Derivative Riemann solvers for systems of conservation laws and ADER methods

We analyze a front tracking algorithm for 2×2 systems of conservation laws with non-genuinely nonlinear characteristic fields. The convergence of the corresponding approximate Riemann solvers is established and the basic interaction estimates for the front tracking approximate solutions are provided.

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A REDUCED STABILITY CONDITION FOR NONLINEAR RELAXATION TO CONSERVATION LAWS

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891604000020 ◽

2004 ◽

Vol 01 (01) ◽

pp. 149-170 ◽

Cited By ~ 20

Author(s):

FRANÇOIS BOUCHUT

Keyword(s):

Conservation Laws ◽

Stability Condition ◽

Hyperbolic Systems ◽

Strong Stability ◽

Riemann Solvers ◽

Systems Of Conservation Laws ◽

Leading Term ◽

Relaxation Systems ◽

The One ◽

Approximate Riemann Solvers

We consider multidimensional hyperbolic systems of conservation laws with relaxation, together with their associated limit systems. A strong stability condition for such asymptotics has been introduced by Chen, Levermore and Liu, namely the existence of an entropy extension. We propose here a new stability condition, the reduced stability condition, which is weaker than the previous one, but still has the property to imply the subcharacteristic or interlacing conditions, and the dissipativity of the leading term in the Chapman–Enskog expansion. This reduced stability condition has the advantage of involving only the submanifold of equilibria, or maxwellians, so that it is much easier to check than the entropy extension condition. Our condition generalizes the one introduced by the author in the case of kinetic, i.e. diagonal semilinear relaxation. We provide an adapted stability analysis in the context of approximate Riemann solvers obtained via relaxation systems.

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HYBRID RIEMANN SOLVERS FOR LARGE SYSTEMS OF CONSERVATION LAWS

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016) ◽

10.7712/100016.2395.11923 ◽

2016 ◽

Author(s):

Birte Schmidtmann ◽

Mariia Astrakhantceva ◽

Manuel Torrilhon

Keyword(s):

Conservation Laws ◽

Riemann Solvers ◽

Systems Of Conservation Laws ◽

Large Systems

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Global Riemann solvers for several 3 × 3 systems of conservation laws with degeneracies

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202518500446 ◽

2018 ◽

Vol 28 (08) ◽

pp. 1599-1626 ◽

Cited By ~ 3

Author(s):

Wen Shen

Keyword(s):

Conservation Laws ◽

Cauchy Problems ◽

Eigenvalues And Eigenvectors ◽

Two Phase ◽

Riemann Solvers ◽

Special Cases ◽

Road Condition ◽

Systems Of Conservation Laws ◽

Lagrangian Coordinate ◽

Linearly Degenerate

We study several [Formula: see text] systems of conservation laws, arising in the modeling of two-phase flow with rough porous media and traffic flow with rough road condition. These systems share several features. The systems are of mixed type, with various degeneracies. Some families are linearly degenerate, while others are not genuinely nonlinear. Furthermore, along certain curves in the domain, the eigenvalues and eigenvectors of different families coincide. Most interestingly, in some suitable Lagrangian coordinate, the systems are partially decoupled, where some unknowns can be solved independently of the others. Finally, in special cases, the systems reduce to some [Formula: see text] models, which have been studied in the literature. Utilizing the insights gained from these features, we construct global Riemann solvers for all these models. Possible treatments on the Cauchy problems are also discussed.

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Systems of conservation laws with discontinuous fluxes and applications to traffic

Annales Universitatis Mariae Curie-Sklodowska sectio A – Mathematica ◽

10.17951/a.2019.73.2.135-173 ◽

2020 ◽

Vol 73 (2) ◽

pp. 135

Author(s):

Massimiliano Rosini

Keyword(s):

Conservation Laws ◽

Common Property ◽

Traffic Modeling ◽

Vehicular Traffic ◽

Riemann Solvers ◽

Density Flow ◽

Systems Of Conservation Laws

In this paper we study \(2\times 2\) systems of conservation laws with discontinuous fluxes arising in vehicular traffic modeling. The main goal is to introduce an appropriate notion of solution. To this aim we consider physically reasonable microscopic follow-the-leader models. Macroscopic Riemann solvers are then obtained as many particle limits. This approach leads us to develop six models. We propose a unified way to describe such models, which highlights their common property of maximizing the density flow across the interface under appropriate physical restrictions depending on the case at hand.

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