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Shock tube problem with sonic rarefaction fan, 400 mesh points, final time T is T = 0.18, the solid red line is the analytic solution.
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In a previous paper, we have achieved the performance analysis of staggered Lagrange-remap schemes, a class of solvers widely used for Hydrodynamics applications. This paper is devoted to the rethinking and redesign of the Lagrange-remap process for achieving better performance using today's computing architectures. As an unintended outcome, the an...
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Citations
... Dans ce chapitre, nous présenterons en détail un schéma numérique eulérien de type INTRODUCTION Lagrange-Flux, tel qu'expliqué dans l'article [27]. Nous compléterons ce schéma en explicitant le solveur de Riemann HLL, tel qu'il est défini dans l'ouvrage [98], pour le calcul de la vitesse et de la pression aux faces du maillage. ...
In this thesis, we develop a numerical method suitable for the simulation of immiscible compressible fluid flows. To model these flows, we analyse an original fully conservative system with six equations, dosed by a stiffened-gas state equation and a pressure equilibrium equation. We also introduce a numerical scheme of order 2, in space and time, specialy designed for capturing interfaces between fluids in multi-dimensional configurations. To each order 2, we are developing a multidimensional slope reconstruction method based on the stability criterion : local extremum diminishing (LED). The order 2 scheme associated with the totaly conservative model leads to the appearance of oscillations in the pressure profiles. To avoid these spurious oscillations, we demonstrate a set of essential properties. First, we find stability conditions of the CFL type, imposed by the slope reconstructions to ensure the positivity of the internal energy. Finally, we develop a single-step numerical method adapted to the simulation of flows involving more than two fluids. All the results presented in this document are illustrated by test cases, in one, two or three dimensions of space.
... Lagrange-Flux scheme are a variant algebraic Lagrange+remap FV method [4]: ...
This work focuses on numerical methods for the dynamics of compressible immiscible multifluid flows. While being intensively studied since three or four decades, there are still issues and open numerical challenges:
- Design a fully conservative robust numerical method that returns accurate solutions without numerical artifacts.
- Minimize as much as possible the unavoidable artificial numerical mixing
- Compute geometrically accurate fluid interfaces.
In this work, we study different computational approaches that address these issues.
... For numerical discretization, stable conservative entropy schemes are searched in order to ensure convergence toward the entropy weak solution. In this work, we propose to use the recent family of Lagrange-flux finite volume schemes [6] that provide both numerical stability and efficiency. The construction of the Lagrange-flux is based on a Lagrangian remapping process with a particular treatment of time discretization. ...
This research is aimed at achieving an efficient digital infrastructure for evaluating risks and damages caused by tsunami flooding. It is mainly focused on the suitable modeling of debris dynamics for a simple (but accurate enough) assessment of damages. For different reasons including computational performance and Big Data management issues, we focus our research on Eulerian debris flow modeling. Rather than using complex multiphase debris models, we rather use an empirical transportation and deposition model that takes into account the interaction with the main water flow, friction/contact with the ground but also debris interaction. In particular, for debris interaction, we have used ideas coming from vehicular traffic flow modeling. We introduce a velocity regularization term similar to the so-called ``anticipation term'' in traffic flow modeling that takes into account the local flow between neighboring debris and makes the problem mathematically well-posed. It prevents from the generation of ``Dirac measures of debris'' at shock waves. As a result, the model is able to capture emerging phenomenons like debris aggregation and accumulations, and possibly to react on the main flow by creating hills of debris and make the main stream deviate. We also discuss the way to derive quantities of interest (QoI), especially ``damage functions'' from the debris density and momentum fields. We believe that this original unexplored debris approach can lead to a valuable analysis of tsunami flooding damage assessment with Physics-based damage functions. Numerical experiments show the nice behaviour of the numerical solvers, including the solution of Saint-Venant's shallow water equations and debris dynamics equations.
... Lagrange-Flux schemes have been derived in [4,3,9] from cell-centered Lagrangeremap schemes (see also [6]). Collocated Lagrangian solvers have been proposed a decade ago by Després-Mazeran [5] and by Maire et al. [8]. ...
... Normal interface velocity (u u u A · ν ν ν A ) and pressure p A can be computed by any approximate Riemann solver in the Lagrangian frame (Lagrangian HLL solver [4] for example) or derived using a pseudo-viscosity approach. One can observe the simplicity of expression (3) which is naturally consistent with the physical flux, and the way pressure terms and convective terms are treated in a separate way. ...
