A weak formulation of roe's approximate riemann solver

Toumi, I.

doi:10.1016/0021-9991(92)90378-c

Cited by 185 publications

(159 citation statements)

References 7 publications

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“…The family of generalized Roe schemes introduced in [28] constitutes a particular case of path-conservative numerical methods. These schemes are based on the following general concept of a Roe linearization for (2.1): given a family of paths Φ, a function A Φ : Ω × Ω → M N ×N (R) is called a Roe linearization if it verifies the following properties:…”

Section: Preliminariesmentioning

confidence: 99%

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On some fast well-balanced first order solvers for nonconservative systems

Castro¹,

Pardo²,

Parés³

et al. 2009

Math. Comp.

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Abstract. The goal of this article is to design robust and simple first order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good wellbalanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods.

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Section: Preliminariesmentioning

confidence: 99%

“…The family of generalized Roe schemes introduced in [28] constitutes a particular case of path-conservative numerical methods. Although the schemes of this family are robust and have good well-balanced properties (see, for instance, [2], [8], [21], [20]) they also present, as their conservative counterpart, some drawbacks:…”

Section: Introductionmentioning

confidence: 99%

On some fast well-balanced first order solvers for nonconservative systems

Castro¹,

Pardo²,

Parés³

et al. 2009

Math. Comp.

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“…See for example Chalmers and Lorin [23] for a discussion on choosing appropriate integration paths. Several conservative numerical schemes and approximate Riemann solvers have been generalised to non-conservative systems based on the theory by Dal Maso et al [22] : Lax-Friedrichs and Lax-Wendroff [24] , Roe's approximate Riemann solver [25] , HLL [26] and the Osher Riemann solver [27] . Parés [28] introduced the concept of path-conservative numerical schemes, as a generalisation of conservative schemes.…”

Section: Spatial Fluxmentioning

confidence: 99%

“…The simpler Lax-Friedrichs method is in our experience not stable enough for the PDEs considered in this article. We settled for a linearised Riemann solver based on Roe's approach [25] , which requires a single numerical evaluation of the eigenvalues and eigenvectors per spatial boundary point, but we replace Roe's matrix with F total s (q av ) , where q av is the average value of the inner and outer trace,…”

Section: Spatial Fluxmentioning

confidence: 99%

Efficient simulation of one-dimensional two-phase flow with a high-order h-adaptive space-time Discontinuous Galerkin method

et al. 2017

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a b s t r a c tOne-dimensional models for multiphase flow in pipelines are commonly discretised using first-order Finite Volume (FV) schemes, often combined with implicit time-integration methods. While robust, these methods introduce much numerical diffusion depending on the number of grid points. In this paper we propose a high-order, space-time Discontinuous Galerkin (DG) Finite Element method with h -adaptivity to improve the efficiency of one-dimensional multiphase flow simulations. For smooth initial boundary value problems we show that the DG method converges with the theoretical rate and that the growth rate and phase shift of small, harmonic perturbations exhibit superconvergence. We employ two techniques to accurately and efficiently represent discontinuities. Firstly artificial diffusion in the neighbourhood of a discontinuity suppresses spurious oscillations. Secondly local mesh refinement allows for a sharper representation of the discontinuity while keeping the amount of work required to obtain a solution relatively low. The proposed DG method is shown to be superior to FV.

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“…We refer the reader to [11,37] for a rigorous definition of Roe matrix and to [39][40][41] for a generalized definition of Roe linearization based on the use of a family of paths.…”

Section: Roe and Roe-type Riemann Solversmentioning

confidence: 99%

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

Pelanti

Bouchut

Mangeney

2011

Journal of Computational Physics

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a b s t r a c tWe present a Riemann solver derived by a relaxation technique for classical single-phase shallow flow equations and for a two-phase shallow flow model describing a mixture of solid granular material and fluid. Our primary interest is the numerical approximation of this two-phase solid/fluid model, whose complexity poses numerical difficulties that cannot be efficiently addressed by existing solvers. In particular, we are concerned with ensuring a robust treatment of dry bed states. The relaxation system used by the proposed solver is formulated by introducing auxiliary variables that replace the momenta in the spatial gradients of the original model systems. The resulting relaxation solver is related to Roe solver in that its Riemann solution for the flow height and relaxation variables is formally computed as Roe's Riemann solution. The relaxation solver has the advantage of a certain degree of freedom in the specification of the wave structure through the choice of the relaxation parameters. This flexibility can be exploited to handle robustly vacuum states, which is a well known difficulty of standard Roe's method, while maintaining Roe's low diffusivity. For the single-phase model positivity of flow height is rigorously preserved. For the two-phase model positivity of volume fractions in general is not ensured, and a suitable restriction on the CFL number might be needed. Nonetheless, numerical experiments suggest that the proposed two-phase flow solver efficiently models wet/dry fronts and vacuum formation for a large range of flow conditions.As a corollary of our study, we show that for single-phase shallow flow equations the relaxation solver is formally equivalent to the VFRoe solver with conservative variables of Gallouët and Masella [T. Gallouët, J.-M. Masella, Un schéma de Godunov approché C.R. Acad. Sci. Paris, Série I, 323 (1996) 77-84]. The relaxation interpretation allows establishing positivity conditions for this VFRoe method.

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A weak formulation of roe's approximate riemann solver

Cited by 185 publications

References 7 publications

On some fast well-balanced first order solvers for nonconservative systems

On some fast well-balanced first order solvers for nonconservative systems

Efficient simulation of one-dimensional two-phase flow with a high-order h-adaptive space-time Discontinuous Galerkin method

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

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