## A

### BibTeX

@MISC{A_a,

author = {V. Dolejˇsí A and T. Gallouët B},

title = {A},

year = {}

}

### OpenURL

### Abstract

numerical study of a particular non-conservative hyperbolic problem ⋆

### Citations

521 |
Approximate riemann solvers, parameter vectors and difference schemes
- Roe
- 1981
(Show Context)
Citation Context ...d not well defined, in our case...), schemes based on approximate Riemann solvers were introduced. For a survey see, e.g., [8], [9], [10], [11], [12]. The linearized approximate Riemann solver of Roe =-=[13]-=- was proposed in 1981 for the numerical solution of hyperbolic conservation laws. A weak formulation of Roe’s approximate Riemann solver, based on the choice of a path in the states space, has been in... |

426 |
Numerical Methods for Conservation Laws
- LeVeque
- 1990
(Show Context)
Citation Context ...l Riemann problem is difficult and expensive for complicated systems (and not well defined, in our case...), schemes based on approximate Riemann solvers were introduced. For a survey see, e.g., [8], =-=[9]-=-, [10], [11], [12]. The linearized approximate Riemann solver of Roe [13] was proposed in 1981 for the numerical solution of hyperbolic conservation laws. A weak formulation of Roe’s approximate Riema... |

330 |
Finite Volume Methods for Hyperbolic Problems
- LeVeque
- 2002
(Show Context)
Citation Context ...mann problem is difficult and expensive for complicated systems (and not well defined, in our case...), schemes based on approximate Riemann solvers were introduced. For a survey see, e.g., [8], [9], =-=[10]-=-, [11], [12]. The linearized approximate Riemann solver of Roe [13] was proposed in 1981 for the numerical solution of hyperbolic conservation laws. A weak formulation of Roe’s approximate Riemann sol... |

237 |
Riemann Solvers and Numerical Methods for Fluid Dynamics
- Toro
- 1997
(Show Context)
Citation Context ...roblem is difficult and expensive for complicated systems (and not well defined, in our case...), schemes based on approximate Riemann solvers were introduced. For a survey see, e.g., [8], [9], [10], =-=[11]-=-, [12]. The linearized approximate Riemann solver of Roe [13] was proposed in 1981 for the numerical solution of hyperbolic conservation laws. A weak formulation of Roe’s approximate Riemann solver, b... |

191 |
Numerical Approximation of Hyperbolic Systems of Conservation
- Godlewski, Raviart
- 1996
(Show Context)
Citation Context ... local Riemann problem is difficult and expensive for complicated systems (and not well defined, in our case...), schemes based on approximate Riemann solvers were introduced. For a survey see, e.g., =-=[8]-=-, [9], [10], [11], [12]. The linearized approximate Riemann solver of Roe [13] was proposed in 1981 for the numerical solution of hyperbolic conservation laws. A weak formulation of Roe’s approximate ... |

74 |
Definition and weak stability of nonconservative products
- Maso, LeFloch, et al.
- 1995
(Show Context)
Citation Context ...) ∂w ∂x (11) lacks meaning as the distribution and the usual theory of hyperbolic systems of conservation laws does not apply. There are several possibilities how to give a unique meaning to (11). In =-=[23]-=-, an integration along a suitable path is used 7for the definition of (11). The approach presented in [24] relies on the form of matrix C: some equations of the system ∂w ∂t + C(w)∂w ∂x = 0, (12) are... |

66 |
A well balanced scheme for the numerical processing of source terms in hyperbolic equations
- Greenberg, Roux
- 1996
(Show Context)
Citation Context ...e terms standard methods can fail in approximating steady or nearly steady flows, see [10]. In order to ensure the approximation of steady flows, the so-called well-balanced schemes were presented in =-=[25]-=-. This approach has been extended in many papers, see, e.g., [20], [26], [27] and the references therein. Since we observed some troubles for approximating a steady state solution, we follow the appro... |

39 | Calculation of interaction of non-steady shock waves with obstacles - Rusanov - 1961 |

37 |
On Godunov-type methods near low densities
- Einfeldt, Munz, et al.
- 1991
(Show Context)
Citation Context ...ce (better than the used one) of the variables for the linearization process could be done. The nonphysical solutions of the linearized Riemann problem appear also for conservative systems, see, e.g, =-=[31]-=- where the vacuum scenario of the Euler equations is investigated. We suppose that the situation from [31] differs from those ones presented in Table 1 since the velocity directions are completely dif... |

31 |
A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms
- GOSSE
(Show Context)
Citation Context ...ady flows, see [10]. In order to ensure the approximation of steady flows, the so-called well-balanced schemes were presented in [25]. This approach has been extended in many papers, see, e.g., [20], =-=[26]-=-, [27] and the references therein. Since we observed some troubles for approximating a steady state solution, we follow the approach from [25] and add to the system of equation (7) one “partial” diffe... |

