Consistency Analysis of a 1D Finite Volume Scheme for Barotropic Euler Models

Berthelin, Florent; Goudon, Thierry; Minjeaud, Sébastian

doi:10.1007/978-3-319-05684-5_8

Cited by 5 publications

(7 citation statements)

References 2 publications

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“…It is worth pointing out that, despite its "kinetic" flavor, the definition of the flux function has a very simple expression and does not need any numerical computation of integrals. The following properties are fundamental for analysing the scheme [5,6,7,25]. Lemma 3.2.…”

Section: Mass Conservation On the Diamond Cellsmentioning

confidence: 99%

“…It uses the velocity u D σ,σ * ,s and the sound speed c(e s ) naturally given on the interface by Definition 2.2, and upwinds the density. The symmetry property (5) implies that…”

Section: Mass Conservation On the Diamond Cellsmentioning

confidence: 99%

“…The following properties are fundamental for analyzing the scheme. [5][6][7][8] Lemma 2. The functions  ± satisfy the following properties:…”

Section: Mass Conservation On the Diamond Cellsmentioning

confidence: 99%

“…In particular, the proposed scheme introduced numerical fluxes strongly inspired from the framework of kinetic schemes. The scheme is shown to be stable, under suitable CFL-conditions, it preserves the entropystructure and the consistency analysis à la Lax-Wendroff can be performed too [5]. This approach is further developped to handle complex systems for mixture flows [7].…”

Section: Introductionmentioning

confidence: 99%

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An explicit finite volume scheme on staggered grids for the Euler equations: Unstructured meshes, stability analysis, and energy conservation

Goudon

Llobell

Minjeaud

2021

Numerical Methods in Fluids

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We set up a numerical strategy for the simulation of the Euler equations, in the framework of finite volume staggered discretizations where numerical densities, energies and velocities are stored on different locations. The main difficulty relies on the treatment of the total energy, which mixes quantities stored on different grids. The proposed method is strongly inspired, on the one hand, from the kinetic framework for the definition of the numerical fluxes, and, on the other hand, from the Discrete Duality Finite Volume (ddfv) framework, which has been designed for the simulation of elliptic equations on complex meshes. The time discretization is explicit and we exhibit stability conditions that guaranty the positivity of the discrete densities and internal energies. Moreover, while the scheme works on the internal energy equation, we can define a discrete total energy which satisfies a local conservation equation. We provide a set of numerical simulations to illustrate the behaviour of the scheme.

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Section: Mass Conservation On the Diamond Cellsmentioning

confidence: 99%

“…It uses the velocity u D σ,σ * ,s and the sound speed c(e s ) naturally given on the interface by Definition 2.2, and upwinds the density. The symmetry property (5) implies that…”

Section: Mass Conservation On the Diamond Cellsmentioning

confidence: 99%

“…The following properties are fundamental for analyzing the scheme. [5][6][7][8] Lemma 2. The functions  ± satisfy the following properties:…”

Section: Mass Conservation On the Diamond Cellsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An explicit finite volume scheme on staggered grids for the Euler equations: Unstructured meshes, stability analysis, and energy conservation

Goudon

Llobell

Minjeaud

2021

Numerical Methods in Fluids

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“…In (1), the unknowns (t, x) → ρ(t, x) and (t, x) → u(t, x) stand respectively for the density and the velocity of a fluid. The quantity µ > 0 is the viscosity of the fluid.…”

Section: Introductionmentioning

confidence: 99%

Numerical investigations of the compressible Navier-Stokes system

et al. 2021

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In this paper we write, analyze and experimentally compare three different numerical schemes dedicated to the one dimensional barotropic Navier-Stokes equations: a staggered scheme based on the Rusanov one for the inviscid (Euler) system,a staggered pseudo-Lagrangian scheme in which the mesh “follows” the fluid,the Eulerian projection (on a fixed mesh) of the preceding scheme. All these schemes only involve the resolution of linear systems (all the nonlinear terms are solved in an explicit way). We propose numerical illustrations of their behaviors on particular solutions in which the density has discontinuities (hereafter called Hoff solutions). We show that the three schemes seem to converge to the same solutions, and we compare the evolution of the amplitude of the discontinuity of the numerical solution (with the pseudo-Lagrangian scheme) with the one predicted by Hoff and observe a good agreement.

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Multifluid Flows: A Kinetic Approach

2015

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We propose a numerical scheme for the simulation of fluid-particles flows with two incompressible phases. The numerical strategy is based on a finite volume discretization on staggered grids, with a flavor of kinetic schemes in the definition of the numerical fluxes. We particularly pay attention to the difficulties related to the volume conservation constraint and to the presence of a close-packing term which imposes a threshold on the volume fraction of the disperse phase. We are able to identify stability conditions on the time step to preserve this threshold and the energy dissipation of the original model. The numerical scheme is validated with the simulation of sedimentation flows

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Consistency Analysis of a 1D Finite Volume Scheme for Barotropic Euler Models

Cited by 5 publications

References 2 publications

An explicit finite volume scheme on staggered grids for the Euler equations: Unstructured meshes, stability analysis, and energy conservation

An explicit finite volume scheme on staggered grids for the Euler equations: Unstructured meshes, stability analysis, and energy conservation

Numerical investigations of the compressible Navier-Stokes system

Multifluid Flows: A Kinetic Approach

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