Preliminary results for the study of the godunov scheme applied to the linear wave equation with porosity at low mach number

Dellacherie, Stéphane; Jung, Jonathan

doi:10.1051/proc/201552006

Cited by 4 publications

(7 citation statements)

References 16 publications

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“…This means that when close to the limit, the accuracy of theses schemes is not sufficient to describe the flow. Many efforts have been done in the recent past in order to correct this main drawback of Godunov schemes, for instance by using preconditioning methods [34] or by splitting and correcting the pressure on the collocated meshes [5], [9,10], [12], [13,14,30], [15], [23,24], [26,27,28], or instead by using staggered grids like in the famous MAC scheme, see for instance [3], [16], [17], [18], [19], [31]. Unfortunately, even if these approaches permit to bypass the consistency problem of Godunov methods, they all need to resolve the scale of the acoustic waves in the fluid in order to remain stable.…”

mentioning

confidence: 99%

Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit

Dimarco¹,

Loubère²,

Vignal³

2017

SIAM J. Sci. Comput.

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Abstract. This article deals with the discretization of the compressible Euler system for all Mach numbers regimes. For highly subsonic flows, since acoustic waves are very fast compared to the velocity of the fluid, the gas can be considered as incompressible. From the numerical point of view, when the Mach number tends to zero, the classical Godunov type schemes present two main drawbacks: they lose consistency and they suffer of severe numerical constraints for stability to be guaranteed since the time step must follow the acoustic waves speed. In this work, we propose and analyze a new unconditionally stable an consistent scheme for all Mach number flows, from compressible to incompressible regimes, stability being only related to the flow speed. A stability analysis and several one and two dimensional simulations confirm that the proposed method possesses the sought characteristics.

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confidence: 99%

Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit

Dimarco¹,

Loubère²,

Vignal³

2017

SIAM J. Sci. Comput.

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“…We also remarked that the accuracy of the Godunov scheme at low Mach number on Cartesian meshes can be recovered by deleting the diffusion term on the velocity field in the Godunov scheme. In [14] we discussed the case with porosity with the help of the modified equation approach; the limitations of this approach is that it only gives hints (but does not provide with a complete proof) on what happens on Cartesian meshes, and does not apply to triangular meshes. Our aim here is to analyse the behavior of the schemes on triangular and rectangular Cartesian meshes by directly studying them rather than their modified equations.…”

Section: The Low Mach Asymptoticsmentioning

confidence: 99%

“…The semi-discrete Godunov scheme applied to the resolution of the linear wave equation is obtained by integrating (10) over each cell Ω i and then solving a Riemann problem on each Γ ij to express interface fluxes as functions of cell-centered values. Details are provided in [14]. This results in…”

Section: Godunov Schemementioning

confidence: 99%

“…The low Mach accuracy problem is then understood and fixed in the linear case for Cartesian meshes, and the reason for its correct behavior on triangular meshes is underlined. In particular, preliminary results obtained in [14] based on a modified equation approach are extended to the discrete Cartesian case. Based on the linear study, a well-balanced scheme accurate at low Mach number for the non-linear system (1) is proposed and numerical tests are performed.…”

Section: Introductionmentioning

confidence: 99%

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Construction of a low Mach finite volume scheme for the isentropic Euler system with porosity

Dellacherie

Jung

2021

ESAIM: M2AN

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Classical finite volume schemes for the Euler system are not accurate at low Mach number and some fixes have to be used and were developed in a vast literature over the last two decades. The question we are interested in in this article is: What about if the porosity is no longer uniform? We first show that this problem may be understood on the linear wave equation taking into account porosity. We explain the influence of the cell geometry on the accuracy property at low Mach number. In the triangular case, the stationary space of the Godunov scheme approaches well enough the continuous space of constant pressure and divergence-free velocity, while this is not the case in the Cartesian case. On Cartesian meshes, a fix is proposed and accuracy at low Mach number is proved to be recovered. Based on the linear study, a numerical scheme and a low Mach fix for the non-linear system, with a non-conservative source term due to the porosity variations, is proposed and tested.

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“…The analysis [40,41,56,2,44,1] and the development of numerical methods [37,33,54,62,42,13,31,51,47,34,30,50,48,16,19,17,15,32,10,28,39,20,11,21] for the passage from compressible to incompressible gas dynamics has been and is still a very active field of research. The compressible Euler equations which describe conservation of density, momentum and energy in a fluid flow become stiff when the Mach number tends to zero.…”

Section: Introductionmentioning

confidence: 99%

Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime

Dimarco

Loubère

Michel-Dansac

et al. 2018

Journal of Computational Physics

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In this work, we consider the development of implicit explicit total variation diminishing (TVD) methods (also termed SSP: strong stability preserving) for the compressible isentropic Euler system in the low Mach number regime. The scheme proposed is asymptotically stable with a CFL condition independent from the Mach number and it degenerates in the low Mach number regime to a consistent discretization of the incompressible system. Since, it has been proved that implicit schemes of order higher than one cannot be TVD (SSP) [29], we construct a new paradigm of implicit time integrators by coupling first order in time schemes with second order ones in the same spirit as highly accurate shock capturing TVD methods in space. For this particular class of schemes, the TVD property is first proved on a linear model advection equation and then extended to the isentropic Euler case. The result is a method which interpolates from the first to the second order both in space and time, which preserves the monotonicity of the solution, highly accurate for all choices of the Mach number and with a time step only restricted by the non stiff part of the system. In the last part, we show thanks to one and two dimensional test cases that the method indeed possesses the claimed properties.

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Preliminary results for the study of the godunov scheme applied to the linear wave equation with porosity at low mach number

Cited by 4 publications

References 16 publications

Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit

Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit

Construction of a low Mach finite volume scheme for the isentropic Euler system with porosity

Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime

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