Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence rate for a semidiscrete approximation of scalar conservation laws

Magnus C. Ørke

TL;DR

This work introduces a semidiscrete particle-path scheme to approximate entropy solutions of one-dimensional scalar conservation laws with nonnegative initial data. By formulating a continuity-equation view with a density-dependent velocity $A$ obtained from interpolated particle velocities, the authors derive an approximate entropy inequality and prove an $L^1$ stability bound that leads to an explicit convergence rate of order $\mathcal{O}(\sqrt{\Delta x^*})$ under standard assumptions. The analysis connects the microscopic Follow-the-Leader model to the macroscopic LWR model for traffic flow and provides a rigorous rate of convergence for this limit, while clarifying differences with front-tracking methods. The results rely on a rigorous PDE formulation, a priori estimates, and Kuznetsov’s lemma, and point toward future work on fully discrete schemes and flow-map convergence to Filippov trajectories. The approach offers a theoretically grounded, numerically implementable route to approximate entropy solutions with provable convergence properties.

Abstract

We propose a semidiscrete scheme for approximation of entropy solutions of one-dimensional scalar conservation laws with nonnegative initial data. The scheme is based on the concept of particle paths for conservation laws and can be interpreted as a finite-particle discretization. A convergence rate of order $1/2$ with respect to initial particle spacing is proved. As a special case, this covers the convergence of the Follow--the--Leader model to the Lighthill--Whitham--Richards model for traffic flow.

Convergence rate for a semidiscrete approximation of scalar conservation laws

TL;DR

This work introduces a semidiscrete particle-path scheme to approximate entropy solutions of one-dimensional scalar conservation laws with nonnegative initial data. By formulating a continuity-equation view with a density-dependent velocity obtained from interpolated particle velocities, the authors derive an approximate entropy inequality and prove an stability bound that leads to an explicit convergence rate of order under standard assumptions. The analysis connects the microscopic Follow-the-Leader model to the macroscopic LWR model for traffic flow and provides a rigorous rate of convergence for this limit, while clarifying differences with front-tracking methods. The results rely on a rigorous PDE formulation, a priori estimates, and Kuznetsov’s lemma, and point toward future work on fully discrete schemes and flow-map convergence to Filippov trajectories. The approach offers a theoretically grounded, numerically implementable route to approximate entropy solutions with provable convergence properties.

Abstract

We propose a semidiscrete scheme for approximation of entropy solutions of one-dimensional scalar conservation laws with nonnegative initial data. The scheme is based on the concept of particle paths for conservation laws and can be interpreted as a finite-particle discretization. A convergence rate of order with respect to initial particle spacing is proved. As a special case, this covers the convergence of the Follow--the--Leader model to the Lighthill--Whitham--Richards model for traffic flow.

Paper Structure

This paper contains 22 sections, 15 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Illustrative example
  4. Comparison to front tracking
  5. Application to traffic modelling
  6. Outline of the paper
  7. Preliminaries
  8. Scalar conservation laws
  9. Particle paths for conservation laws
  10. Continuity equations
  11. The scheme
  12. Approximation of initial data
  13. Particle dynamics
  14. Resolution of collisions
  15. Existence and uniqueness of the approximation
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $u_0$ be a nonnegative function in $\mathop{\mathrm{BV}}\nolimits \cap L^1(\mathbb{R})$ and assume that $f$ is Lipschitz. Then the particle path scheme generates a unique approximation $v$ which satisfies where $u$ is the entropy solution of the conservation law eq:conservation_law.

Figures (2)

  • Figure 1: Approximations of \ref{['eq:entropy_sol_example']} using different numbers of initial particles. The particle paths were computed with the forward Euler method using a small time step. In the particle path plots (right), the background shading represents the magnitude of the approximate solution.
  • Figure 2: Particles $x^i$ and $x^{i+1}$ collide at time $t_1$. The particle $x^{i}$ and the local density $v^i$ are deleted before the system is restarted.

Theorems & Definitions (29)

  • Theorem 1.1: Main Theorem
  • Corollary 1.2
  • Theorem 2.1: Fjordholm, Mæhlen, Ørke fjordholm_maehlen_oerke
  • Remark 2.2
  • Lemma 2.3: Forward flow
  • Lemma 2.4: Representation formula
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • ...and 19 more