ACCEPTED PAPERS
FVCA4, MARRAKECH, MOROCCO, JULY 4 - JULY 8, 2005
http://averoes.math.univ-paris13.fr/fvca4

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

Paper 1, Code: A001

Authors: F. Bachmann

Abstract:
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

Abstract:

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

Abstract:
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
2005]

Direct numerical simulation of convective turbulence

Paper 4, Code: C001

Authors: I. Palymsky

Abstract:
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

Authors:
J.H.M ten Thije Boonkkamp

Abstract:
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

Abstrac:
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

Abstract:
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

Abstract:
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

Abstract:
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

Authors:  M. KARKRI  A. NACHAOUI  Y. JARNY  P. MOUSSEAU

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

Abstract:
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

Abstract:
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

Abstract:
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

Abstract:
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

Abstract:
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

Abstract:
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.

A MULTIDIMENSIONAL MULTIPHASE FLOW FINITE VOLUME SOLVER USING A LAGRANGIAN PHASE AND A PROJECTION TECHNIQUE

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

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

Abstract:
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

Abstract:
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
Abstract:

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

Abstract:
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

Abstract:
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 Burgers equations as the approximating or coarse model. A formulation for the model error is derived using the dual-weighted residual method. We illustrate the importance of adjoint boundary conditions for the model-error estimator by two cases with different initial conditions.

A finite volume method for the modeling of CO2 storage

Paper 29, Code: C007

Authors: A. Michel, E. Tillier and L. Trenty

Abstract:
This paper describes a fully coupled simulator based on the finite volume method for the simulation of CO2 storage. The simulator takes into account convective and diffusive flow, and chemicals reactions (speciation and mineral precipitation and dissolution). Currently, the simulator considers only the aqueous phase in one dimension. The continuous problem and the discrete equations are presented. Finally, two numerical examples are given.

Key words: reactive transport, finite volume, CO2 storage

A Finite Volume Scheme with Weighted Least Square Reconstruction

Paper 30, Code: S013

Authors: Jiri  Furst
Abstract:
This article describes development of one variant of ENO scheme with the so-called weighted least-square reconstruction. The cell-wise interpolation is achieved by a minimization of the interpolation error for fixed stencil with data-dependent weights. The stability of the reconstruction is analyzed theoretically for simple scalar case and several numerical experiments are done both for multidimensional scalar case as well as for the case of compressible flows in 2D and 3D.

Key words :  Finite volumes, ENO, least-squares, total variation, unstructured mesh

Numerical modeling of flow in fractured porous media

Paper 32, Code: C004

Authors: P. Angot, F. Boyer  and F. Hubert

Abstract:
This study proposes some zero- or first-order models to compute the flow outside and inside fractures supposed to have a small thickness with respect to the macroscopic lenght scale, and hence reduced to immersed polygonal fault interfaces. A cell-centered finite volume scheme on general polygonal meshes which fit the interfaces is derived to solve the set of equations with the additional algebraic or differential transmission conditions linking both pressure and normal velocity jumps through the interfaces. The correction for the angle a  corners between the connected interfaces is addressed. The models are then numerically experimented for immersed fractures. Some numerical results are reported which show different kinds of flows in the case of impermeable or permeable fractures, or also for intermediate cases.

Keywords: Porous media flow, impermeable fractures, permeable fractures, immersed polygonal fault interfaces, cell-centered finite volumes.

Numerical Modelling with Finite Volumes of the Isolated Current Leads for Cryogenic Applications

Paper 33, Code: C005

Authors: Ioan POPA, Florian ŞTEFĂNESCU, Dan POPESCU

Abstract
The article presents the mathematical model and the results obtained by the numerical simulation of the thermal state, of isolated electrical leads  in the framework of  cryogenic applications.  the mathematical Model is based on  1D and 2D equations of thermal conduction. For the numerical resolution of the thermal conduction equation one uses the finite volumes method. The elaborated program enables to get the distribution of the temperature and the heat flux along the leads. The  obtained results are used for the optimization of these electrical leads.

Key words: finite volumes method, electrical leads, cryogenic applications.

A finite Volume scheme for dynamic reliability studies

Paper 34, Code: A012

Authors: C. Cocozza-Thivent, R. Eymard, S. Mercier

Abstract:
The dynamic reliability study of an industrial device implies the determination of the probability for this device to be in a given state at a given time. We consider in this paper a model of dynamic reliability based on the Chapman-Kolmogorov equations, which can be seen as a system of linear scalar hyperbolic equations coupled by the right hand side, the solution of which are probability measures. We propose a finite volume scheme for the approximation of these measures, and we prove its weak convergence. A numerical example illustrates this property.

