Finite volume schemes for a nonlinear hyperbolic conservation law with a flux function involving discontinuous coefficients

Paper 1, Code: A001

Authors: F. Bachmann

A model for two phase flow in porous media with distinct permeabilities leads to a non linear hyperbolic conservation law with discontinuous flux function.  One presents in this paper for such a problem, the notion of entropy solution, proves existence,  uniqueness and convergence of finite volume scheme.  One remarks that no hypothesis of convexity or genuine non linearity on the  flux function is assumed, which is a new point in comparaison with preceding  works. This brings a new difficulty because one can not consider the trace of  the solution along the line of discontinuity of the flux function.

Key words: discontinuous flux, entropy solution, finite volume scheme.

Variational Smoothing of Volumetric Medical Images

Paper 2, Code: A002

Authors:  A. Ben Hamza


We propose an information-theoretic variational model for volumetric medical image smoothing. It is a result of minimizing a functional subject to some noise constraints, and takes a hybrid form of a negative-entropy variational integral for small gradient magnitudes and a total variational integral for large gradient magnitudes. The core idea behind this approach is to use geometric insight in helping construct regularizing functionals and avoiding a subjective choice of a prior in maximum a posteriori estimation. Illustrating experimental results demonstrate a much improved performance of the approach in the presence of noise.

Key words : variational smoothing, information theory, volumetric medical imaging.

Boundary conditions for Petrov-Galekin Finite Volumes

Paper 3, Code: S001

Authors: S. Borel, F. Dubois, C. Le Potier and M.M. Tekitek

We consider the Petrov-Galerkin finite volumes method based on dual Raviart-Thomas basis functions and the least square method. We propose a numerical scheme for various boundary conditions. First numerical tests indicates good convergence properties. [FVCA4, april

Direct numerical simulation of convective turbulence

Paper 4, Code: C001

Authors: I. Palymsky

Turbulent convective flow of water in horizontal layer with free horizontal boundaries, arising at heating from below,  is numerically simulated by spectral method using the Boussinesq model without any semiempirical relationships (DNS) in 2-D case.  It is shown that numerical results have good agreement with experimental data and results of numerical simulations with rigid boundary  conditions at enough high supercriticality and that this simulation reflects the transition to hard turbulence. We study also the role of boundary conditions (free-rigid).

A complete flux scheme for one dimensional combustion simulation

Paper 5, Code: C002

J.H.M ten Thije Boonkkamp

A discretisation scheme for the conservation equations describing a one-dimensional laminar flame is presented. The scheme is a finite volume discretisation combined with a generalized exponential scheme, the so-called complete flux scheme, for the computation of the  numerical fluxes. The numerical flux contains two terms, viz. one corresponding to the homogeneous and one corresponding to the particular solution of the differential equation. Its derivation is based on the advection-diffusion-reaction equation.  As an example, numerical results for a methane/air flame are presented.

Finite Volume approximation for an oblique derivative boundary value problem

Paper 6, Code: A003

Authors: A. Bradji and T. Gallouet
Abstract :

In this paper, we consider the Laplace equation with oblique boundary conditions, on an open bounded polygonal connected  domain. We introduce  an admissible mesh and we develop  a finite volume scheme. We prove that the  finite volume solutions arising from this scheme  converge to the weak solution of the problem, when the mesh size tends  to zero.
Key words : Oblique derivative, non structured mesh, diffusion equation.

A space-time conservative Finite Volume scheme for Hyperbolic conservation laws

Paper 7, Code: S003

Authors: Q. U. Ain, S. Qamar and G. Warnecke

We present a second order scheme for the numerical solution of hyperbolic systems which treats space and time in a unified manner. The flow variables and their slopes are the basic unknowns in the scheme.  The scheme utilizes the advantages of the space-time conservation  element and solution element (CE/SE) method (Chang 1995) as well as central schemes (Nessyahu and Tadmor 1990). However, unlike the  CE/SE method the present scheme is Jacobian-free and hence like the central schemes can also be applied to any hyperbolic system. In Chang's method they used a finite difference approach for the slope calculation in case of nonlinear hyperbolic equations. We propose to propagate the slopes by a scheme even in the case of nonlinear systems. By introducing a suitable limiter for the slopes of flow variables, we can apply the  same scheme to linear and non-linear problems with discontinuities.  The scheme is simple, efficient and has a good resolution especially at contact discontinuities. We derive the scheme for the one and two space dimensions. In two-space dimensions we use triangular meshes. The second order accuracy of the scheme has been verified by numerical experiments. Several numerical test computations presented in this article validate the accuracy and robustness of the present scheme.

