Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of the non-staggered Nessyahu-Tadmor scheme for coupled systems of one-dimensional nonlocal balance laws

Sanjibanee Sudha, Jan Friedrich, Samala Rathan

TL;DR

The paper develops a second-order, non-staggered Nessyahu–Tadmor central scheme for coupled one-dimensional nonlocal balance laws. It proves an L∞ bound and, under linear flux/source assumptions, weak-* convergence to weak solutions, with strong convergence to entropy weak solutions under stronger kernel regularity. BV and L1_loc techniques yield entropy-convergence results and a robust convergence theory, complemented by extensive numerical experiments across Keyfitz–Kranzer, nonlinear scalar flux, multilane traffic, nonlocal Euler, and GARZ-type models. The results demonstrate the scheme’s accuracy, flexibility, and applicability to a broad class of nonlocal balance laws in applications such as traffic flow and fluid dynamics.

Abstract

We derive a second-order accurate, non-staggered central scheme based on the well-known Nessyahu-Tadmor scheme to approximate solutions of coupled systems of nonlocal balance laws. We show that the approximate solutions stay bounded by an exponential $L^\infty$ bound in time. Under linearity assumptions on the flux and source terms the approximate solutions converge weakly-$*$ to weak solutions of the nonlocal balance laws. Assuming stronger regularity, in particular on the convolution kernel, we show strong convergence towards entropy weak solutions in the nonlinear case. Numerical examples validate our results and demonstrate its applicability to various systems of nonlocal problems.

Convergence of the non-staggered Nessyahu-Tadmor scheme for coupled systems of one-dimensional nonlocal balance laws

TL;DR

The paper develops a second-order, non-staggered Nessyahu–Tadmor central scheme for coupled one-dimensional nonlocal balance laws. It proves an L∞ bound and, under linear flux/source assumptions, weak-* convergence to weak solutions, with strong convergence to entropy weak solutions under stronger kernel regularity. BV and L1_loc techniques yield entropy-convergence results and a robust convergence theory, complemented by extensive numerical experiments across Keyfitz–Kranzer, nonlinear scalar flux, multilane traffic, nonlocal Euler, and GARZ-type models. The results demonstrate the scheme’s accuracy, flexibility, and applicability to a broad class of nonlocal balance laws in applications such as traffic flow and fluid dynamics.

Abstract

We derive a second-order accurate, non-staggered central scheme based on the well-known Nessyahu-Tadmor scheme to approximate solutions of coupled systems of nonlocal balance laws. We show that the approximate solutions stay bounded by an exponential bound in time. Under linearity assumptions on the flux and source terms the approximate solutions converge weakly- to weak solutions of the nonlocal balance laws. Assuming stronger regularity, in particular on the convolution kernel, we show strong convergence towards entropy weak solutions in the nonlinear case. Numerical examples validate our results and demonstrate its applicability to various systems of nonlocal problems.
Paper Structure (16 sections, 10 theorems, 124 equations, 6 figures, 5 tables)

This paper contains 16 sections, 10 theorems, 124 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Second-order central scheme
  3. Main results
  4. Non-staggered Nessyahu-Tadmor scheme
  5. Linf-estimate
  6. Weak-$*$ convergence
  7. $L^1_{loc}$ convergence
  8. Numerical examples
  9. Keyfitz-Kranzer-Model
  10. Nonlinear flux: Scalar case
  11. Nonlocal systems with source term
  12. Multilane traffic flow model
  13. Nonlocal Euler equations
  14. Nonlocal in velocity
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 2.4

Let the Assumption set2 hold and let ${\bm{\rho}}_\Delta$ be the sequence of approximate solutions generated by the NT scheme with the slopes eq:slopesmodconv satisfying the CFL conditions eq:CFLmaxNT. Then, as $\Delta x, \Delta t \to 0$, there exists a sufficiently small time $T^*\in (0,T]$ and a f Moreover, $\rho$ is a weak solution of the nonlocal model eq:generalsystemsimple in the sense of De

Figures (6)

  • Figure 1: NT scheme reconstruction from non-staggered to staggered grid for fixed $k\in\{1,\dots,N\}$, compare Figure 3.1 from jiang1998high
  • Figure 2: Numerical solutions of the nonlocal Keyfitz-Kranzer model at $T=0.3$ obtained by different numerical schemes for $\Delta x= \frac{1}{40}$.
  • Figure 3: Zoom into the numerical solutions of the Arrhenuis-type look ahead dynamics at $T=1.5$ obtained by different numerical schemes with $\Delta x= \frac{1}{160}$.
  • Figure 4: Numerical solutions of the nonlocal multilane model at $T=0.5$ obtained by different numerical schemes $\Delta x=5\cdot 10^{-3}$.
  • Figure 5: Numerical solutions of the nonlocal Euler equations at $T=0.5$ obtained by different numerical schemes for $\eta=2\cdot 10^{-3}$.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Definition 2.2
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • Remark 2.11
  • Lemma 2.12
  • proof
  • ...and 18 more