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.

翻译：我们提出并分析了一种数值逼近方法，用于求解由Caffarelli与Vázquez引入的分数阶势压力多孔介质方程： \[ \partial_t u = \nabla \cdot (u^{m-1}\nabla (-\Delta)^{-\sigma}u) \qquad \text{其中} \qquad m\geq2 \quad \text{且} \quad \sigma\in(0,1). \] 该格式适用于一维空间情形及正解$u$。其核心思路为：对$v(x,t)=\int_{-\infty}^xu(x,t)dx$所满足的拟线性非散度形式方程 \[ \partial_t v= -|\partial_x v|^{m-1} (- \Delta)^{s} v \qquad \text{其中} \qquad s=1-\sigma, \] 进行数值求解，再通过数值微分计算$u=v_x$。我们创新性地运用迎风思想，构造了针对$v$方程的新型简单单调且$L^\infty$-稳定的逼近格式，并证明了该格式局部一致收敛至唯一不连续粘性解。借助概率论方法，进一步证明$u$的逼近在弱-*意义下收敛，更精确地说，在归一化处理后，收敛于$C(0,T; P(\mathbb{R}))$空间——其中$P(\mathbb{R})$为装备Rubinstein-Kantorovich度量的概率测度空间。分析还涵盖了$u$初值为Dirac质量的基本解情形。数值试验验证了理论结果。该格式是此类问题的首个数值方案。