Bound-preserving OEDG schemes for Aw-Rascle-Zhang traffic models on networks

TL;DR

The paper develops bound-preserving oscillation-eliminating DG schemes for the adapted pressure ARZ (AP ARZ) and original ARZ traffic models on networks. It introduces invariant-domain analysis, a generalized Lax–Friedrichs splitting via geometric quasilinearization, and an oscillation-eliminating damping operator, integrated with high-order DG and SSP-RK time stepping. A bound-preserving limiter enforces positivity of density and bounds on the Riemann invariants $w$ and $c$ (which indirectly control $v$), while local/global approximate invariant domains manage tight constraints at boundaries and junctions. Numerical experiments on single roads and networks demonstrate third-order accuracy, robust BP properties, and accurate wave-coupling across junctions, including near-vacuum states. This framework enhances the reliability and fidelity of macroscopic traffic simulations that respect essential physical bounds.

Abstract

Physical solutions to the widely used Aw-Rascle-Zhang (ARZ) traffic model and the adapted pressure (AP) ARZ model should satisfy the positivity of density, the minimum and maximum principles with respect to the velocity $v$ and other Riemann invariants. Many numerical schemes suffer from instabilities caused by violating these bounds, and the only existing bound-preserving (BP) numerical scheme (for ARZ model) is random, only first-order accurate, and not strictly conservative. This paper introduces arbitrarily high-order provably BP DG schemes for these two models, preserving all the aforementioned bounds except the maximum principle of $v$, which has been rigorously proven to conflict with the consistency and conservation of numerical schemes. Although the maximum principle of $v$ is not directly enforced, we find that the strictly preserved maximum principle of another Riemann invariant $w$ actually enforces an alternative upper bound on $v$. At the core of this work, analyzing and rigorously proving the BP property is a particularly nontrivial task: the Lax-Friedrichs (LF) splitting property, usually expected for hyperbolic conservation laws and employed to construct BP schemes, does not hold for these two models. To overcome this challenge, we formulate a generalized version of the LF splitting property, and prove it via the geometric quasilinearization (GQL) approach [Kailiang Wu and Chi-Wang Shu, SIAM Review, 65: 1031-1073, 2023]. To suppress spurious oscillations in the DG solutions, we employ the oscillation-eliminating (OE) technique, recently proposed in [Manting Peng, Zheng Sun, and Kailiang Wu, Mathematics of Computation, in press], which is based on the solution operator of a novel damping equation. Several numerical examples are included to demonstrate the effectiveness, accuracy, and BP properties of our schemes, with applications to traffic simulations on road networks.

Figures (13)

• Figure 1: Flowchart of key theoretical findings towards the BP property of proposed schemes.
• Figure 2: Example \ref{['exam2']}, the numerical results of density and velocity obtained by using the BP-OEDG, BPDG, and conventional DG schemes, $t = 0.4$, $\Delta x = 1/300$.
• Figure 3: Example \ref{['exam2b']}, BP-OEDG scheme, $t=0.1$, $\Delta x = 1/300$. Reference solutions are depicted in black dashed lines.
• Figure 4: Example \ref{['exam2b']}, BP-OEDG scheme, $t=0.1$, $\Delta x = 1/300$. Reference solutions are depicted in black dashed lines.
• Figure 5: Example \ref{['exam2b']}, Test T2b, $\gamma = 0$, BP-OEDG scheme compared with BP-OEDG scheme without enforcing the constraint $w \le w_{\max}$, $t =0.0309$, $\Delta x = 1/300$.

Theorems & Definitions (44)

• Remark 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4: GQL representation of $\Omega_2$
• proof
• Lemma 2.5