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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...
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