Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol103.Issue1.1999:

Journal of Computational and Applied Mathematics 103 (1999) 187–205

Finite volume simulation of a droplet
ame ignition on
unstructured meshes
I. Elmahi a , F. Benkhaldoun b , R. Vilsmeier a; ∗ , O. Gloth a , A. Patschull a , D. Hanel a
a

LMI UPRES-A CNRS 6085, INSA de Rouen, France
b
IVG, University of Duisburg, Duisburg, Germany
Received 29 October 1999

Abstract
This work deals with the numerical simulation on an unstructured mesh of the ignition and burning of an isolated fuel
droplet modelled as a porous cylindrical wall. The reaction is assumed to be described by the equation A + B −→ P.
The complexity of the physical model considered, including multi-scale feature and the presence of sti propagating
fronts, justies manifold developments to overcome the diculties connected with corresponding simulations. Within this
report, two recently developed methods are employed and compared. Both methods rely on nite volume approaches, but
show distinct dierences in data-structures, choice of control volumes, meshing-strategy and even programming language.
c 1999 Elsevier Science B.V. All rights reserved.

Keywords: Droplet ignition; Finite volumes; Unstructured meshes; Roe; Gradient interpolation; Object-oriented numerics;
C ++

1. Introduction
The numerical simulation of combustion phenomena continues to be a subject of great interest.
Flame propagation processes play an essential role in most technical combustion devices, such as
combustion engines, conventional power plants or house heating. Studying such processes to understand the physical phenomena is a prerequisite for their optimization, thus having an important
impact on a reduced production of pollutants in a wide range of technological applications.
However, the simulation of combustion problems faces diculties of several origins. A rst problem is how to discretize and solve the full Navier–Stokes equations including reaction mechanisms
eciently. These equations contain a hyperbolic part related to transport phenomena, a parabolic part
due to diusion and viscosity as well as nonlinear and often sti source terms modelling chemical
reactions.
∗

Corresponding author.

c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 5 2 - 0

188

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

A second problem is how to deal with the time and space multi-scale feature of the physical
problem. The thinness of the reaction zone requires cells with very small scale lengths, at least in
the normal direction. Since computational resources are restricted, adaptive, specially anisotropically
adaptive meshing is very protable.
A third and throughgoing question, which is however not of major importance for the present
studies, is the formulation of schemes, able to treat reacting
ow problems with embedded shock
waves. The development and improvement of upwind solvers, limiter formulations and extension to
higher-order schemes is thus a continuous challenge.
According to the variety of diculties, many dierent methods and corresponding codes were developed. Within the present paper, two numerical codes arising from dierent origins of development
are presented and compared for the given physical problem. The aim is to evolve toward a unique
code beneting of the advantages of both approaches.

2. Mathematical model
Considering an homogeneous mixture of N species A1 ; : : : ; A

and assuming
that their combustion
PN
P
process can be reduced to a one-step overall chemical reaction Ni=1 i Ai −→ Ni=1 i Ai , the conservation equations can be expressed in the following general form:
@(Fnv + Fv )(W ) @(Gnv + Gv )(W )
@W
+
+
= S(W );
@t
@x
@y

(1)

where
W = (; u; v; E; Y1 ; : : : YN )t ;

t

Fnv (W ) = u; u2 + P; uv; (E + P)u; uY1 ; : : : ; uYN ;

t

Gnv (W ) = v; uv; v2 + P; (E + P)v; vY1 ; : : : ; vYN ;
@YN
@Y1
; : : : ; −D
0; −xx ; −yx ; −(uxx + vyx − qx ); −D
@x
@x

Fv (W ) =



Gv (W ) =



t

@Y1
@YN
0; −xy ; −yy ; −(uxy + vyy − qy ); −D
; : : : ; −D
@y
@y

;

t

;

S(W ) = (0; 0; 0; !
Q; M1 (1 − 1 ) !; : : : ; MN (N − N ) !)t :
In the above equations, ; u; v; P; E are the density of the mixture, velocity components, pressure and
total energy per unit volume, respectively. Yi and Mi represent the mass fraction and the molecular
mass of the species Ai .
Q indicates the heat of the chemical reaction at a reference temperature
T0 . The components of the viscous tensor are
@u