Keywords: Conservation laws, hyperbolic systems, space-time control volumes, finite volume schemes, high order accuracy, shock solutions.

Solution of Unsteady Natural Convection Flows using Finite Volume Modified Method of Characteristics

Paper 8, Code: C003

Authors:  Mofdi El-Amrani and Mohammed Seaid

This paper proposes a finite volume modified method of characteristics for the numerical solution of natural convection flows. The unsteady, viscous incompressible, and thermal flows are modelled by a set of coupled Navier-Stokes/Boussinesq equations involving conservation of mass, momentum, and energy equations. To develop an accurate and efficient solver for the model, we implement a modified method of characteristics in the finite volume framework. The presented method is unconditionally stable, independent of the Rayleigh number, and can be easily implemented on unstructured meshes. Numerical results are shown for the two benchmark problems: the model problem of natural convection in a square cavity and also for heated flow over a cylinder.

Key words : Incompressible Navier-Stokes Equations; Boussinesq Approximation; Modified Method of Characteristics; Finite Volume Method; Natural Convection flow Simulation.

A finite volume method for the approximation of highly anisotropic diffusion operators on unstructured meshes

Paper 10, Code S004

Author: C Le Potier

We introduce a finite volume method for highly anisotropic diffusion operators on unstructured meshes. The main idea is to calculate the gradient on each cell vertex using the cell-centered unknown and other unknowns calculated on the cell faces. These degrees of freedom are eliminated imposing flux continuity conditions. The resulting global matrix is symmetric and positive definite. We show the robustness and the accuracy of the method in comparison with analytical solutions and results obtained with other numerical schemes.

Stability and Convergence of a Finite Volume Method for a Reaction-Diffusion system of Equations in Electro-Cardiology

Paper 11, Code: A004

Authors: Y. Coudiere and C. Pierre

In the field of Electro-cardiology, the mono-domain equations model the electrical phenomena occurring in the he muscle, considered as an anisotropic conducting medium.  Is is  a coupled system of one semi-linear parabolic PDE of reaction-diffusion type and one or severa  ODEs. Numerical difficulties are due to the different time and space scales in the equations. Actually, the solution exhibit very sharp fronts propagating at high speeds. Simulations needs unstructured, fine and/or locally refined meshes.
Hence, we propose a Finite Volume methods, and address here the difficult problem of numerical stability for two simple time-stepping methods. Convergence results are also proved.

Experimental and numerical investigation on heat transfer for polyethylene flowing through extrusion die

Paper 13, Code C005


Abstract:Experimental and numerical study is carried out on the conjugate thermal transport in polyethylene (PE) flowing through extrusion die. This paper deals with the experimental design and with SIMPLE (Semi-Implicit Method for Pressure-Linked Equations Patankar and Spalding 1972) method for solving the continuity, momentum and energy equations. The simulation is performed to determine the influence of conduction through the die wall and the pressure drops on the transport in the polymer and on the temperature profiles in the interface channel-wall. The computational model has been validated with the experimental results, and good agreement has been observed.

Key words : Finite element method, SIMPLE algorithm, Die design, Polymer flow, extrusion die

On Finite Volume Schemes for Nonconservative Hyperbolic Problems

Paper 14, Code: S006

Authors: V. Dolejsi, T. Gallouet

We deal with a numerical solution of a nonconservative system of hyperbolic equations arising from a °ow simulation of a solid-liquid-gas mixture. This system has at least one eigenvalue which changes its sign in the interior of the computational domain and it can cause a fail of some numerical schemes. We discuss a use of several Roe-type ¯nite volume methods, which are based on a solution of linearized Riemann problems. We show that the mentioned property of the system can completely spoil a solution of the linearized Riemann problem and consequently a collapse of a computational process. We propose a numerical scheme which does not su®er from the mentioned drawbacks and which e±ciently simulates flows of solid-liquid-gas slurries.

