A random sampling method for a family of temple-class systems of conservation laws
type: Item
instance of: scholarly article
title: A random sampling method for a family of temple-class systems of conservation laws
zbMATH Open document ID: 1405.65139
author: F. Betancourt
author: Raimund Bürger
author: Ch. Chalons
author: Stefan Diehl
author: Sebastian Farås
published in: Numerische Mathematik
publication date: 2018-01-26T00:00:00Z
review text: 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 Rie…
Mathematics Subject Classification ID: 65M99
Mathematics Subject Classification ID: 90B20
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M75
zbMATH DE Number: 6830164
zbMATH Keywords: conservation law
zbMATH Keywords: random sampling method
zbMATH Keywords: temple-class system
describes a project that uses: HLLE
reviewed by: Temur Jangveladze
MaRDI profile type: Publication
full work available at URL: s00211-017-0900-z
OpenAlex ID: W2734329673
cites work: Resurrection of "Second Order" Models of Traffic Flow
cites work: Random sampling remap for compressible two-phase flows
cites work: A note on the convergence of Godunov type methods for shock reflection problems
cites work: An instability of the Godunov scheme
cites work: Antidiffusive and Random-Sampling Lagrangian-Remap Schemes for the Multiclass Lighthill--Whitham--Richards Traffic Model
cites work: Numerical Approximation of a Macroscopic Model of Pedestrian Flows
cites work: Computing material fronts with a Lagrange-Projection approach
cites work: Transport equilibrium schemes for computing contact discontinuities in traffic flow modeling
cites work: Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling
cites work: General constrained conservation laws. Application to pedestrian flow modeling
cites work: Glimm’s Method for Gas Dynamics
cites work: A \(2\times 2\) hyperbolic traffic flow model
cites work: Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model
cites work: The Aw-Rascle traffic model with locally constrained flow
cites work: Solutions in the large for nonlinear hyperbolic systems of equations
cites work: The Aw-Rascle vehicular traffic flow model with phase transitions
cites work: EXISTENCE OF SOLUTIONS FOR THE AW–RASCLE TRAFFIC FLOW MODEL WITH VACUUM
cites work: On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws
cites work: Front tracking for hyperbolic conservation laws
cites work: Analysis of a singular hyperbolic system of conservation laws
cites work: A system of non-strictly hyperbolic conservation laws arising in elasticity theory
cites work: Stability of a resonant system of conservation laws modeling polymer flow with gravitation
cites work: An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model
cites work: Q3538687
cites work: New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations
cites work: Q4889859
cites work: Stability of Godunov's Method for a Class of 2 × 2 Systems of Conservation Laws
cites work: Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow
cites work: A Comparison of Convergence Rates for Godunov’s Method and Glimm’s Method in Resonant Nonlinear Systems of Conservation Laws
cites work: The entropy condition and the admissibility of shocks
cites work: Suppression of Oscillations in Godunov’s Method for a Resonant Non-Strictly Hyperbolic System
cites work: Existence of global bounded weak solutions to a Keyfitz-Kranzer system
cites work: A Hybrid Lagrangian Model Based on the Aw–Rascle Traffic Flow Model
cites work: Q3274322
cites work: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation
cites work: Discontinuous Galerkin finite element scheme for a conserved higher-order traffic flow model by exploring Riemann solvers
cites work: Q4746129
cites work: Global solution of the Cauchy problem for a class of 2x2 nonstrictly hyperbolic conservation laws
cites work: Systems of Conservation Laws with Invariant Submanifolds
cites work: Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions
DOI: 10.1007/S00211-017-0900-Z
description: scientific article; zbMATH DE number 6830164