110 research outputs found
All-regime Lagrangian-Remap numerical schemes for the gas dynamics equations. Applications to the large friction and low Mach coefficients
In this talk, we propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics
equations. By all regime, we mean that the numerical scheme is able to compute accurate
approximate solutions with an under-resolved discretization with respect to the Mach number
M, i.e. such that the ratio between the Mach number M and the mesh size or the time step is
small with respect to 1. The key idea is to decouple acoustic and transport phenomenon and
then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error
in term of M. This modified scheme is conservative and endowed with good stability properties
with respect to the positivity of the density and the internal energy. A discrete entropy inequality
under a condition on the modification is obtained thanks to a reinterpretation of the modified
scheme in the Harten Lax and van Leer formalism. A natural extension to multi-dimensional
problems discretized over unstructured mesh is proposed. Then a simple and efficient semi
implicit scheme is also proposed. The resulting scheme is stable under a CFL condition driven
by the (slow) material waves and not by the (fast) acoustic waves and so verifies the all regime
property. Numerical evidences are proposed and show the ability of the scheme to deal with
tests where the flow regime may vary from low to high Mach values.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec
Computing material fronts with a Lagrange-Projection approach
This paper reports investigations on the computation of material fronts in
multi-fluid models using a Lagrange-Projection approach. Various forms of the
Projection step are considered. Particular attention is paid to minimization of
conservation errors
Beyond pressureless gas dynamics: Quadrature-based velocity moment models
Following the seminal work of F. Bouchut on zero pressure gas dynamics which
has been extensively used for gas particle-flows, the present contribution
investigates quadrature-based velocity moments models for kinetic equations in
the framework of the infinite Knudsen number limit, that is, for dilute clouds
of small particles where the collision or coalescence probability
asymptotically approaches zero. Such models define a hierarchy based on the
number of moments and associated quadrature nodes, the first level of which
leads to pressureless gas dynamics. We focus in particular on the four moment
model where the flux closure is provided by a two-node quadrature in the
velocity phase space and provide the right framework for studying both smooth
and singular solutions. The link with both the kinetic underlying equation as
well as with zero pressure gas dynamics is provided and we define the notion of
measure solutions as well as the mathematical structure of the resulting system
of four PDEs. We exhibit a family of entropies and entropy fluxes and define
the notion of entropic solution. We study the Riemann problem and provide a
series of entropic solutions in particular cases. This leads to a rigorous link
with the possibility of the system of macroscopic PDEs to allow particle
trajectory crossing (PTC) in the framework of smooth solutions. Generalized
-choc solutions resulting from Riemann problem are also investigated.
Finally, using a kinetic scheme proposed in the literature without mathematical
background in several areas, we validate such a numerical approach in the
framework of both smooth and singular solutions.Comment: Submitted to Communication in Mathematical Science
On the Eulerian Large Eddy Simulation of disperse phase flows: an asymptotic preserving scheme for small Stokes number flows
In the present work, the Eulerian Large Eddy Simulation of dilute disperse
phase flows is investigated. By highlighting the main advantages and drawbacks
of the available approaches in the literature, a choice is made in terms of
modelling: a Fokker-Planck-like filtered kinetic equation proposed by Zaichik
et al. 2009 and a Kinetic-Based Moment Method (KBMM) based on a Gaussian
closure for the NDF proposed by Vie et al. 2014. The resulting Euler-like
system of equations is able to reproduce the dynamics of particles for small to
moderate Stokes number flows, given a LES model for the gaseous phase, and is
representative of the generic difficulties of such models. Indeed, it
encounters strong constraints in terms of numerics in the small Stokes number
limit, which can lead to a degeneracy of the accuracy of standard numerical
methods. These constraints are: 1/as the resulting sound speed is inversely
proportional to the Stokes number, it is highly CFL-constraining, and 2/the
system tends to an advection-diffusion limit equation on the number density
that has to be properly approximated by the designed scheme used for the whole