High order Finite Volume scheme for phase transition

Paper 35, Code: S015

Authors: A. Semma M. El Ganaoui and A. Cheddadi

Abstract:
A numerical model is achieved for the analysis of phase change problems including natural convection effects. In this work, the momentum and energy equations, written in terms of its primitive variables, are solved using a specific continuum formulation associated to a finite volume method. A third–order QUICK scheme is used for the convective terms. The time discretization is implicit, and a second order Euler scheme is employed. Simulations show the ability of high order finite volume method to predict  phase change phenomena. Good agreement with the experimental measurements for materials with high and low energy-momentum coupling is obtained.

Key words : Finite volumes, flux limiter, phase change, enthalpy method.

Discrete-Duality Finite Volume Method for Second Order Elliptic Problems

Paper 36, Code: S016

Authors: S. Delcourte, K. Domelevo, P. Omnes

Abstract:
This paper deals with applications of the “Discrete-Duality Finite Volume” approach to a variety of elliptic problems. This is a new finite volume method, based on the derivation of discrete operators obeying a Discrete-Duality principle. An appropriate choice of the degrees of freedom allows one to use arbitrary meshes. We show that the method is naturally related to finite and mixed finite element methods.

An Artificial compression technique applied to shallow water equations with pollutant

Paper 39, Code: S017

Authors: E.D. Fernandez and V. Martinez

Abstract:
In this work we present an artificial compression technique to shallow water equations with pollutant. Our purpose is to avoid the numerical diffusion in contact discontinuities. This happens when the considered system has linearly degenerated fields. This technique obtains a new partial differential system where there are not linearly degenerated fields but only genuinely nonlinear fields, in such a way the solution does not vary. The difficulty of applying artificial compression to shallow water equations with pollutant is that the primitive variable corresponding to the linearly degenerated field is zero. For Euler Equations, for example, this is not the case. Finally we present several tests which apply this technique with Roe and HLL schemes.

Key words : artificial compression, shallow water, pollutant.

Parallel Implementation of a Dynamic Overset Unstructured Grid Using Finite Volume Method

Paper 40, Code: C010

Abstract:
The purpose of this study is to develop an ecient parallel computational algorithm for complex geometries using overset unstructured grid technique. The strategy adopted in the parallelization of the overset grids method including the use of hierarchical data structure and communication, is described. Numerical results are presented to demonstrate the eciency of the resulting parallel overset grid method.

Key words : Overlapping grids, Chimera method, Unstructured , Overset grid, MPI, CFD.

General fictitious domain method with nonconforming structured meshes
Paper 41, Code: S018
Authors: P. Angot, H. Lomenede and I. Ramiere

Abstract:
This study addresses a new fictitious domain method  for elliptic problems  in order to handle general embedded boundary conditions (E.B.C.) :  Fourier, Neumann and Dirichlet conditions on an immersed interface.  The main interest  is to use a simple structured  mesh, possibly uniform, which does not generally fit the interface  but defines an approximate one.  A cell-centered finite volume scheme  with a non-conforming structured mesh  is derived to solve the set of equations with the additional  algebraic transmission conditions linking both flux and solution  jumps through the immersed approximate interface.  Hence, a local correction is devised to take account of the relative surface ratios  in each control volume for the Fourier or Neumann boundary condition.  Then, the numerical scheme conserves the first-order accuracy with respect to the mesh step,  as observed in the numerical results reported here.  This opens the way to combine the E.B.C. method with the multi-level FIC solver  to increase the precision in the vicinity of the interface by a local mesh refinement.  Such a fictitious domain method is expected to be very efficient.

Key words : Fictitious domain method, embedded boundary conditions, cell-centered finite volumes, multi-level local mesh refinement.

Diffusion with dissolution and precipitation in a porous media: approximation by a finite volume scheme
Paper 44, Code: S020
Authors: N. Bouillard, R. Eymard, R. Herbin, P. Montarnal

Abstract:
This paper study a finite volumes scheme for nonlinear diffusion and dissolution-precipitation equations with non homogeneous Dirichlet boundary conditions. The approximate solution is shown to converge to a weak solution which existence is thus proved. The chemical reaction is kinetics controlled. It involves two species in liquid phase and one species in solid phase. Some numerical tests are shown.

