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Deterministic particle method for nonlinear nonlocal scalar balance equations

Emanuela Radici, Federico Stra

Abstract

We study a deterministic particle scheme to solve a scalar balance equation with nonlocal interaction and nonlinear mobility used to model congested dynamics. The main novelty with respect to "Radici-Stra [SIAM J. Math. Anal. 55.3 (2023)]" is the presence of a source term; this causes the solutions to no longer be probability measures, thus requiring a suitable adaptation of the numerical scheme and of the estimates leading to compactness.

Deterministic particle method for nonlinear nonlocal scalar balance equations

Abstract

We study a deterministic particle scheme to solve a scalar balance equation with nonlocal interaction and nonlinear mobility used to model congested dynamics. The main novelty with respect to "Radici-Stra [SIAM J. Math. Anal. 55.3 (2023)]" is the presence of a source term; this causes the solutions to no longer be probability measures, thus requiring a suitable adaptation of the numerical scheme and of the estimates leading to compactness.

Paper Structure

This paper contains 5 sections, 14 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Theoretical analysis
  3. Properties of the numerical approximation
  4. Compactness
  5. Convergence to entropy solutions

Key Result

Theorem 1.1

Let $v,V,W,f$ satisfy the ass and let $\rho_0\in\mathscr{M}_+(\mathbb{R})\cap L^\infty(\mathbb{R})\cap BV(\mathbb{R})$ be a compactly supported non-negative measure. For every $N\in\mathbb{N}_+$, let $\bar{\rho}^N$ be the piecewise constant density associated via eq:rhobar to particles $(x_0,\dots which is an entropy solution of eq:pde in the sense of def:entropy-solution with initial datum $\rh

Figures (1)

  • Figure 1: Piecewise constant density $\bar{\rho}^N$ associated to particles $(x_0,\dots,x_N)$ and masses $(q_1,\dots,q_N)$.

Theorems & Definitions (30)

  • Definition 1: Entropy solution, Kruzkov
  • Theorem 1.1
  • Lemma 1: Initial datum
  • proof
  • Lemma 2
  • proof
  • Lemma 3: First and second differences of the free velocity
  • proof
  • Proposition 1: Uniform bound of the mass
  • proof
  • ...and 20 more