2
@v
;
xx =  2 −
3
@x @y




yy

@v
2
@u
=  2
;
−
3
@y @x





@u
@v
xy = yx = 
:
+
@y @x




I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

189

The components of the heat conduction
ux are given from Fourier’s law as

q = −∇T

with the thermal conductibility

=

Cp
;
Pr

where Pr is the Prandtl number and Cp the specic heat of the gaseous mixture at constant pressure.
We assume that the molecular diusion coecient is the same for all species which leads us to
admit that the diusion
ux of each species is given by the Fick’s law. The global reaction rate !
is assumed to be described by an Arrhenius law,
−Ta
! = B(T ) exp
T


Y

N 
 Yi i

Mi

i=1

;

where B represents an algebraic function of the temperature and Ta the activation temperature of the
reaction. Finally, for the problem to be closed, the equation of state has to be added to the above
set:
P=

N
X
i=1

Pi = RT

N
X
Yi
i=1

Mi

;

where Pi is the partial pressure of Ai and R the perfect gas constant.

3. Sketch of the algorithms employed
In the following the algorithmic approaches are referred to as alg1 and alg2. Both methods rely
on nite volumes for logically unstructured grids. Their main purposes may be described in brief as
in Table 1.
In the table, alg1 is especially designed for the physical problem described here, while alg2
corresponds to a multi-purpose development. Therefore, alg2 is still missing a reliable dynamic
adaption tool as oered by alg1.

In the further sections of this paper, discretization methods of both algorithms will be described in
more detail. However, before continuing, a short overview of the algorithmic structures is outlined.
3.1. The sequential algorithm, alg1
This code is especially designed for the physical problem described here and is thus highly optimized. As the vast majority of numerical methods for the simulation of complex physical problems,
it relies on a procedural programming language, that is FORTRAN. The major advantage of such
languages is the availability of very good compilers for Fortran as well as for C. Besides, Fortran codes are being developed since several decades and new versions are usually based on older
ones.
As usual for sequential programming, the code splits up in subroutines, performing the dierent
tasks. For the present system, we may summarize these as follows:
• Initial mesh generation. This task is fully decoupled from the rest of the system.

190

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205
Table 1
Basic issues of algorithms employed.

Discretization method
Control volumes
Element types
Dynamic adaptivity
Element orientation
eData structure (solve)
Projection
Time integration
Order in time
Acceleration
3-D extension
Programming
Programming language

∗1

: in development or unused property within the current paper

alg1:

alg2:

Finite volumes
Cell centered
Simplex
Yes, conservative
Isotropic
Elements
MUSCL, 2nd order
Expl. Runge Kutta
Second
None
Yesa
Sequential
FORTRAN

Finite volumes
Node centered
Hybrid
Noa
Anisotropic∗1a
Edges
MUSCL, second or quasi-thirda order
Expl. Runge Kutta or dual time-stepa
First or second
Multi-sequencea or multi-grida
Yesa
Objective
C++

• Advance in time. This part performs the discretization and integration for an interval.
• Mesh adaptation. This task performs a rearrangement by discrete operations.
The latter two items are interchangeable: that is a mesh rearrangement is performed in cycles after
a set of (or a single) time step(s).
To conclude, we may say that procedural languages are very useful for the solution of a specic
problem. However, the corresponding algorithms are less
exible. In practice, this results in large
numbers of program versions when solving dierent physical problems.
3.2. The objective algorithm, alg2
The aim of this development is the construction of a reliable platform for a variety of applications.
At the site of the authors these range from ship-hydrodynamics to the simulation of the structure of
detonation waves.
Dealing with the same grid types, data structures and iteration methods, a very high partition
of the code is independent from a concrete physical problem to be simulated. The advantage of
objective programming in this context is that it allows a very consequent dissection in modules for
logical and physical parts of the program.
The whole software is therefore split in three major programming levels:
• Basic storage classes. These classes include dynamic data handling. These objects are independent
of the kind of problems treated and thus do not even relate to the solution of dierential equations.
• Basic functionality. These objects include the solution concepts, mesh generators, handling of data
structures as well as some utility routines and hardware-dependent loop iterators.
• The physical level. Herein a user may allocate data and create the processing methods for his
specic application. All calls to procedures are virtual. Using a method of the basic functionality
suitable for its application does not require knowing much about these.
As an example, consider an explicit single time-step scheme to be performed on the discretized
equation (3). For this very simple task, the basic functionality provides the corresponding logic,
consisting of the computation of Res
and the advance in time. Some of the operations are automatic,