On two fractional step finite volume and finite element schemes for reactive low Mach number  flows

Paper 45, Code: S021

Authors: F. Babik, T. Gallouet, J.C. Latche, S. Suard and D. Vola

Abstract:

We address in this paper a class of physical problems which can be set under the form of the momentum and mass balance equations, supplemented by the balance equation of an independent unknown field z. The fluid density is supposed to be given as a nonlinear function of this latter unknown. In particular, governing equations of some reactive low Mach number flows enter this framework. We present and compare two original fractional steps schemes devoted to the solution of this system of equations. They combine a particular finite volume discretization of the balance of z, which $\LS^\infty$ and $\LS^2$ stability is proven, with a non-conforming finite element pressure correction method for the solution of the Navier-Stokes equations. Via numerical experiments, these schemes are shown  to be almost optimally convergent in time and space and to present remarkable stability properties, in view of the strong non-linearity of the problems in hand.

Key words : Low Mach number  flows, reactive  flows, finite volumes, mixed finite elements

Eulerian Finite Volume Methods for hydraulic fracture problems

Paper 46, Code: C011

Author: A. Peirce

Abstract:
Hydraulic fracturing is a process that is often used in the oil and gas industry to enhance the flow of hydrocarbons by generating a fracture within a reservoir. The fracture is propagated by injecting a viscous fluid into a bore-hole under a sufficiently high pressure to overcome the tensile strength of the rock and the ambient geological stresses. We describe an Eulerian Finite Volume scheme based on a fixed rectangular grid to solve the coupled system of hydraulic fracture nonlinear integro-partial differential equations. We illustrate the accuracy of the algorithm by comparing the Finite Volume solution to the exact solution for a radial solution evolving in a homogeneous elastic medium.

Estimates of blow-up time for the radial symmetric semilinear heat equation in the “open-spectrum” case

Paper 48, Code: A015

Authors:  N.I. Kavallaris, C.V. Nikolopoulos, D. E. Tzanetis

Abstract:
We estimate the blow-up time for the reaction diffusion equation u_t = Delta u + L f(u), when f is a positive, increasing and convex function growing fast enough and L>L* whereL* is a  critical value for the parameter  L. Estimates of the blow-up time are obtained by using comparison methods. Some numerical results are also presented.

Numerical Computation of Laminar Mixed Convection Flow Combined with Mass Transfer in a Duct

Paper 49, Code: C012

Authors:  B. Benhamou, N. Galanis, M. El-Ganaoui

Abstract:
A numerical study is conducted to investigate a laminar mixed convection flow in a vertical parallel plates channel. The plates are isothermal and iso-concentration. The adopted mathematical model includes axial diffusion of momentum, heat and mass transfer. The numerical solution of the PDE set modelling the flow field and transfers is based on the control volumes method. Both upward and downward flows are considered. Results show that heat and mass transfers are significantly affected by the direction and the conditions of the flow.

Key words: channel, laminar flow, heat transfer, mass transfer, mixed convection.

Unstructured finite volume method for modelling the flood wave propagation: Application to the Ourika valley

Paper 50, Code: C013

Authors: H. Belhadj, D. Ouazar, A. Taik

Abstract:
This paper is devoted to the modeling of the wave propagation resulting from the flood phenomenon. The flood physical problem is modeled by using the free surfac  flow equations in the shallow water case. The two dimensional obtained system is generated from the three-dimensional Navier Stokes equations by integrating the state variables over the vertical dimension. This system is writen in a conservative form with hyperbolic homogeneous part. The discretisation is carried out by the use of the finite volume method on unstructured mesh. For the numerical experiment, we have studied the realistic case of Ourika valley which is located in Morocco. The flood occured on August 1995 is simulated and results gives velocities and free surface elevations at different stopped times of the simulation.

Evaluation of Numerical Fluxes for a Locally Exact Finite Volume Scheme Using Hypergeometric Functions

Paper 51, Code: S023

Author:
J. Fuhrmann

Abstract:
For a recently proposed finite volume scheme for certain nonlinear convection diffusion problems based on the solution of local Dirichlet problems, in the special case of power law nonlinearities, we present a method for the evaluation of the flux functions which uses the Gauss hypergeometric function.

