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A deterministic particle method for the porous media equation

Amina Amassad, Datong Zhou

TL;DR

This work analyzes a deterministic particle method for the porous media equation in dimension $d$ by recasting it as a diffusion-velocity transport problem with mollified kernels. The authors establish a quantitative convergence rate in the Wasserstein-2 distance between the mollified particle-transport solution and the true porous media solution, driven by a novel commutator estimate across an intermediate scale and a gradient-flow framework in Wasserstein space. In general dimensions, they prove $\sup_{t\in[0,T]} W_2(u,u^\varepsilon) \le C\varepsilon^r$ with $r<\frac{1}{d(4k+2)}$, where $k$ encodes kernel regularity, and provide a corollary giving an $L^2$ error bound; in 1D, stronger rates are obtained under convexity assumptions on the kernel. The results have practical implications for explicit, efficient numerical schemes based on convolution kernels and gradient-flow structure, and they quantify how mollification error propagates through the nonlinear diffusion-velocity dynamics.

Abstract

This paper deals with the deterministic particle method for the equation of porous media (with p = 2). We establish a convergence rate in the Wasserstein-2 distance between the approximate solution of the associated nonlinear transport equation and the solution of the original one. This seems to be the first quantitative rate for diffusion-velocity particle methods solving diffusive equations and is achieved using a novel commutator estimate for the Wasserstein transport map.

A deterministic particle method for the porous media equation

TL;DR

This work analyzes a deterministic particle method for the porous media equation in dimension by recasting it as a diffusion-velocity transport problem with mollified kernels. The authors establish a quantitative convergence rate in the Wasserstein-2 distance between the mollified particle-transport solution and the true porous media solution, driven by a novel commutator estimate across an intermediate scale and a gradient-flow framework in Wasserstein space. In general dimensions, they prove with , where encodes kernel regularity, and provide a corollary giving an error bound; in 1D, stronger rates are obtained under convexity assumptions on the kernel. The results have practical implications for explicit, efficient numerical schemes based on convolution kernels and gradient-flow structure, and they quantify how mollification error propagates through the nonlinear diffusion-velocity dynamics.

Abstract

This paper deals with the deterministic particle method for the equation of porous media (with p = 2). We establish a convergence rate in the Wasserstein-2 distance between the approximate solution of the associated nonlinear transport equation and the solution of the original one. This seems to be the first quantitative rate for diffusion-velocity particle methods solving diffusive equations and is achieved using a novel commutator estimate for the Wasserstein transport map.

Paper Structure

This paper contains 19 sections, 13 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. A particle method for the porous media equation
  3. Gradient flow structure in Wasserstein space
  4. Choice of kernel
  5. Main results
  6. Sketch of the proofs
  7. Well-posedness of equations
  8. Tools from optimal transport
  9. Proof of Theorem \ref{['thm:1d']}
  10. Proof in general dimensions
  11. Fundamental a priori estimates
  12. Kernel decomposition
  13. Key commutator estimate
  14. Bound of $C_{u, \tilde{u}_\eta}^{(1)}$
  15. Bound of $C_{u, \tilde{u}_\eta}^{(2)}$
  16. ...and 4 more sections

Key Result

Theorem 1.2

Assume initial data $u_0 \in L^\infty(\mathbb{T}^d)$ and an admissible kernel $R$ in the sense of Definition defi:potential. Let $u^\varepsilon \in L^\infty([0,T], L^\infty(\mathbb{T}^d))$ be the weak solution of eq:num with mollifying kernel $R_\varepsilon$ given by eq:R_torus. Let $u \in L^\infty( where $k$ comes from Definition defi:potential. The explicit formula of $C$ depends on $T > 0$, the

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 11 more