22 |
Coq. On the numerical solution to two-fluid models via a cell centered finite volume method
- Ghidaglia, Kumbaro, et al.
(Show Context)
Citation Context ...considered model (defined in Section 2) gives no chance to apply this approach since we are not able to evaluate an integration along a suitable path in the states space. Another approach was used in =-=[16]-=- for the solution of a two-fluids model, for comparison see [17]. Also for this method, the complexity of the considered model does not enable to perform some matrix operations analytically and their ... |

17 |
A sequel to a rough Godunov scheme: application to real gases, Comput. Fluids 29
- Buffard, Gallouët, et al.
- 2000
(Show Context)
Citation Context ...f a twophase flow. Indeed, we use a variant of this scheme, called VFRoe-ncv (“volumes finis” Roe with non-conservative variables) scheme, which was applied for the solution of the Euler equations in =-=[19]-=-, for the solution of shallow-water equations with topography in [20] and for two-phase flow in [21]. VFRoe-ncv scheme is based on the solution of the linearized Riemann problem at each mesh interface... |

16 | A rough finite volume scheme for modeling two-phase flow in a pipeline - Faille, Heintze - 1999 |

15 |
On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems, ESAIM-Math
- Parés, Castro
- 2004
(Show Context)
Citation Context ...ows, see [10]. In order to ensure the approximation of steady flows, the so-called well-balanced schemes were presented in [25]. This approach has been extended in many papers, see, e.g., [20], [26], =-=[27]-=- and the references therein. Since we observed some troubles for approximating a steady state solution, we follow the approach from [25] and add to the system of equation (7) one “partial” differentia... |

14 |
Nejad, Towards very high order Godunov schemes, GodunovMethods
- Toro, Millington, et al.
- 2001
(Show Context)
Citation Context ... is difficult and expensive for complicated systems (and not well defined, in our case...), schemes based on approximate Riemann solvers were introduced. For a survey see, e.g., [8], [9], [10], [11], =-=[12]-=-. The linearized approximate Riemann solver of Roe [13] was proposed in 1981 for the numerical solution of hyperbolic conservation laws. A weak formulation of Roe’s approximate Riemann solver, based o... |

12 | Numerical modeling of two-phase flows using the two-fluid two-pressure approach
- Gallouët, Hérard, et al.
(Show Context)
Citation Context ...th non-conservative variables) scheme, which was applied for the solution of the Euler equations in [19], for the solution of shallow-water equations with topography in [20] and for two-phase flow in =-=[21]-=-. VFRoe-ncv scheme is based on the solution of the linearized Riemann problem at each mesh interface in non-conservative variables. A similar approach was applied in [22] for modelling of two-phase fl... |

9 |
A weak formulation of Roe’s approximate Riemann solver
- Toumi
- 1992
(Show Context)
Citation Context ...d in 1981 for the numerical solution of hyperbolic conservation laws. A weak formulation of Roe’s approximate Riemann solver, based on the choice of a path in the states space, has been introduced in =-=[14]-=-. This weak formulation was applied in order to build a Roe-averaged matrix for a conservative system governing a homogeneous equilibrium two-phase flow. A generalization of this formulation to a hype... |

9 | On an approximate Godunov scheme
- Masella, Faille, et al.
- 1999
(Show Context)
Citation Context ...erations analytically and their numerical realization leads to unsatisfactory results. Therefore, we deal with an alternative to the Roe scheme, the “Finite volume Roe scheme” which was introduced in =-=[18]-=- to approximate solution of a twophase flow. Indeed, we use a variant of this scheme, called VFRoe-ncv (“volumes finis” Roe with non-conservative variables) scheme, which was applied for the solution ... |

8 |
An Approximate Linearized Riemann Solver for a TwoFluid Model
- Toumi, Kumbaro
- 1996
(Show Context)
Citation Context ...ive system governing a homogeneous equilibrium two-phase flow. A generalization of this formulation to a hyperbolic non-conservative system modelling a two-component two-phase flow was carried out in =-=[15]-=-. However, the complexity of the considered model (defined in Section 2) gives no chance to apply this approach since we are not able to evaluate an integration along a suitable path in the states spa... |

5 |
Nonconservative products in bounded variation functions
- Colombeau, Heibig
- 1992
(Show Context)
Citation Context ...aws does not apply. There are several possibilities how to give a unique meaning to (11). In [23], an integration along a suitable path is used 7for the definition of (11). The approach presented in =-=[24]-=- relies on the form of matrix C: some equations of the system ∂w ∂t + C(w)∂w ∂x = 0, (12) are considered in a strong sense and the other in a weak sense. In [6] and [7], the author adds to (12) a diff... |