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

191

Fig. 1. Sketch of a simple explicit time step. P: pointwise, E: edgewise, A: automatic, V: virtual. Grey elds: dependent
on a specic application.

that is, regardless of the variables in use, the corresponding method is always available. For the
projection of variables from the nodes to the cell interfaces, a choice of automatic methods is
available. The user of the system may thus concentrate on the computation of the time step length
and the
ux-formulation for his application (Fig. 1).
The dotted arrows indicate virtual function calls. These require an additional amount of computational time. To minimize this overhead it is useful to provide a sucient amount of computational
work within the virtually called methods. Therefore, these methods contain loops for subsets of
points or edges, respectively. Since data are passed between subsequent virtual calls, it is not useful
to perform the inner loops over all points or edges, but only for moderate-sized groups, thus saving
memory.
The one-step time integration is the simplest version. However, all other explicit integration strategies are implemented in the same way. Therefore, once a
ux and a time-step calculation for a
physical problem is formulated, other more complex integration methods are accessible as well.

4. Finite volume schemes and discretizations
4.1. The nite volume scheme
Integration of system (1) over a control volume Vi and Gauss divergence formulas leads to
Z

Vi

@W
+
@t

Z

Fnv (W; n) d +
@Vi

Z

Fv (W; n) d =
@Vi

Z

S(W ) dV;
Vi

where
Fnv (W; n) = nx Fnv (W ) + ny Gnv (W );
Fv (W; n) = nx Fv (W ) + ny Gv (W );
(nx ; ny ) are the components of the outward unit normal to @Vi .

(2)

192

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 2. Cell–centred control volume.

The discrete form of the above equations for a control volume Vd reads

W 
+ Res
;Vd = 0:

t Vd


(3)

Since unstructured grids do not oer a natural ordering of nodes and elements, an articial ordering, i.e., a data structure is required. Further on an element of the data structure will be called
a molecule. To perform a time step for the equation given above, the discrete residual Res
;Vd has
to be constructed upon such molecules. Introducing the discrete
ux functions  and v for the
nonviscous and viscous
uxes yields
Res
;Vd =

nr(Vd)
1 X
( + v )j(˜ j) nj(˜ j)
Aj(˜ j) − SVd ;
VVd j=1

(4)

where nr(Vd) is the number of molecules contributing to the residual, VVd is the corresponding
volume, nj(˜ j) represents the outward normal vector and
Aj(˜ j) the area of the control interface
˜
supported by a molecule j(j).
alg1 employs cell-centred control volumes. Corresponding molecules are the common edges between two neighbouring cells and the normal vectors are the normals ni(i)
to these
˜
interfaces (Fig. 2).
By way of contrast, alg2 employs a nodal arrangement of nonoverlapping control volumes, requiring the denition of polyhedral regions and corresponding cell interfaces between neighbouring
nodes (Fig. 3).

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

193

Fig. 3. Decomposition of a 3-D control volume and edge-based molecule.

A mixed discrete-analogue edge-based data structure is employed. For each edge of the mesh
the addresses of the two ending nodes K1; K2 and the control interface area-normal vectors n
A
between the control volumes of both nodes are stored.
For a
ux evaluation at a cell interface, a projection of the variables stored at the cells (alg1),
respectively nodes (alg2), onto the corresponding cell interfaces is required. The formal spatial order
of the algorithm is determined by this projection. Both algorithms employ MUSCL projection.
For the cell-centred approach of alg1, the cell gradients are evaluated by minimizing the quadratic
functional
LVd (X; Y ) =

