A convex numerical scheme for the Evans-Gangbo transport-density system, in one, two, and three dimensions

We give a constructive numerical treatment of Monge's optimal transport problem in 1D, 2D, 3D, following Evans-Gangbo: the map is built from a Kantorovich potential $u$ and density $a$ solving $-\mathrm{div}(a\nabla u) = f^+-f^-$, via $\dot T = -a\nabla u(T)/[(1-t)f^+(T)+tf^-(T)]$, $T(0,x)=x$. In 1D