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

Abstract

Physical solutions to the widely used Aw–Rascle–Zhang (ARZ) traffic model and the adapted pressure 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 discontinuous Galerkin (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 approach (Wu and Shu, 2023 [47]). To suppress spurious oscillations in the DG solutions, we incorporate the oscillation-eliminating technique, recently proposed in (Peng et al., 2024 [34]), 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.