X

2

|WVd + (xk − xVd ) X + (yk − yVd ) Y − Wk | ;

k∈K(Vd)

where K (Vd) is the set of indices of neighbourhood cells that have a common edge or vertex with
the control volume Vd, (xVd ; yVd ) and (xk ; yk ) are respectively the barycentre coordinates of cells Vd
and Vk .
Nodal gradients of the conservative variables, required for the projections to be performed for
alg2, are obtained by a Green–Gauss-type integration at the control volumes themselves:
∇WVd

nr(Vd)
1 X
W ˜ n ˜
A ˜ :
=
VVd j=1 j( j) j( j) j( j)

(5)

At present, both schemes rely on projection limiters. However, concerning the specic application
here, these play no essential role, since solutions are continuous (Fig. 4).

194

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 4. Decomposition of a 2-D control volume and edge-based molecule.

4.2. Flux formulations for the hyperbolic part
For the present simulations, both schemes employ Roe’s approximate Riemann
ux function.
Consider the hyperbolic part of Eqs. (1):
Wt + Fnv (W )x + Gnv (W )y = 0

(6)

classically called the Euler equations.
The hyperbolic contribution to the residual, requires the evaluation of the term:
Z

Fnv (W; n) d
ij

or discretized:

nr(Vd)
1 X
 ˜ n ˜
A ˜ ;
VVd j=1 j( j) j( j) j( j)

where j(˜ j) is calculated upon the left- and right-hand side projected variables Wl and Wr . Rather
˜
than regarding an edge j(j),
we will thus refer to the left and right sides of the cell interface l and
r. Roe proposed the following particular choice of (Wl ; Wr ) [6]:
(Wl ; Wr ) · nlr =

1
[Fnv (Wl ; nlr ) + Fnv (Wr ; nlr )]
2

1 

− A(W˜ ; nlr ) (Wr − Wl );
2

(7)

where A(W˜ ; nlr ) is an average Jacobian matrix constructed to satisfy the following properties, which
traduce respectively the consistency with jump conditions, positiveness and convective
ux denition,
(i) Fnv (Wr ; nlr ) − Fnv (Wl ; nlr ) = A(W˜ ; nlr ) (Wr − Wl );
(ii) A(W˜ ; n) is diagonalisable with real eigenvalues,
(iii) (W; W ) · n = Fnv (W; n):

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

195

4.2.1. Simplied Roe scheme
Remembering that |A| = A+ − A− , the conservativity relation (i) allows a simplication of the
numerical
ux form which can be written in the following two ways:
(Wl ; Wr ) · nlr = Fnv (Wr ; nlr ) − A+ (W˜ ; nlr )(Wr − Wl );

(8)

(Wl ; Wr ) · nlr = Fnv (Wl ; nlr ) + A− (W˜ ; nlr )(Wr − Wl ):

(9)

The great advantage of this formulation, specially in the context of multi-species
ows, is the
low computational eort. The reason for this fact is that at most one eigenvalue has a sign dierent
to all others. Since A+ contains all positive and A− all negative eigenvalues, one of the two terms
A+ (W˜ ; nlr )(Wr − Wl ) or A− (W˜ ; nlr )(Wr − Wl ) contains at most one eigenvalue, thus becoming very
simple to evaluate. For details we may refer here to [2].
4.2.2. Entropy correction
One disadvantage of Roe’s linearization is that the resulting approximate Riemann solution consists
of only discontinuities which can lead to nonentropic shocks. In this case, entropic modication is
required. The approach used here due to Harten and Hyman [4] consists of introducing a rarefaction
wave wherever the Roe scheme produces nonadmissible shocks. A correction is required, if the left
and right sonic eigenvalues of same type have dierent signs, while representing a rarefaction:
l = (Wl ) ¡ 0 ¡ r = (Wr ):
This situation may correspond to a nonphysical shock. To overcome the problem, the eigenvalues
are modied. Depending on the simplication chosen, one of the following modications are required:
∗

˜ − l
r − l

!

if Eq. (8) is used,

∗

r − ˜
r −  l

!

if Eq. (9) is used.