Key words : nonconservative hyperbolic equations, nonconservative Roe Finite volume schemes, numerical flux.

A well balanced scheme for gas flows in protoplanetary nebulae

Paper 15, Code: S007

Authors:  H. Guillard, E. Daniel

This work deals with the study of the long time evolution of the gas and dust protoplanetary nebula where planets are supposed to form. In a preliminary step, we show here how to construct a well-balanced FV scheme preserving the equilibrium state of a gas flow rotating in the gravitational field of a central star.

Key words :
Origin solar system, planetary formation, hydrodynamics, finite volume methods, well-balanced schemes

Conservative Locally Moving Mesh Method for Multifluid Flows

Paper 16, Code S008

Authors:  A. Chertock, A. Kurganov

We present a conservative locally moving mesh finite-volume-particle method for computing compressible multifluids flows. The idea behind the new method is to use different schemes for the flow and the interface tracking: the Euler equations are numerically integrated using a finite-volume scheme, while a particle method is used to track the moving interface and obtain a subcell information needed to create an adaptive locally moving mesh such that the material interface always coincides with the moving cell boundary. The method does not generate significant oscillations across the material interface and provides an enhanced resolution of the contact discontinuities.

Convergence of the water-air flow model to Richards model

Paper 17, Code: A008

Authors: R. Eymard, M. Ghilani and M. Marhraoui

The aim of this work is to prove the convergence of the system which models the air-water flow in the soils to the Richards model when the air mobility tends to infinity. The convergence is obtained in two steps. In the first one we prove the convergence of the discrete solution obtained by the implicit finite volumes scheme to the weak solution of the continues problem independently of the air mobility . This step is obtained thanks to the priori estimates of the water saturation and the gradient of the water and capillary pressures. In the second step we prove the convergence of the weak solutions as the mobility tends to infinity.

A novel Cartesian cut-cell approach

Paper 18, Code: S009

Authors: I. Wenneker and M. Borsboom

An alternative to the Cartesian cut-cell method is introduced. Its formulation is fundamentally different from the formulations usually encountered in literature, since it avoids discretization of the boundary conditions on cut cells. In the approach presented here, the boundary condition is extrapolated by means of a Taylor series expansion to a nearby gridline of the Cartesian grid. A simple Cartesian grid method uses then this extrapolated boundary condition. This leads to a simple yet accurate discretization of the boundary conditions. The resulting geometric issues are very limited in number and straightforward. Moreover, the method can be applied to more complicated boundary conditions than closed boundary conditions.

Key words : cut cells, Cartesian grids, inviscid flows.

An optimal a priori error analysis of the Finite Volume method for linear convection problems

Paper 19, Code: A005

Authors: D. Bouche, J.M. Ghidaglia, and F. Pascal

This paper investigates the order of convergence of the upwind Finite volume method for solving linear steady convection problem on a bounded domain and with natural boundary conditions. In order to overcome the non consistency in the fiite differences sense of the scheme, we introduce a sequence of what we call geometric correctors and which is associated with every finite volume mesh and every constant convection vector. Under a local quasi-uniformity condition for the triangulation and if the continuous solution is regular enough, we first establish a link between the convergence of the Finitevolume scheme and these geometric correctors. Hence the study of this corrector in the case of uniformly re ned triangular meshes in two dimensions leads to the proof of the optimal order of convergence for the Finite volume scheme. We then focus the second part of the paper on the Peterson case where for a non uniformly refined mesh and a convection direction parallel to one side of the domain, a loss of accuracy was proved.

Key words:  Finite Volume Method ; Consistency and Accuracy ; Geometric Corrector ; Unstructured Meshes.

A Finite volume scheme for the computation of erosion limiters

Paper 20, Code: A006

Authors: R. Eymard and T. Gallouet

Abstract :
Sedimentation and erosion processes, in sedimentary basins, can be modeled by a scalar hyperbolic equation, under an erosion limiting condition. We present in this paper a Finite volume scheme for the approximation of this problem and the proof of its convergence. This proof is based on the uniqueness of a weaker solution to this problem, involving Lagrange multipliers, which is shown to be the limit of the numerical scheme.

