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

Félix del Teso; Espen R. Jakobsen

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arXiv · 2303.05168

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

Félix del Teso·Espen R. Jakobsen

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Abstract

We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and Vázquez: \[ \partial_t u = \nabla \cdot (u^{m-1}\nabla (-Δ)^{-σ}u) \qquad \text{for} \qquad m\geq2 \quad \text{and} \quad σ\in(0,1). \] 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 \[ \partial_t v= -|\partial_x v|^{m-1} (- Δ)^{s} v \qquad \text{where} \qquad s=1-σ, \] and then computing $u=v_x$ by numerical differentiation. Using upwinding ideas in a novel way, we construct a new and simple, monotone and $L^\infty$-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.

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BibTeX RIS

Félix del Teso, Espen R. Jakobsen. 2025-12-18. A convergent finite difference-quadrature scheme for the porous medium equation with nonlocal pressure. https://arxiv.org/abs/2303.05168

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