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

Abstract

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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.

How to cite

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Westdickenberg, 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 -

References

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