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Influx ratio preserving coupling conditions for the networked Lighthill-Whitham-Richards model

Niklas Kolbe

TL;DR

The paper studies coupling conditions for the LWR traffic model on networks, introducing an influx-ratio preserving rule at merging junctions and embedding it into both the classical LWR framework and a Jin–Xin relaxation system. It develops two Riemann solvers for 2-to-1 mergers: a relaxation-based solver that enforces Kirchhoff and influx-ratio constraints in the relaxation limit, and an influx-ratio preserving solver for the unrelaxed model that either maximizes flow in free flow or distributes capacity according to local inflow ratios in congestion. Theoretical results provide solvability conditions and a selection criterion among multiple coupling states, while numerical experiments demonstrate that the relaxation-based solver can predict both free-flow and congested regimes without flow maximization, though it may introduce inefficiencies in complex congestion cases. The work offers a robust alternative to traditional flow-maximizing junction models and suggests practical implications for traffic prediction at merges, with a framework that preserves key physical quantities and enables efficient computational solvers.

Abstract

A new coupling rule for the Lighthill-Whitham-Richards model at merging junctions is introduced that imposes the preservation of the ratio between inflow from a given road to the total inflow into the junction. This rule is considered both in the context of the original traffic flow model and a relaxation setting giving rise to two different Riemann solvers that are discussed for merging 2-to-1 junctions. Numerical experiments are shown suggesting that the relaxation based Riemann solver is capable of suitable predictions of both, free-flow and congestion scenarios without relying on flow maximization.

Influx ratio preserving coupling conditions for the networked Lighthill-Whitham-Richards model

TL;DR

The paper studies coupling conditions for the LWR traffic model on networks, introducing an influx-ratio preserving rule at merging junctions and embedding it into both the classical LWR framework and a Jin–Xin relaxation system. It develops two Riemann solvers for 2-to-1 mergers: a relaxation-based solver that enforces Kirchhoff and influx-ratio constraints in the relaxation limit, and an influx-ratio preserving solver for the unrelaxed model that either maximizes flow in free flow or distributes capacity according to local inflow ratios in congestion. Theoretical results provide solvability conditions and a selection criterion among multiple coupling states, while numerical experiments demonstrate that the relaxation-based solver can predict both free-flow and congested regimes without flow maximization, though it may introduce inefficiencies in complex congestion cases. The work offers a robust alternative to traditional flow-maximizing junction models and suggests practical implications for traffic prediction at merges, with a framework that preserves key physical quantities and enables efficient computational solvers.

Abstract

A new coupling rule for the Lighthill-Whitham-Richards model at merging junctions is introduced that imposes the preservation of the ratio between inflow from a given road to the total inflow into the junction. This rule is considered both in the context of the original traffic flow model and a relaxation setting giving rise to two different Riemann solvers that are discussed for merging 2-to-1 junctions. Numerical experiments are shown suggesting that the relaxation based Riemann solver is capable of suitable predictions of both, free-flow and congestion scenarios without relying on flow maximization.
Paper Structure (11 sections, 1 theorem, 31 equations, 2 figures)

This paper contains 11 sections, 1 theorem, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Coupling conditions for the networked Lighthill–-Whitham-–Richards model
  3. Mass conservation
  4. Outgoing roads
  5. Incoming roads
  6. Demand and supply conditions
  7. A relaxation based Riemann solver
  8. Merging 2-to-1 junctions
  9. The relaxation based Riemann solver at merging junctions
  10. An influx ratio preserving Riemann solver for the unrelaxed model
  11. Numerical experiments

Key Result

Lemma 4.1

Suppose that $v_0^1 + v_0^2>0$ as well as the bounds hold then eq:2to1final has at least one real solution. The same consequence follows if the converse inequalities of both, eq:lemc1 and eq:lemc2 hold.

Figures (2)

  • Figure 1: Merging junction with two incoming and one outgoing road.
  • Figure 2: Numerical solutions to Experiments 1--3 showing the incoming vehicle densities $\rho^1$ (blue lines) and $\rho^2$ (red lines) on the left and the outgoing vehicle density $\rho^3$ on the right at the final time ($T=0.75$ in Experiment 1 and $T=1$ in Experiments 2 and 3). In Experiment 3 the solution obtained by the relaxation based Riemann solver is shown using dotted lines.

Theorems & Definitions (4)

  • Remark 2.1
  • Lemma 4.1
  • proof
  • Remark 4.2