Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gradient flows of interacting Laguerre cells as discrete porous media flows

Andrea Natale

TL;DR

The article develops a rigorous framework linking gradient flows of Laguerre-cell-based energies to porous medium-type PDEs in the high-particle limit. By exploiting a modulated energy and semi-discrete optimal transport tools, it proves quantitative convergence of discrete densities μ̄_N to smooth densities ρ solving ∂_t ρ - div(ρ ∇U'(ρ)) = 0, along with convergence of the discrete flow to the Lagrangian velocity field. The work also extends to external potentials and provides a time-discretized numerical scheme with empirical tests on Barenblatt-type dynamics and quadratic potentials, confirming the theoretical rates. This establishes a solid bridge between particle-based Laguerre flows and continuum porous media dynamics, with practical implications for Lagrangian discretizations and numerical approximations of nonlinear diffusion processes.

Abstract

We study a class of discrete models in which a collection of particles evolves in time following the gradient flow of an energy depending on the cell areas of an associated Laguerre (i.e. a weighted Voronoi) tessellation. We consider the high number of cell limit of such systems and, using a modulated energy argument, we prove convergence towards smooth solutions of nonlinear diffusion PDEs of porous medium type.

Gradient flows of interacting Laguerre cells as discrete porous media flows

TL;DR

The article develops a rigorous framework linking gradient flows of Laguerre-cell-based energies to porous medium-type PDEs in the high-particle limit. By exploiting a modulated energy and semi-discrete optimal transport tools, it proves quantitative convergence of discrete densities μ̄_N to smooth densities ρ solving ∂_t ρ - div(ρ ∇U'(ρ)) = 0, along with convergence of the discrete flow to the Lagrangian velocity field. The work also extends to external potentials and provides a time-discretized numerical scheme with empirical tests on Barenblatt-type dynamics and quadratic potentials, confirming the theoretical rates. This establishes a solid bridge between particle-based Laguerre flows and continuum porous media dynamics, with practical implications for Lagrangian discretizations and numerical approximations of nonlinear diffusion processes.

Abstract

We study a class of discrete models in which a collection of particles evolves in time following the gradient flow of an energy depending on the cell areas of an associated Laguerre (i.e. a weighted Voronoi) tessellation. We consider the high number of cell limit of such systems and, using a modulated energy argument, we prove convergence towards smooth solutions of nonlinear diffusion PDEs of porous medium type.
Paper Structure (28 sections, 8 theorems, 153 equations, 4 figures)

This paper contains 28 sections, 8 theorems, 153 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem description
  3. Relation with Laguerre tessellations and other discrete models
  4. Relation with Lagrangian discretizations of porous media flow
  5. Relation with semi-discrete optimal transport
  6. Continuous limit
  7. Analysis of the discrete model
  8. Internal energy
  9. Dual formulation
  10. Discrete dynamical system
  11. Convergence towards smooth solutions
  12. Preliminary lemmas
  13. Main assumptions and relative entropy
  14. Time derivative of the relative entropy
  15. Uniform estimates
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $U:[0,\infty) \rightarrow \mathbb{R}$ be a smooth strictly convex function, with $U(0)=0$, satisfying the assumptions of Lemma lem:Abound and suppose that there exist $R,\alpha>1$ and $\beta>0$, such that Suppose that $\rho:[0,T]\times \Omega \rightarrow [0,\infty)$ is a strong solution of eq:pde0, such that $\rho^0 : \Omega \rightarrow [\rho_{min},\infty)$ with $\rho_{min}>0$ is of class $C^

Figures (4)

  • Figure 1: An example of optimal tessellation with cells constructed via equation \ref{['eq:Lsxw']}.
  • Figure 2: Error $\Delta \varphi$ defined in equation \ref{['eq:deltaphi']} as a function of $1/\sqrt{N}$, for $\gamma = 1.5$ (left), $\gamma =2$ (center) and $\gamma=4$ (right). The curves are compared to $h_N^p \sim N^{-p/2}$, corresponding to $p$th order of convergence, with $p$ evaluated over the last refinement step.
  • Figure 3: Scatter plot of the particle positions at different times (from left to right, $t=0, 0.05, 0.2, 8$) for the quadratic potential test case. The color scale refers to the density, computed for each particle as $m^0_i/|L_i|$.
  • Figure 4: Energy evolution for the quadratic potential test case: (a) $F_\varepsilon(X)$; (b) $\sum_i U(m^0_i/|L_i|)|L_i|$; (c) internal energy of the equilibrium density $\int U(\rho_\infty)$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • ...and 7 more