  We introduce and analyze a numerical approximation of the porous medium
equation with fractional potential pressure introduced by Caffarelli and
V\'azquez: \[ \partial_t u = \nabla \cdot (u^{m-1}\nabla (-\Delta)^{-\sigma}u)
\qquad \text{for} \qquad m\geq2 \quad \text{and} \quad \sigma\in(0,1). \] Our
scheme is for one space dimension and positive solutions $u$. It consists of
solving numerically the equation satisfied by
$v(x,t)=\int_{-\infty}^xu(x,t)dx$, the quasilinear non-divergence form equation
\[ \partial_t v= -|\partial_x v|^{m-1} (- \Delta)^{s} v \qquad \text{where}
\qquad s=1-\sigma, \] and then computing $u=v_x$ by numerical differentiation.
Using upwinding ideas in a novel way, we construct a new and simple, monotone
and $L^\infty$-stable, approximation for the $v$-equation, and show local
uniform convergence to the unique discontinuous viscosity solution. Using ideas
from probability theory, we then prove that the approximation of $u$ converges
weakly-$*$, or more precisely, up to normalization, in $C(0,T; P(\mathbb{R}))$
where $P(\mathbb{R})$ is the space of probability measures under the
Rubinstein-Kantorovich metric.The analysis include also fundamental solutions
where the initial data for $u$ is a Dirac mass. Numerical tests are included to
confirm the results. Our scheme seems to be the first numerical scheme for this
type of problems.
