We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and Vázquez: \( \partial _t u = \nabla \cdot (u^{m-1}\nabla (-\Delta )^{-\sigma }u) \qquad \text {for} \qquad m\ge 2 \quad \text {and} \quad \sigma \in (0,1). \) Our scheme is for one space dimension and positive solutions u. It consists of solving numerically the equation satisfied by \(v(x,t)=\int _{-\infty }^xu(y,t)dy\) , the quasilinear nondivergence 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. The full scheme then becomes a conservative up-wind finite volume approximation for the u-equation. We show local uniform convergence to the unique discontinuous viscosity solution for the v-problem, and using ideas from probability theory, we prove that the approximation of u converges up to normalization in \(C(0,T; P(\mathbb {R}))\) where \(P(\mathbb {R})\) is the space of probability measures under the Rubinstein-Kantorovich (bounded Lipschitz) metric. The analysis includes also fundamental solutions where the initial data for u is a Dirac mass. Numerical tests are included to support the results. Our scheme seems to be the first numerical scheme for this type of problems.
