Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fully discrete follow-the-leader approximation of one-dimensional scalar conservation laws with vacuum

M. Di Francesco, S. Fagioli, V. Iorio, M. D. Rosini

TL;DR

The paper develops a fully discrete Follow-the-Leader particle scheme for 1D scalar conservation laws with vacuum, using a $\theta$-method in time and a density reconstruction from particles. It proves global well-posedness, $L^{\infty}$ and BV stability, and time-compactness under a CFL-type condition, enabling convergence to Kruzhkov entropy solutions without requiring a strictly positive initial density. With additional time-step constraints, the scheme converges to the entropy solution, ensuring numerical entropy consistency even in vacuum. The results provide a robust, vacuum-tolerant Lagrangian framework that unifies discrete particle methods with entropy theory for hyperbolic conservation laws.

Abstract

We present a fully discrete particle approximation for one-dimensional scalar conservation laws. Under suitable monotonicity assumptions on the macroscopic velocity, we construct a vacuum-compatible family of time-discrete particle equations and show that an appropriate piecewise-constant density reconstruction from the particle setting converges to the unique entropy weak solution of the macroscopic scalar conservation law.

Fully discrete follow-the-leader approximation of one-dimensional scalar conservation laws with vacuum

TL;DR

The paper develops a fully discrete Follow-the-Leader particle scheme for 1D scalar conservation laws with vacuum, using a -method in time and a density reconstruction from particles. It proves global well-posedness, and BV stability, and time-compactness under a CFL-type condition, enabling convergence to Kruzhkov entropy solutions without requiring a strictly positive initial density. With additional time-step constraints, the scheme converges to the entropy solution, ensuring numerical entropy consistency even in vacuum. The results provide a robust, vacuum-tolerant Lagrangian framework that unifies discrete particle methods with entropy theory for hyperbolic conservation laws.

Abstract

We present a fully discrete particle approximation for one-dimensional scalar conservation laws. Under suitable monotonicity assumptions on the macroscopic velocity, we construct a vacuum-compatible family of time-discrete particle equations and show that an appropriate piecewise-constant density reconstruction from the particle setting converges to the unique entropy weak solution of the macroscopic scalar conservation law.
Paper Structure (13 sections, 14 theorems, 127 equations, 1 figure)

This paper contains 13 sections, 14 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The numerical particles scheme and main results
  3. The scheme
  4. Main result
  5. Well-posedness for the particle scheme and stability estimates
  6. Uniform bound for
  7. -estimate
  8. Time oscillations and compactness
  9. Convergence to entropy solution
  10. Compactness
  11. Convergence to a weak solution
  12. Convergence to the entropy solution
  13. An interpolation inequality

Key Result

Theorem 2.2

Consider a velocity function $v \colon [0,R] \to \mathbb{R}$ and an initial density $\overline{\rho} \colon \mathbb{R} \to [0,R]$ satisfying e:vel and e:inirho, respectively. Fix a time horizon $T>0$, a parameter $\theta \in [0,1]$ and two positive integers $M,N\in\mathbb{N}$ satisfying the followin where $L = \left\|\overline{\rho}\right\|_{L^1(\mathbb{R})}$. In this case, the discrete density e:

Figures (1)

  • Figure 1: Approximate solution $\rho_n$ plotted at different times, corresponding to $v(\rho)=\frac{1}{2}-\rho$, with initial data $\bar{\rho}(x)=0.8 \, \mathbf{1}_{[-1,0)}(x) + 0.4 \, \mathbf{1}_{[0,1)}(x)$ (left) and $\bar{\rho}(x) = 0.4 \, \mathbf{1}_{[-1,0)}(x) + 0.8 \, \mathbf{1}_{[0,1)}(x)$ (right). We use the implicit version ($\theta=0$) of the scheme described in \ref{['e:theta_intro']} below, with $\ell=0.012$ and $\tau=0.01$. The density $\rho_n$ attains values both greater than and less than $1/2$; hence the velocity $v(\rho_n)$ takes both negative and positive values. Moreover, $\rho_n$ approximates the exact solution to \ref{['e:CP']}, showing the correct upwinding in the scheme.

Theorems & Definitions (31)

  • Remark 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 21 more