

month = {3},

number = {1},

pages = {133-166},

publisher = {EDP Sciences},

title = {Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations},

url = {http://eudml.org/doc/250812},

volume = {44},

year = {2010},

}

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

DA - 2010/3//

PB - EDP Sciences

VL - 44

IS - 1

SP - 133

EP - 166

AB -
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.

LA - eng

KW - Optimal transport; Wasserstein metric;
isentropic Euler equations; porous medium equation; numerical methods; optimal transport; isentropic Euler equations

UR - http://eudml.org/doc/250812

ER -

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