Félix del Teso, Espen R. Jakobsen
Abstract We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and Vázquez: $$ \partial _t u = \nabla \cdot (u^{m-1}\nabla (-\Delta )^{-\sigma }u) \qquad \text {for} \qquad m\ge 2 \quad \text {and} \quad \sigma \in (0,1). $$ ∂ t u = ∇ · ( u m - 1 ∇ ( - Δ ) - σ u ) for m ≥ 2 and σ ∈ ( 0 , 1 ) . Our scheme is for one space dimension and positive solutions u . It consists of solving numerically the equation satisfied by $$v(x,t)=\int _{-\infty }^xu(y,t)dy$$ v ( x , t ) = ∫ - ∞ x u ( y , t ) d y , the quasilinear nondivergence form equation $$ \partial _t v= -|\partial _x v|^{m-1} (- \Delta )^{s} v \qquad \text {where} \qquad s=1-\sigma , $$ ∂ t v = - | ∂ x v | m - 1 ( - Δ ) s v where s = 1 - σ , and then computing $$u=v_x$$ u = v x by numerical differentiation. Using upwinding ideas in a novel way, we construct a new and simple, monotone and $$L^\infty $$ L ∞ -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 $$C(0,T; P(\mathbb {R}))$$ <mml:math xmlns:mml="http://www.w3.org/199