A convergent finite difference-quadrature scheme for the porous medium equation with nonlocal pressure
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We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and Vázquez: ∂_t u = ∇· (u^m-1∇ (-Δ)^-σu) 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)=∫_-∞^xu(x,t)dx, the quasilinear non-divergence form equation ∂_t v= -|∂_x v|^m-1 (- Δ)^s v where s=1-σ, 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^∞-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(ℝ)) where P(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.
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