 = r

 = l

4.3. Formulation of diusive terms
The discretization of the diusive
ux contributions in the context of nite volume methods require
the discretization of terms of the following type:
Z

ij



a′

@b
@a
nx + b′ ny
@x
@y



d;

(10)

where, in the context of the present paper, a and b represent either u, v, T , Y1 · · · or YN and a′ ,
respectively b′ represent either u, v, P, Y1 · · · or YN .
The discretization of these diusive
uxes on unstructured meshes is still a challenging problem.
Many theoretical investigations and mathematical analysis have been performed in this way. Vignal
et al. [7] have used and analysed a four-points cell-centred nite volume scheme to discretize the
elliptic part of a coupled elliptic-hyperbolic system. The convergence of the approximate solution

196

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 5. Diamond shaped co-volume.

towards the entropy solution was proved under some stability condition and some requirements on
the mesh: the angles of each triangle must be lower than 12 .
The algorithms presented here use two-step formulations to compute the
ux contribution. In a
rst step, the gradients required at the cell interfaces are computed. In a second step an ordinary
nite volume type discretization is performed. For the computation of gradients, Green–Gauss-type
integrations are employed for both algorithms.
4.3.1. Diamond shaped co-volume
The cell-centred approach of alg1 uses an edge centred co-volume, as sketched in Fig. 5. The
gradients, which are supposed to be constants on the co-volume Cdec , are computed upon the values
of the data on the nodes Gi , Gj , N , and S as follows:
@a 
@x 


≃
ij

Z
X 1
1
nx ” d;
(aN1 + aN2 )
meas(Cdec ) ”∈@C 2
”
dec

where N1 and N2 are the nodes of an edge ” of @Cdec . aN1 and aN2 are respectively the values of the
state a on the node N1 and N2 .
To obtain the values at a node N of the mesh, a specic linear interpolation based on the set of
cells sharing the vertex N is employed, ensuring weak consistency of the scheme (see [3]).
4.3.2. Direct nite volume method
Since already required for the higher-order projections, alg2 provides a set of conservative gradients at the nodes, as computed in Eq. (5). These gradients are thus reused for the formulation of
viscous
uxes.
In a rst step, average gradients are obtained at the cell interfaces. Following the left–right notation,
where the states l and r correspond to the nodes at both the ends of the edge regarded, the preliminary
gradient reads:
∇W˜ lr = 12 (∇Wl + ∇Wr ):

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 6. alg1: reaction rate isolines of the mixture at dierent time steps t1 ; t2 ; t3 :

197

198

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 6. Continued

Since in the direction of the edge a central formulation of the gradients is possible, the preliminary
conservative gradient may be corrected in order to reduce the range of in
uence:
∇Wlr = ∇W˜ lr +



Wr − W1
− ∇W˜ lr · mlr mlr ;
Llr


where Llr is the length of the corresponding edge, mlr the unit vector aligned with.
The primitive gradients, required for the evaluation of the terms 10, are then computed at the
cell interface employing dierentiation rules. For example, the gradient of the speed u in x-direction
reads:


@u
1 @(u)
@
=
−u
:
@x

@x
@x
5. Mesh adaption
The phenomena under study exhibits a high degree of stiness due to the very dierent space and
time scales. This stiness appears essentially through thinness of the reaction zone. So, to simulate
the ignition of the droplet accurately enough and hence to obtain a correct physical behaviour, a mesh
adaptation method is needed during the calculation. Here, we present brie
y an adaptive procedure
based on multi-level renement and de-renement, aimed at constructing an adaptive mesh which
dynamically follows the
ame front during its motion around the droplet. For more details about
this technique, the reader is referred to Benkhaldoun et al. [1] and Maman [5].

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 7. alg1: adaptive mesh at dierent time steps t1 ; t3 :

199

200

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 8. alg2: reaction rate isolines of the mixture at dierent time steps t1 ; t2 ; t3 :

Following the local values of some criterion, dened in our case by the reaction rate of the
mixture, one establishes a list S of triangles which need to be rened and those to be unrened,
by lling, for all the macro-element K, an integer array called IADIV. At time t n = n
t and for a
macro-element K; IADIV(K) = m means that K has to be divided into 4m triangles, or equivalently

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 9. alg1: mass fraction isolines of fuel at time t3 :

Fig. 10. alg1: mass fraction isolines of oxidizer at time t3 :

201

202

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 11. alg2: mass fraction isolines of the fuel at time t3 :

that K has to be divided into 4 sons, and that each of these sons has to be divided into 4m−1
sub-triangles. The algorithm is then based upon a multi level hierarchical tree data structure.