Key words :
Finite Volumes, Nonlinear Diffusion-Convection Problems, Hypergeometric Function

A finite volume solver based on matrix sign for non homogeneous systems

Paper 52, Code: S023

Authors:  Slah Sahmim, Fayssal Benkhaldoun, Francisco Alcrudo

Abstract:
This article is devoted to the analysis, and 2D extension of a finite volume scheme proposed recently to handle a class of non homogeneous systems. The stability analysis of the considered scheme leads to a new formulation which includes the Jacobian matrix sign. This formulation is identical to the VFRoe scheme introduced by Gallouët, in the homogeneous case, and has a natural extension here to non homogeneous systems. Comparative numerical experiments for Shallow Water equations with source terms  and two phase problem (Ransom faucet) are presented to validate the scheme.
Key words : Finite volume method, SRNHS scheme, Shallow water equations, Source terms, Riemann Problems, Tow-phase flow.

1D and 2D Numerical Solution of Two Phase Flow Problem Using a Dissipative Lax-Wendroff Scheme

Paper 54, Code: CO15

Authors: Jan Halama and Fayssal Benkhaldoun

Abstract:
This work deals with the numerical simulation of one and two dimensional two phase flows using a Lax-Wendroff finite volume scheme. An artificial viscosity term is used to handle the physical and numerical stiffness of the system. One shows that a good choice of this term can lead to valuable results, even on very refined meshes.

Non-Oscillatory Third Order Fluctuation Splitting Schemes
Paper 55, Code: S024
Authors: M.Hubbard

Abstract:
This paper addresses the issue of constructing non-oscillatory, higher than second order, multidimensional, fluctuation splitting methods on unstructured triangular meshes. It highlights the reasons why current approaches fail and proposes a potential solution to these problems. The results presented for a simple steady state scalar advection problem show significant improvements on previous methods.

Key words : fluctuation splitting, monotonicity, high order.

A Finite Volume two steps flux scheme for 1D and 2D non homogeneous systems

Authors:  Fayssal Benkhaldoun, Kamel Mohamed  and Laure Quivy
Paper 56, Code: S025
Abstract:
In this work we consider a two steps finite volume scheme, recently developed to solve nonhomogeneous systems. The first step of the scheme depends on a diffusion control parameter which we modulate, using the limiters theory. Results on Shallow water equations and two phase flows are presented.

A new Finite Volume formulation for diffusion problems in anisotropic non-homogeneous media

Paper 57
, Code: S026
Authors:  I.M. Nguena, A. Njifenjou

Abstract:
In this paper, we present a new finite volume method that displays a strong capacity to handle flow problems in anisotropic non-homogeneous porous media. The conservativity at cell level, the flux continuity and the consistency of the approximation of the flux are the main features of the proposed method.

Kinetic Modelisation of Tumor and Immune System Competition
Paper 58, Code: A016
Authors:  L. Derbel

Abstract:
In this work, we present mathematical models that describe the interactions and competition between tumors and the immune system. The evolution of the system may end up either with the blow-up of the host (with inhibition of the immune system), or with the suppression of the host due to the action of the immune system. We particularly develop the model introduced in [DeAngelis 01], where the capacity of the body to repair cells damaged is not taken into account. The idea is that for a long time range the body produces new cells to replace the dead ones in order to try to reach its normal healthy state. We are particularly interested by the asymptotic behavior o  solutions when time goes to infinity. We also give some numerical results that show that the model we introduce is suitable for chronic diseases.

3D finite volume approximations for stiff fluid mechanics problems
Paper 59, Code: C016
Authors: Gilbert Accary, Mohammed El Ganaoui , Isabelle Raspo , Rachid Bennacer

Abstract:
Two illustrations of Finite Volume (FV) use for predicting complex 3D flows are developed. The first concerns the unsteady Rayleigh-Bénard convection in a fluid close to its gas-liquid critical point. The second concerns the convection inside a fluid of a low Prandtl number heated from below. Such situation is particularly encountered during directional solidification process.

Hydrodynamic numerical simulation of the lake of Bouregreg using finite volume method
Paper 60, Code: C017
Authors:  A. Taik, H. Mouies, D. Ouazar

Abstract:
This work deals with the numerical simulation of unsteady circulation model in lake of Bouregreg in order to allow the delimitation of dead zones where the eutrophication phenomena may occur. We use the finite volume method with unstructured mesh. The cell-centred finite volume scheme considered here uses a generalised Roe's approximate Riemann solver with MUSCL technique for the conservative part and Green-Gauss type interpolation for the viscous part.

Modelling suspended load with moment equations and linear concentration profiles
Paper 61, Code: S027
Authors:  Khuat Duy B., Dewals B.J., Archambeau P. 1, Erpicum S., Detrembleur S., Pirotton M.

