A positive and entropy-satisfying finite volume scheme for the Baer-Nunziato model

doi:10.1016/J.JCP.2016.11.017

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A positive and entropy-satisfying finite volume scheme for the Baer-Nunziato model

Journal Article•DOI•

A positive and entropy-satisfying finite volume scheme for the Baer-Nunziato model

Frédéric Coquel¹, Jean-Marc Hérard, Khaled Saleh²•Institutions (2)

École Polytechnique¹, University of Lyon²

01 Feb 2017-Journal of Computational Physics (Academic Press)-Vol. 330, pp 401-435

TL;DR: This is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition.

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About: This article is published in Journal of Computational Physics.The article was published on 2017-02-01 and is currently open access. It has received 35 citations till now. The article focuses on the topics: Godunov's scheme & Relaxation (approximation).

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Journal Article•DOI•

HLLC-type Riemann solver with approximated two-phase contact for the computation of the Baer-Nunziato two-fluid model

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Hippolyte Lochon¹, Hippolyte Lochon², Frédéric Daude¹, Pascal Galon¹, Jean-Marc Hérard² - Show less +1 more•Institutions (2)

Université Paris-Saclay¹, Centre national de la recherche scientifique²

01 Dec 2016-Journal of Computational Physics

TL;DR: A new HLLC-type Riemann solver is built based on the solver proposed by Tokareva & Toro, and the key idea lies in an approximation of the two-phase contact discontinuity of the Baer–Nunziato model, which allows to bypass the resolution of a non-linear equation based on those Riem Mann invariants.

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27 citations

Cites methods from "A positive and entropy-satisfying f..."

...Finally, relaxation approaches have been applied to both the isentropic and the complete Baer-Nunziato model [2, 9, 10]....
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Journal Article•DOI•

Multi-dimensional shear shallow water flows: Problems and solutions

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Sergey Gavrilyuk¹, Sergey Gavrilyuk², Kseniya Ivanova², Nicolas Favrie²•Institutions (2)

Novosibirsk State University¹, Aix-Marseille University²

01 Aug 2018-Journal of Computational Physics

TL;DR: The capacity of the model to describe the full transition scenario as commonly seen in the formation of roll waves: from uniform flow to 1D roll waves, and, finally, to 2D transverse ‘fingering’ of the wave profiles, is shown.

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21 citations

Cites background or methods from "A positive and entropy-satisfying f..."

...The third one is based on the relaxation technique where the studied system is approximated by a new hyperbolic system with all linearly degenerate in the sense of Lax eigenfields [9, 14]....
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...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])....
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Journal Article•DOI•

A fast, robust, and simple Lagrangian–Eulerian solver for balance laws and applications

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Eduardo Abreu¹, John E. Pérez²•Institutions (2)

State University of Campinas¹, ITM University, Gurgaon, Haryana²

01 May 2019-Computers & Mathematics With Applications

TL;DR: An improvement of the Lagrangian–Eulerian space–time tracking forward scheme to deal with balance laws and related applications, and a new strategy of discretization of conservation laws, starting from the scalar case in one space dimension, extending it to systems and to the multi-dimensional setting.

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Abstract: In this work, we present an improvement of the Lagrangian–Eulerian space–time tracking forward scheme to deal with balance laws and related applications. This extended algorithm is shown in the most simple setting and it is a result of our previous works. We describe and explain a new strategy of discretization of conservation laws, starting from the scalar case in one space dimension, extending it to systems and to the multi-dimensional setting. The computations are fast, accurate and stable with good resolution. This algorithm is very easy to implement in a computer to address the delicate well-balancing between the first-order hyperbolic flux and the source term. We do not use approximate or exact Riemann solvers, nonlinear reconstructions, or upwind source term discretizations. The scheme is written into the classical theory of monotone schemes, which produces a scheme that converges to entropy solutions linked to the purely hyperbolic counterpart. This method can produce well-balanced approximations of solutions for nonlinear balance laws. Numerical experiments also demonstrate the robustness of the forward tracking to solve related problems involving systems and two-dimensional models.

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19 citations

Journal Article•DOI•

A flux splitting method for the Baer-Nunziato equations of compressible two-phase flow

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Svetlana Tokareva¹, Eleuterio F. Toro²•Institutions (2)

University of Zurich¹, University of Trento²

15 Oct 2016-Journal of Computational Physics

TL;DR: The Toro-Vazquez flux vector splitting approach is extended to the Baer-Nunziato equations of compressible two-phase flow, and the construction of a second-order scheme is presented.

