Mathematics – Numerical Analysis
Scientific paper
2008-07-22
ESAIM: M2AN 44 (2010) 133-166
Mathematics
Numerical Analysis
36 pages, 9 figures; re-wrote introduction, added 6 references, added discussion of diagonally implicit Runge-Kutta schemes, m
Scientific paper
10.1051/m2an/2009043
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.
Westdickenberg Michael
Wilkening Jon
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