Abstract:
In numerical simulations, it is always necessary to find an optimum between the simplicity of the model and a good representation of real phenomena. In this context, the models using depth-averaged and moment equations are an interesting compromise between full 3D and simple depth-averaged models. This paper presents the use of a moment equation for suspended load transport. A simple but representative model for the sediment concentration profiles is developed. This original bi-linear concentration profile is compared to the traditional Rouse-profiles and shows a good correspondence despite its great simplicity. Advective and diffusive sediment fluxes are developed analytically and lead to a concise formulation, which is an asset for practical use. A differential equation for the sediment concentration moment is also fully developed, and a special attention is cast to the source term. The finite volume scheme has been chosen to implement the model, because it is particularly well suited for highly advective transport equations, it is conservative and it makes the choice of the upwinding easier. 1D simulations show the capacity of the model to reproduce laboratory experiments described in the literature.
Key words: sediment transport, suspended load, moment equations, depth-averaged, finite volume
Wiener-Ito Chaos expansions and FV solution of Stochastic Advection Equation
Paper 62, Code: S028
Authors:  Mostafa Zahri, Mohammed Seaıd, Hassan Manouzi, Mofdi El-Amrani

Abstract:
Stochastic uncertainties in advection flow problems can appear in the advection operator, domain boundary, or initial condition. In the present work, we propose a robust algorithm for approximating numerical solution of stochastic advection equations of Wick type. The method consists on combining the Wiener-Itˆo chaos expansions for random representation with a finite volume method for space discretization. The time integration is carried out using a second order I-stable explicit scheme. As applications, we consider linear free-advection and the nonlinear inviscid Burgers equations. Preliminary results, obtained for examples on stochastic problems in two space dimensions, show that the proposed method is efficient, stable and highly accurate.
Key words :
Wick-Stochastic Advection Problems, Wiener-Ito Chaos Expansions, Finite Volume Method, Stochastic Simulation, White Noise.

Multidimensional extension of the reservoir technique for linear advection equations
Paper 63, Code: S029
Authors: François Alouges, Gérard Le Coq, and Emmanuel Lorin

Abstract:
In this paper we present an extension of the reservoir technique for multidimensional advection equations with non constant velocities. The purpose is to make decrease the numerical diffusion correcting the numerical directions of propagation of the information, using a so-called corrector vector combined with the reservoirs.
Key words :
Reservoirs, finite volume schemes, numerical diffusion.

Axisymmetric finite volumes for the numerical simulation of CO2 transport and assimilation in a leaf
Paper 64, Code: S030Authors: Emily Gallouet and Raphaele Herbin
Abstract:
This paper deals with the numerical simulation of CO2 transport in the leaf. We study a mathematical model of the diffusion and photosynthezing processes, and present the implementation of an axisymmetric cell centered finite volume scheme for their numerical simulation. The resulting code enables the computation of the diffusion coefficient in the leaf porous medium, from experimental measurements which yield the pointwise value of internal CO2 concentration. Hence our model and numerical code allow the analysis of the role of the internal diffusion in the photosynthesis process.
Keywords: diffusion, photosynthesis, porous media, finite volumes.

A Finite volume simulation of a supersonic laminar flow: Application to a flat plate and compression corner model

Paper 65, Code: C018
Authors: Bahia Driss, Salhi Najimy, Boulerhcha Mohammed, and Elmahi  Imad
Abstract:
In this paper, the finite volume method is used for the computation of a supersonic laminar flow. As an application, we consider the flow over a flat plate and compression corner with different angles. The inviscid fluxes are approximated using Steger and Warming Flux Vector Splitting method with the MUSCL approach to increase the spatial accuracy. The transport properties (viscosity and conductivity) which are functions of two independent thermodynamic parameters (T,P), are calculated using a centered scheme. The performance of the methods used here is shown by some comparisons with other results.
Keywords: Navier-Stokes, Numerical analysis, Supersonic flow, Finite volume, Flat plate, Compression corner.

A diffuse interface model for the numerical simulation of three-component flows

Paper 66, Code: A017
Authors: Franck Boyer, Céline LapuertaAbstract:
In this paper we propose a new diffuse interface model for the study  of three-component incompressible viscous flows. The model is based on  the Cahn-Hilliard free energy approach. In particular, the free energy we  consider is built in such a way that the model is well-posed and consistent  with two-component models. The hydrodynamic of the mixture is taken into  account by coupling the Cahn-Hilliard system with a Navier-Stokes equation supplemented.
Keywords: Multicomponent flows, Cahn-Hilliard equations