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

TL;DR

This work develops the first convergent numerical method for the porous medium equation with fractional potential pressure in one dimension by leveraging the integrated formulation v(x,t)=∫_{-∞}^x u and a monotone upwind finite-difference-quadrature scheme for the nonlocal operator. The authors prove local uniform convergence of the integrated variable and weak-* convergence of the density to the true solution, handling measure-valued and delta initial data, and establish a robust convergence framework via viscosity solutions and half-relaxed limits. They provide a careful CFL-based stability analysis, a finite-volume interpretation, and comprehensive numerical experiments (including explicit m=2 solutions and delta data) to validate accuracy and convergence rates. The methodology offers a stable, monotone, and scalable approach for nonlocal PME with fractional pressure, with potential extensions to higher dimensions and more general nonlocal operators.

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.

Figures (5)

• Figure 1: Errors in experiment 1 measured in $E_v(h)$, $E_u(h)$, $E_u^w(h)$ respectively.
• Figure 2: Errors in experiment 2 measured in $E_v(h)$, $E_u(h)$, $E_u^w(h)$ respectively.
• Figure 3: Errors in experiment 3 measured in $E_v(h)$, $E_u(h)$, $E_u^w(h)$ respectively.
• Figure 4: Example of no comparison principle: $u_1$ (dotted blue) and $u_2$ (solid red).
• Figure 5: Example of comparison for the integrated problem: $v_1$ (dotted blue) and $v_2$ (solid red).

Theorems & Definitions (47)

• Definition 2.1
• Theorem 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• Theorem 2.4
• proof : Proof of Theorem \ref{['thm:maineq']}\ref{['thm:maineq-itemtight']}
• Remark 3.1
• Theorem 3.2
• Remark 3.3