A random sampling method for a family of temple-class systems of conservation laws
The Aw-Rascle-Zhang (ARZ) traffic model, a model of sedimentation, and other applications lead to nonlinear \(2 \times 2\) systems of conservation laws that are governed by a single scalar system velocity. Such systems are of the Temple class since rarefaction wave curves and Hugoniot curves coincide. This paper is focused on the following initial value problem: \[ \begin{pmatrix} \phi \\ k\phi \end{pmatrix}_t + \left( V(\phi ,k)\begin{pmatrix} \phi \\ k\phi \end{pmatrix}\right)_x = \begin{pmatrix} 0 \\ 0\end{pmatrix}\quad (x,t) \in \mathbb{R}\times (0,T),\quad T > 0,\tag{1} \] \[ \phi (x,0) = \phi _0 (x),\quad k(x,0) = k_0 (x),\quad x \in \mathbb{R},\tag{2} \] where the unknowns are \(\phi = \phi (x,t)\) and \(k = k(x,t)\). The first main novelty of the present work is a so-called anti-diffusive numerical scheme, which relies on averaging (Av) and random sampling (RS) and is referred to as the Av-RS scheme. The second main novelty is an explicit treatment of the delicate vacuum case. The paper is divided into sections as follows. The first section is an introduction, where the investigated Problem (1), (2) is stated and applications to sedimentation and to traffic flow is given. Related works and an outline of this work are also described. The basic attributes such as eigenvalues, Riemann invariants, Hugoniot curves of System (1), and some central notations are given in Section 2. In Section 3, solutions to the Riemann problem associated with (1) are stated and analyzed. Section 4 is devoted to conservative methods. The intention is to demonstrate that a large family of such schemes, including Godunov's method, produces inadequate approximations near contact discontinuities, and thereby to motivate why the authors turn to alternative methods. In Section 4, a family of two step averaging-remap schemes having a common averaging technique are considered. The Av-RS scheme is a member of this family. Another member, whose remap relies on a convex combination (CC) of Riemann invariants is also defined. The resulting scheme is called the Av-CC scheme. In Section 5, convergence of the entire family of averaging-remap schemes is proved, and then it is shown that the limit function obtained with the Av-RS scheme is in fact a weak solution to (1), (2). The performances of the Godunov, Av-RS and Av-CC schemes are numerically investigated on five sample problems in Section 6. In Section 7, the authors return to the vacuum case and argue that a new concept of the unstable Riemann solution, called ``type B'' Riemann solution, provides more realistic solutions, especially in the context of the ARZ model in certain situations involving vacuum ahead. Finally, some concluding remarks are given in Section 8.
scientific article; zbMATH DE number 6830164
scientific article