Structural Stability of Shock Solutions of Hyperbolic Systems in Nonconservation Form via Kinetic Relations

Audebert, Bruno; Coquel, Fré́dé́ric

doi:10.1007/978-3-540-75712-2_36

Cited by 2 publications

(11 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One can insert this function into the general expression of f 2 . One has now : f 2 (h, P) = φ 1 h, P 11 P 22 − P 2 12 + n 1 h (P 11 + P 22 ) .…”

Section: Appendix a : Conservation Lawsmentioning

confidence: 99%

“…For mathematical reasons, the choice of P is more convenient than that of R. For simplicity, both R and P will be further refereed to as the Reynolds stress tensor. 1 Equations (1) admit the energy conservation law: ∂ ∂t h 1 2 |u| 2 + e i + e T + div hu 1 2 |u| 2 + e i + e T + gh 2 2 I + hP u = 0,…”

Section: Introductionmentioning

confidence: 99%

“…Since the number of scalar unknowns is six (h, u and three independent components of P), the system is not in conservative form. The definition and computation of discontinuous solutions for non-conservative hyperbolic equations is a challenging problem (see examples of non-conservative systems in compressible turbulence [7,2], multi-layer shallow water flows [34,4,32,5,1,30], multi-phase fluid flows [3,24,40,41,42,19,14], solid-fluid systems [17,33]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multi-dimensional shear shallow water flows: Problems and solutions

Gavrilyuk

Ivanova

Favrie

2018

Journal of Computational Physics

107

View full text Add to dashboard Cite

The mathematical model of shear shallow water flows of uniform density is studied. This is a 2D hyperbolic non-conservative system of equations which is reminiscent of a generic Reynoldsaveraged model of barotropic turbulent flows. The model has three families of characteristics corresponding to the propagation of surface waves, shear waves and average flow (contact characteristics). The system is non-conservative : for six unknowns (the fluid depth, two components of the depth averaged horizontal velocity, and three independent components of the symmetric Reynolds stress tensor) one has only five conservation laws (conservation of mass, momentum, energy and mathematical 'entropy'). A splitting procedure for solving such a system is proposed allowing us to define a weak solution. Each split subsystem contains only one family of waves (either surface or shear waves) and contact characteristics. The accuracy of such an approach is tested on exact 2D solutions describing the flows where the velocity is linear with respect to the space variables, and 1D solutions. The capacity of the model to describe the full transition observed in the formation of roll waves : from uniform flow to one-dimensional roll waves, and, finally, to 2D transverse 'fingering' of roll wave profiles is shown.

show abstract

“…One can insert this function into the general expression of f 2 . One has now : f 2 (h, P) = φ 1 h, P 11 P 22 − P 2 12 + n 1 h (P 11 + P 22 ) .…”

Section: Appendix a : Conservation Lawsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multi-dimensional shear shallow water flows: Problems and solutions

Gavrilyuk

Ivanova

Favrie

2018

Journal of Computational Physics

107

View full text Add to dashboard Cite

show abstract

“…Approximate relations (10) can also be obtained from transport equations written in pseudoconservative variables R + u ⊗ u instead of R introduced in Section 1.2, as explained in [22, p. 18 and Appendix 4].…”

Section: Shock Wavesmentioning

confidence: 99%

“…[3][4][5][6]). When focusing on turbulent compressible flows, a stable numerical strategy has been proposed, which is grounded on the hyperbolic structure of the full-convective system embedded in the model (see [7][8][9][10]). More recently, this methodology has been extended to shallow-water models [6,[11][12][13].…”

Section: Introductionmentioning

confidence: 99%