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

We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and Vázquez: $${\mathrm{\partial }}_{t}u=\mathrm{\nabla }\cdot (u^{m-1}\mathrm{\nabla }(-\mathrm{\Delta }{)}^{-\sigma }u)\text{for}m\ge 2\text{and}\sigma \in (0,1)\mathrm{.}$$ Our scheme is for one space dimension and positive solutions $u$. It consists of solving numerically the equation satisfied by $v(x,t)={\int }_{-\mathrm{\infty }}^{x}u(x,t)dx$, the quasilinear non-divergence form equation $${\mathrm{\partial }}_{t}v=-\mathrm{\mid }{\mathrm{\partial }}_{x}v{\mathrm{\mid }}^{m-1}(-\mathrm{\Delta }{)}^{s}v\text{where}s=1-\sigma ,$$ and then computing $u=v_x$ by numerical differentiation. Using upwinding ideas in a novel way, we construct a new and simple, monotone and $L^{\mathrm{\infty }}$-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-$\ast$, 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.