range of Stokes numbers. Then, the present work proposes a numerical scheme
that is able to handle both. Relying on the ideas introduced in a different
context by Chalons et al. 2013: a Lagrange-Projection, a relaxation formulation
and a HLLC scheme with source terms, we extend the approach to a singular flux
as well as properly handle the energy equation. The final scheme is proven to
be Asymptotic-Preserving on 1D cases comparing to either converged or
analytical solutions and can easily be extended to multidimensional
configurations, thus setting the path for realistic applications
A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations
This work focuses on the numerical approximation of the Shallow Water
Equations (SWE) using a Lagrange-Projection type approach. We propose to extend
to this context recent implicit-explicit schemes developed in the framework of
compressibleflows, with or without stiff source terms. These methods enable the
use of time steps that are no longer constrained by the sound velocity thanks
to an implicit treatment of the acoustic waves, and maintain accuracy in the
subsonic regime thanks to an explicit treatment of the material waves. In the
present setting, a particular attention will be also given to the
discretization of the non-conservative terms in SWE and more specifically to
the well-known well-balanced property. We prove that the proposed numerical
strategy enjoys important non linear stability properties and we illustrate its
behaviour past several relevant test cases
Convergent and conservative schemes for nonclassical solutions based on kinetic relations
We propose a new numerical approach to compute nonclassical solutions to
hyperbolic conservation laws. The class of finite difference schemes presented
here is fully conservative and keep nonclassical shock waves as sharp
interfaces, contrary to standard finite difference schemes. The main challenge
is to achieve, at the discretization level, a consistency property with respect
to a prescribed kinetic relation. The latter is required for the selection of
physically meaningful nonclassical shocks. Our method is based on a
reconstruction technique performed in each computational cell that may contain
a nonclassical shock. To validate this approach, we establish several
consistency and stability properties, and we perform careful numerical
experiments. The convergence of the algorithm toward the physically meaningful
solutions selected by a kinetic relation is demonstrated numerically for
several test cases, including concave-convex as well as convex-concave
flux-functions.Comment: 31 page
A Godunov-type method for the seven-equation model of compressible two-phase flow
We are interested in the numerical approximation of the solutions of the compressible seven-equation two-phase flow model. We propose a numerical srategy based on the derivation of a simple, accurate and explicit approximate Riemann solver. The source terms associated with the external forces and the drag force are included in the definition of the Riemann problem, and thus receive an upwind treatment. The objective is to try to preserve, at the numerical level, the asymptotic property of the solutions of the model to behave like the solutions of a drift-flux model with an algebraic closure law when the source terms are stiff. Numerical simulations and comparisons with other strategies are proposed
Fast Relaxation Solvers for Hyperbolic-Elliptic Phase Transition Problems
International audiencePhase transition problems in compressible media can be modelled by mixed hyperbolicelliptic systems of conservation laws. Within this approach phase boundaries are understood as shock waves that satisfy additional constraints, sometimes called kinetic relations. In recent years several tracking-type algorithms have been suggested for numerical approximation. Typically a core piece of these algorithms is the usage of exact Riemann solvers incorporating the kinetic relation at the location of phase boundaries. However, exact Riemann solvers are computationally expensive or even not available. In this paper we present a class of approximate Riemann solvers for hyperbolic-elliptic models that relies on a generalized relaxation procedure. It preserves in particular the kinetic relation for phase boundaries exactly and gives for isolated phase transitions the correct solutions. In combination with a novel sub-iteration procedure the approximate Riemann solvers are used in the tracking algorithms. The efficiency of the approach is validated on a barotropic system with linear kinetic relation where exact Riemann solvers are available. For a nonlinear kinetic relation and a thermoelastic system we use the new method to gain information on the Riemann problem. Up to our knowledge an exact solution for arbitrary Riemann data is currently not available in these cases
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