  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.
