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

Published online by Cambridge University Press:  16 December 2009

Michael Westdickenberg  and
Jon Wilkening
Michael Westdickenberg
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA. mwest@math.gatech.edu
Jon Wilkening
Affiliation:
Department of Mathematics, University of California, 1091 Evans Hall #3840, Berkeley, CA 94720-3840, USA. wilken@math.berkeley.edu
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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.

Keywords

Optimal transportWasserstein metricisentropic Euler equationsporous medium equationnumerical methods
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 1 , January 2010 , pp. 133 - 166
Copyright
© EDP Sciences, SMAI, 2009

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