... Following the construction of the Lagrange-flux schemes presented in [4], we now consider the Lagrange-remap scheme described in the two above subsections and observe what happens when ∆t n tends to 0. We get a semi-discrete Lagrange-flux scheme written in the form (4), (5). From (20), at the limit ∆t n → 0 we get the semidiscrete entropy inequality ...
The Lagrange-Flux schemes are Eulerian finite volume schemes that make use of an approximate Riemann solver in Lagrangian description with particular upwind convective fluxes. They have been recently designed as variant formulations of Lagrange-remap schemes that provide better HPC performance on modern multicore
processors, see~[De Vuyst et al., OGST 71(6), 2016]. Actually Lagrange-Flux schemes show several advantages compared to Lagrange-remap schemes, especially for multidimensional problems: they do not require the computation of deformed Lagrangian cells or mesh intersections as in the remapping process. The paper focuses on the entropy property of Lagrange-Flux schemes in their semi-discrete in space form, for one-dimensional problems and for the compressible Euler equations as example. We provide pseudo-viscosity pressure terms that ensure entropy production of order $O(|\Delta u|^3)$, where $|\Delta u|$ represents
a velocity jump at a cell interface. Pseudo-viscosity terms are also designed to vanish into expansion regions as it is the case for rarefaction waves.
... Getting further into the numerical field, we now deal with numerical scheme derivations, and again we propose to link it with HPC issues. In the contribution of F. De Vuyst, T. Gasc, R. Motte, M. Peybernes and R. Poncet [7] the Lagrange-Flux schemes are reformulated in a "HPCcompatible" way. This paper evaluates the algorithmic aspects of the scheme with a performance model. ...
... Nous poursuivons l'exploration des strates numériques en nous intéressant maintenant à la dérivation de schémas, sans pour autant perdre le lien avec les problématiques de calcul intensif. Dans l'article de F. De Vuyst, T. Gasc, R. Motte, M. Peybernes et R. Poncet [7] les schémas Lagrange-flux sont reformulés pour une meilleure efficacité computationnelle. Cette contribution évalue les algorithmes de ces schémas grâce à un modèle de performance en charge de prédire la performance en temps de calcul de l'algorithme et sa consommation mémoire à partir d'informations sur l'architecture cible (bande passante et puissance crête). ...
... Lagrange-Flux schemes have been derived in [4,3,9] from cell-centered Lagrangeremap schemes (see also [6]). Collocated Lagrangian solvers have been proposed a decade ago by Després-Mazeran [5] and by Maire et al. [8]. ...
... Normal interface velocity (u u u A · ν ν ν A ) and pressure p A can be computed by any approximate Riemann solver in the Lagrangian frame (Lagrangian HLL solver [4] for example) or derived using a pseudo-viscosity approach. One can observe the simplicity of expression (3) which is naturally consistent with the physical flux, and the way pressure terms and convective terms are treated in a separate way. ...
... Following the construction of the Lagrange-flux schemes presented in [4], we now consider the Lagrange-remap scheme described in the two above subsections and observe what happens when ∆t n tends to 0. We get a semi-discrete Lagrange-flux scheme written in the form (4), (5). From (20), at the limit ∆t n → 0 we get the semidiscrete entropy inequality ...
The Lagrange-Flux schemes are Eulerian finite volume schemes that make use of an approximate Riemann solver in Lagrangian description with particular upwind convective fluxes. They have been recently designed as variant formulations of Lagrange-remap schemes that provide better HPC performance on modern multicore processors, see~[De Vuyst et al., OGST 71(6), 2016]. Actually Lagrange-Flux schemes show several advantages compared to Lagrange-remap schemes, especially for multidimensional problems: they do not require the computation of deformed Lagrangian cells or mesh intersections as in the remapping process. The paper focuses on the entropy property of Lagrange-Flux schemes in their semi-discrete in space form, for one-dimensional problems and for the compressible Euler equations as example. We provide pseudo-viscosity pressure terms that ensure entropy production of order $O(|\Delta u|^3)$, where $|\Delta u|$ represents a velocity jump at a cell interface. Pseudo-viscosity terms are also designed to vanish into expansion regions as it is the case for rarefaction waves.
... In a previous work [2], we established that the so-called ECM model [3] has the capability to accurately predict Lagrange-Remap solver performance on multicore CPUs. The main contribution of this paper is to show how performance modeling using the ECM model leads us to extract the computational bottlenecks of the Lagrange-Remap algorithm, link them to properties of the numerical method, and naturally propose the Lagrange-Flux algorithm (originally introduced in [4] from a numerical analysis point view) as an alternative. Superior computational performance of this new algorithm — from both a scalability and absolute performance point of view — is then inferred from theoretical analysis, and verified by numerical experiments. ...