4 |
Traveling-wave solutions of convection-diffusion systems in nonconservation form
- Sainsaulieu
- 1996
(Show Context)
Citation Context ...T. Gallouët). 2In our non-conservative problem, the jump conditions are not prescribed. These jump conditions could be given, for instance, with the form of a neglected viscosity term (as in [6] and =-=[7]-=-). But this viscosity term seems to be not available. Since the model is interesting from a practical point of view, we deal with its numerical solution although the relevant mathematical theory does ... |

4 |
The split coefficient matrix method for hyperblic systems of gas dynamics equations
- Chakravarthy, Anderson, et al.
- 1980
(Show Context)
Citation Context ... ( W 2 k i k,∗,P − W i+ 1 ) − B 2 k i− 1 ( 2 W k,∗,P i− 1 2 − W k )] i ,(47) where k,∗,P represents either k,∗,L or k,∗,R or k,∗,LR . The scheme (47) is the scheme of Chakravarthy et al. presented in =-=[29]-=-. Although the schemes (46) and (47) are analytically identical the numerical examples in Section 4 show that their implementations produce rather different numerical solutions. We have introduced six... |

3 |
Traveling waves solution of convection-diffusion systems whose convection terms are weakly nonconservative: application to the modeling of two-phase fluid flows
- Sainsaulieu
- 1995
(Show Context)
Citation Context ...mrs.fr (T. Gallouët). 2In our non-conservative problem, the jump conditions are not prescribed. These jump conditions could be given, for instance, with the form of a neglected viscosity term (as in =-=[6]-=- and [7]). But this viscosity term seems to be not available. Since the model is interesting from a practical point of view, we deal with its numerical solution although the relevant mathematical theo... |

3 |
On the use of symmetrizing variables for vacuums, Calcolo 40
- Gallouët, Hérard, et al.
- 2003
(Show Context)
Citation Context ... satisfactory the tasks C1 – C3. The fact that the solution of the linearized Riemann problem can be unphysical may depend on the choice of the variable for the linearization process in VFRoencv, see =-=[30]-=-. But for the considered model, as we mentioned in Section 3.2.2, we do not know what natural choice (better than the used one) of the variables for the linearization process could be done. The nonphy... |

2 |
Two flux schemes for computing two phases flows throught multidimensional finite volume methods
- GHIDAGLIA, COQ, et al.
- 1999
(Show Context)
Citation Context ... this approach since we are not able to evaluate an integration along a suitable path in the states space. Another approach was used in [16] for the solution of a two-fluids model, for comparison see =-=[17]-=-. Also for this method, the complexity of the considered model does not enable to perform some matrix operations analytically and their numerical realization leads to unsatisfactory results. Therefore... |

2 |
Hyperbolic equations and systems with discontinuous coefficients or source terms, These proceedings
- Gallouët
- 2006
(Show Context)
Citation Context ...a of the VFRoe-ncv scheme is the following. We consider a transformation of variables U → Y = Ψ(U), where Ψ is a smooth function. Although the function Ψ does not need to be necessary invertible (see =-=[28]-=-), for simplicity we assume that that there exists the inverse mapping Ψ −1 . Then system (25) reads in the non-conservative form ∂Y ∂t + ¯ B(Y ) ∂Y ∂x = 0, where ¯ B ≡ ( ) DΨ Df DU DU ( ) −1 DΨ . (28... |

1 |
Unsteady slurry pipe flow with virtual mass effect, institute of Water Problems
- Bournaski, Ivanov
(Show Context)
Citation Context ...gas slurry, which can be used, e.g., for the transport of a mixture of sand, oil and air in a pipe. The complete model is based on the so-called virtual mass effect. It can be found, for instance, in =-=[1]-=-, [2]. This model belongs to the class of non-conservative hyperbolic problems with source terms, which can be written (for one space dimension) in the form ∂w ∂t + C(w)∂w ∂x = S(w), x ∈ (0, L), t ∈ (... |

1 |
Instationäre Strömung von Wasser-Feststoff-Gemischen
- Vogel, Bechteler, et al.
- 1983
(Show Context)
Citation Context ...lurry, which can be used, e.g., for the transport of a mixture of sand, oil and air in a pipe. The complete model is based on the so-called virtual mass effect. It can be found, for instance, in [1], =-=[2]-=-. This model belongs to the class of non-conservative hyperbolic problems with source terms, which can be written (for one space dimension) in the form ∂w ∂t + C(w)∂w ∂x = S(w), x ∈ (0, L), t ∈ (0, T)... |

1 |
On a numerical simulation of three-phase flow with a virtual mass effect, Engineering Mechanics 12 (6
- Dolejˇsí, Gallouët
- 2005
(Show Context)
Citation Context ...L) × (0, T) → IR N2 is a N ×N matrix and S(w) : (0, L)×(0, T) → IR N is a vector representing source terms. Firstly we solved this problem with the aid of the centred Rusanov scheme [3] in the papers =-=[4]-=- and [5]. Although systems of type (1) describe many physical processes (e.g., multiphase flow, shallow-water problems) our model is rather particular. For flow regimes of interest, various numerical ... |