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We introduce and analyze a numerical approximation of the porous mediumequation with fractional potential pressure introduced by Caffarelli andV\'azquez: \[ \partial_t u = \nabla \cdot (u^{m-1}\nabla (-\Delta)^{-\sigma}u)\qquad \text{for} \qquad m\geq2 \quad \text{and} \quad \sigma\in(0,1). \] Ourscheme is for one space dimension and positive solutions $u$. It consists ofsolving 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, monotoneand $L^\infty$-stable, approximation for the $v$-equation, and show localuniform convergence to the unique discontinuous viscosity solution. Using ideasfrom probability theory, we then prove that the approximation of $u$ convergesweakly-$*$, or more precisely, up to normalization, in $C(0,T; P(\mathbb{R}))$where $P(\mathbb{R})$ is the space of probability measures under theRubinstein-Kantorovich metric.The analysis include also fundamental solutionswhere the initial data for $u$ is a Dirac mass. Numerical tests are included toconfirm the results. Our scheme seems to be the first numerical scheme for thistype of problems.

A convergent finite difference-quadrature scheme for the porous medium equation with nonlocal pressure