| Abstract: |
| We carefully construct and prove convergence of a numerical discretization of the porous medium equation with fractional pressure,
\begin{equation}\label{FPE}
\frac{\partial u}{\partial t} - \nabla \cdot \left( u^{m-1} \nabla (-\Delta)^{-\sigma}u \right) = 0,
\end{equation}
for $\sigma \in (0,1)$ and $m \geq 2$. The model, introduced by Caffarelli and V\`azquez in 2011, is currently one of two main nonlocal extensions of the local porous medium equation. It has finite speed of propagation, but as opposed to the other extension, it does not satisfy the comparison principle. We exploit the fact that the \emph{cumulative density} $v(x,t) = \int_{-\infty}^x u(y,t)\,dy$ satisfies
\begin{equation*}
\frac{\partial v}{\partial t} + |\partial_x v|^{m-1} (-\Delta)^s v = 0, \quad s = 1 - \sigma,
\end{equation*}
which is a nonlocal quasilinear parabolic equation in non-divergence form that can be analyzed through viscosity solution methods.
The numerical method consists in discretizing this equation with a difference quadrature scheme with upwinding ideas and then computing the solution $u$ of \eqref{FPE} via numerical differentiation. Our results cover both absolutely continuous and Dirac or point mass initial data, and in the latter case, machinery for discontinuous viscosity solutions is needed in the analysis. |
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