The Aronson-B\'enilan estimate for a Lagrangian particle discretization of the Porous Medium Equation

TL;DR

This paper proves that a particle-based discretization of the 1D porous medium equation satisfies key continuum estimates and converges to the true solution for general initial data, ensuring reliability of the numerical scheme.

Contribution

It establishes a discrete Aronson-Bénilan estimate for a particle scheme and proves its convergence to the PDE solution for broad initial conditions.

Findings

Discrete Aronson-Bénilan estimate proven for particle scheme

Uniform support growth and decay estimates established

Convergence of the scheme to the PDE solution in L^1 for general initial data

Abstract

We consider a nearest neighbor, Lagrangian particle discretization of the one dimensional porous medium equation. We prove that the particle model satisfies a discrete analog of the celebrated Aronson-B\'enilan estimate, which we use to prove a growth estimate for the evolution of the support and an $L^\infty$ decay estimate which are both known to hold in the continuum. These estimates are uniform with respect to the number of particles. We also prove convergence of the scheme towards the solution to the porous medium equation in the full generality of $L^1$ initial data.