... The detailed construction of the Lagrange-Flux algorithm is given in [4]. Hereafter, we give key ideas of its construction and provide a schematic description. ...
This paper is a practical example of co-design between numerical analysis and
high performance computing, applied to compressible fluid mechanics. We consider a legacy
numerical method, based on a Lagrange-Remap solver for the compressible Euler equations,
use tools from analytical performance modeling to quantitatively understand its behavior on
recent multicore CPUs, and extract its computational bottlenecks. This analysis inspires us to
propose a new numerical method, called Lagrange-Flux. Experimental results show that this
new method yields an algorithm that is more computationally efficient, and at the same time
retains the good numerical properties of the original Lagrange-Remap solver.
... with the vector of variables W = ([z ] , u, E) other a time step ∆t n (see De Vuyst et al. [11]). ...
In this paper, stable and "low-diffusive" multidimensional interface capturing (IC) schemes using slope limiters are discussed. It is known that direction-by-direction slope-limited MUSCL schemes create geometrical artifacts and thus return a poor accuracy. We here focus on this particular issue and show that the reconstruction of gradient directions are an important factor of accuracy. The use of a multidimensional limiting process (MLP) added with an adequate time integration scheme leads to an artifact-free and instability-free interface capturing (IC) approach. Numerical experiments like the reference Kothe-Rider forward-backward advection case show the accuracy of the approach. We also show that the approach can be extended to the more complex compressible multimaterial hydrodynamics case, with potentially an arbitrary number of fluids. We also believe that this approach is appropriate for multicore/manycore architecture because of its SIMD feature, which may be another asset compared to interface reconstruction approaches.
arXiv:1605.07091
This work is dedicated to the design of a robust compressible multifluid conservative finite volume scheme using stiffened gas equations of state for the different fluids. The starting point is the use of a five-equation model [1]. Then different original ingredients are used. First, an algebraic Lagrangian remapping approach referred to as Lagrange-flux scheme [3] is used. Second, in the same goal as in [2], we derive an approximate Riemann solver (a HLL solver in Lagrange form) that ensures both positiveness of internal energy and hyperbolicity. Finally, we use the so-called MLP-UB shape-preserving interface capturing method [4] for solving the different advection equations of volume fractions. We also derive a renormalization strategy in order to keep the sum of volume fractions equal to 1. Second-order accuracy is obtained by the use of a MUSCL approach. Energy consistency conditions (ECC) [5] in order not to create oscillations at material interfaces are also discussed.
We present a novel numerical scheme for the efficient and accurate solution of the isothermal two-fluid (electron + ion) equations coupled to Poisson's equation for low-temperature plasmas at low-pressure. The model considers electrons and ions as separate fluids, comprising the electron inertia and space charge regions. The discretization of this system with standard explicit schemes is constrained by very restrictive time steps and cell sizes related to the resolution of the Debye length, electron plasma frequency, and electron sound waves. Both sheath and electron inertia are fundamental to fully explain the physics in low-pressure and low-temperature plasmas. However, most of the phenomena of interest for fluid models occur at speeds much slower than the electron thermal speed and are quasi-neutral, except in small charged regions. A numerical method that is able to simulate efficiently and accurately all these regimes is a challenge due to the multiscale character of the problem. In this work, we present a scheme based on the Lagrange-projection operator splitting that preserves the asymptotic regime where the plasma is quasi-neutral with massless electrons. As a result, the quasi-neutral regime is treated without the need of an implicit solver nor the resolution of the Debye length and electron plasma frequency. Additionally, the scheme proves to accurately represent the dynamics of the electrons both at low speeds and when the electron speed is comparable to the thermal speed. In addition, a well-balanced treatment of the ion source terms is proposed in order to tackle problems where the ion temperature is very low compared to the electron temperature. The scheme significantly improves the accuracy both in the quasi-neutral limit and in the presence of plasma sheaths when the Debye length is resolved. In order to assess the performance of the scheme in low-temperature plasmas conditions, we propose two specifically designed test-cases: a quasi-neutral two-stream periodic perturbation with analytical solution and a low-temperature discharge that includes sheaths. The numerical strategy, its accuracy, and computational efficiency are assessed on these two discriminating configurations.