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17 citations

Journal Article•DOI•

An HLLC-type Riemann solver and high-resolution Godunov method for a two-phase model of reactive flow with general equations of state

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M. Hennessey¹, Ashwana K. Kapila¹, Donald W. Schwendeman¹•Institutions (1)

Rensselaer Polytechnic Institute¹

15 Mar 2020-Journal of Computational Physics

TL;DR: An assessment of the approximate Riemann solvers in terms of the accuracy and efficiency of numerical solutions of the two-phase model of reactive flow considers the state-sensitive behavior in different regimes of the solutions in a reverse-impact configuration.

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14 citations

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Journal Article•DOI•

On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws

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Amiram Harten¹, Peter D. Lax², Bram van Leer•Institutions (2)

Tel Aviv University¹, New York University²

01 Jan 1983-Siam Review

TL;DR: This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes.

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Abstract: This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes. Special attention is given to the Godunov-type schemes that result from using an approximate solution of the Riemann problem. For schemes based on flux splitting, the approximate Riemann solution can be interpreted as a solution of the collisionless Boltzmann equation.

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3,133 citations

"A positive and entropy-satisfying f..." refers methods in this paper

...3 The relaxation scheme for the auxiliary model In this section, the exact Riemann solver Wr(ξ;WL,WR) for the relaxation system (25) is used to derive an approximate Riemann solver of Harten, Lax and van Leer [28] for the simulation of the auxiliary system (13)....
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...In this section, the exact Riemann solver Wr(ξ;WL,WR) for the relaxation system (25) is used to derive an approximate Riemann solver of Harten, Lax and van Leer [28] for the simulation of the auxiliary system (13)....
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Book•

Numerical Approximation of Hyperbolic Systems of Conservation Laws

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Edwige Godlewski, Pierre-Arnaud Raviart

27 Sep 1996

TL;DR: In this paper, the authors define and define nonlinear hyperbolic systems in one space dimension and define finite difference schemes for one-dimensional systems in the case of multidimensional systems.

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Abstract: From the contents: Introduction: Definitions and Examples.- Nonlinear hyperbolic systems in one space dimension.- Gas dynamics and reaction flows.- Finite Difference Schemes for one-dimensional systems.- The case of multidimensional systems.- An Introduction to Boundary conditions.

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1,386 citations

Journal Article•DOI•

A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials

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Melvin R. Baer¹, J.W. Nunziato¹•Institutions (1)

Sandia National Laboratories¹

01 Nov 1986-International Journal of Multiphase Flow

TL;DR: In this article, a two-phase mixture theory is presented which describes the deflagration-to-detonation transition (DDT) in reactive granular materials, based on the continuum theory of mixtures formulated to include the compressibility of all phases and the compaction behavior of the granular material.

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1,155 citations

"A positive and entropy-satisfying f..." refers background or methods in this paper

...In one-space dimension, the convective part of the model introduce in [9] reads:...
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...This paper is concerned with the Baer-Nunziato two-phase flow model introduced in [9], and studied in various papers [20, 6, 12, 24, 33]....
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Journal Article•DOI•

The relaxation schemes for systems of conservation laws in arbitrary space dimensions

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Shi Jin¹, Zhouping Xin²•Institutions (2)

Georgia Institute of Technology¹, Courant Institute of Mathematical Sciences²

01 Jan 1995-Communications on Pure and Applied Mathematics

TL;DR: A linear hyperbolic system is constructed with a stiff lower order term that approximates the original system with a small dissipative correction and can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally.

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Abstract: We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1-D and 2-D problems are presented. The second-order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.

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928 citations

"A positive and entropy-satisfying f..." refers background in this paper

...In the spirit of [32], the relaxation approximation consists in considering an enlarged system involving two additional unknowns Tk, associated with linearizations πk of the phasic pressure laws....
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Journal Article•DOI•

A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows

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Richard Saurel¹, Rémi Abgrall²•Institutions (2)

Centre national de la recherche scientifique¹, University of Bordeaux²

10 Apr 1999-Journal of Computational Physics

TL;DR: A new model and a solution method for two-phase compressible flows is proposed that provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.

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906 citations

"A positive and entropy-satisfying f..." refers background in this paper

...One may mention various closure laws for the interfacial velocity and pressure [23, 38, 25] or extensions to multi-phase flows [29, 27, 36]....
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...Various examples of such problems were introduced in [39], [38, 3] or [42]....
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