Key words : Hyperbolic inequations, Finite volume methods, erosion and sedimentation models.


Paper 21, Code S010

Authors: B. Desjardins, J. Francescatto, J.M. Ghidaglia

Abstract :
On our way to design a numerical platform able to reproduce complex turbulent mixing  flows, we report here on some numerical benchmarks of the code. We shall first rapidly present our numerical method which essentially relies on a  Finite Volume approach and a splitting in 3 steps for the time scheme. Then we present 3 new test cases towards the validation of the scheme.

Key words : Multiphase Flow, Lagrange + Remap, spherical test cases.

Homogeneous two-phase flow models: coupling by finite volume methods

Paper 22, Code: S011

A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutiere, P.A.  Raviart, N. Seguin

The simulation of complex configurations often requires to use specific models in the different parts of the domain of computation in order to account for specific behaviors of the flow. The models must be coupled to obtain a complete and coherent description of the system as a whole. We study here two separate domains sharing a fixed interface. In each one, a different model is used to describe the flow. The main difficulty is to give appropriate conditions at the interface in a way to obtain a coherent description of the unsteady flow according to physical considerations. In this work, we present the coupling of two different hyperbolic systems. As an example, the coupling of the homogeneous equilibrium model and the homogeneous relaxation model is studied and illustrated by numerical results.

Key words : Model coupling, hyperbolic systems, two-phase flow, phase transition.

Convergence of a finite volume scheme on a MAC mesh for the Stokes problem with right-hand side in H−1

Paper 23, Code: A007

Authors: P. Blanc

We consider a finite volume scheme on MAC meshes for the Sto equations with right-hand side in H−1 and we prove the convergence the scheme.
Key words: Finite volume scheme, MAC mesh, Stokes equations.

Finite-volume schemes for nonlinear elliptic problems on general 2D meshes

Paper 24, Code: S012

Authors: Boris Andreianovy 1, Franck Boyer 2, Florence Hubert 2

Finite volume schemes on general meshes are studied for nonlinear diffusion problems with non homogeneous Dirichlet boundary condition. These schemes allow the discretization nonlinear  fluxes in such a way that the discrete operator inherits the key properties of the continuous one. The schemes being nonlinear, we show that the approximate solution exists and is unique. The convergence rate is analyzed for W_2^p solutions.

Key words : Finite-volume methods, Error estimates, Leray-Lions operators.

Mesh Partitioning for Parallel Computational Fluid Dynamics Applications on a Grid

Paper 26, Code: C006

Authors: Y. Mesri, H. Digonnet, H. Guillard

The problem of partitioning unstructured meshes on a homogeneous architecture is largely studied. However, existing partitioning schemes fail when the target architecture introduces heterogeneity in resource characteristics. With the advent of heterogeneous architecture as the Grid, it becomes imperative to study the partitioning problem taking into account the heterogeneous platforms. In this work, we present a new mesh partitioning scheme, that takes into account the heterogeneity of CPU and networks. Our load balancing mesh partition strategy improves the performance of parallel applications running in a heterogeneous environment. The use of these techniques are applied to some model problems in CFD. Experimental results con rm that these techniques can improve the performance of applications on a computational Grid.

Key words: Grid computing, mesh partitioning, Distributed computing, Load balancing, performance study.

Model-error estimation for goal-oriented model adaptation in flow simulations

Paper 28, Code: A010

Authors: J.M. Cnossen

We derive a model-error estimator for output-oriented model adaptation in flow simulations. Model adaptation is expected to save CPU time in aerospace engineering problems requiring large number of CFD computations. In this paper we derive the model error estimator for a well-known model problem for the Euler vs. Navier-Stokes equations: the unsteady 1-D Burgers problem. In this problem the viscous Burgers equation acts as the sophisticated model and the inviscid 51, 153);">yhst est/spanumarsesophis. Atroduced. Itsn MAC mee derive therobhis pally inteent pumat styledes anipumafinite em as studied an obtainorts the peadjo, tzation of the boundasn MAC mee der-ve the model errion wont 3 neokes pling of t

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