6. Numerical experiments
For the comparison of the two algorithms described above, the simulation of the ignition and
burning of a fuel droplet is considered. Therefore a simplied combustion model was used consisting
of three species, including a one-step overall chemical reaction A + B −→ P. The fuel indicated by
the symbol A reacts with the oxidizer B to form a new species P. The density of the mixture, initially
consisting only of the oxidizer (YF (x; y; 0) = 1, YO (x; y; 0) = 0), was assumed to be (x; y; 0) = 1.
The
ow was initially at rest having a temperature of T (x; y; 0) = 300 K. On the droplet surface the
mass fraction of the fuel was set to YF;drop = 1 so that YO;drop = 0.
The leakage of the fuel through the droplet surface is obtained by relating the conditions specied
on the droplet surface to the maximum pressure Pmax inside the computational domain. Initiated by a
temperature of T = 1100 K at the right boundary of the domain the mixture was ignited. For this planar wall the Neumann condition has been applied to the mass fractions @YF =@n = @YO =@n = 0. On the
boundaries at the top and the bottom of the domain, the velocities were assumed to be u = 0 ; v = 0.
The values of constants taken are as follows:
= 1:4; Cv = 717 J kg−1 K −1 ; 0 = 0:03 kg m−1 s−1 ;
C0 = 110:4; Pr = 0:5; Sc = 1 K = 1:5 · 1013 s−1 ; Ta = 1 · 104 K;
Q = 4 · 107 .
The results of the simulations are shown in Figs. 6–14.

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 12. alg2: mass fraction isolines of the oxidizer at time t3 :

Fig. 13. Temperature isolines for alg1 at time t3 :

203

204

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

Fig. 14. Temperature isolines for alg2 at time t3 :

7. Conclusion
Two dierent methods, developed at dierent sites, have been studied to simulate the problem of
a droplet ignition. Both codes rely on nite volume approaches and employ a simplied version of
Roe’s approximate Riemann solver. Beside these similarities, both methods dier signicantly. It is
thus interesting to note that the results of both methods are in good agreement, although they were
not even applied on the same computational mesh.
Upon further development we plan to combine the advantages of the dierent methods. Concerning
the discretization and meshing techniques, a 3-D extension of the methods is a challenging problem.
Adaptive or not, both meshing concepts employed for the present studies where isotropic. Due to
the high computational complexity, such concepts are not useful in 3-D. Anisotropic concepts, by
way of contrast, most often result in additional diculties for the discretization of viscous terms,
crucial for the problem presented.

References
[1] F. Benkhaldoun, T. Fernandez, B. Larrouturou, P. Leyland, A dynamical adaptive method based on local renement
and unrenement for triangular nite-element meshes : Preliminary results, Rapport de Recherche no. 1271, INRIA,
1990.
[2] T. Buard, J.M. Herard, Methodes de resolution pour les systemes hyperboliques en forme conservative sur maillage
non structure, Rapport EDF, HE-41=90:43, 1990.
[3] Y. Coudiere, J.P. Vila, P. Villedieu, Convergence of a nite volume scheme for a diusion problem, Proc. 1st Internat.
Symp. on Finite Volumes for Complex Applications, Rouen, 1996, pp. 161–168.

I. Elmahi et al. / Journal of Computational and Applied Mathematics 103 (1999) 187–205

205

[4] A. Harten, J.M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comp.
Phys. 50 (1983) pp. 235–269.
[5] N. Maman, Algorithmes d’adaptation dynamique de maillages en e lement nis. Application a des e coulements reactifs
instationnaires, These, Universite Paul Sabatier, Toulouse, 1991.
[6] P.L. Roe, Approximate Riemann solvers, parameter vectors, and dierence scheme, J. Comp. Phys. 43 (1981) 357–372.
[7] M.H. Vignal, Convergence of a nite volume scheme for an elliptic-hyperbolic system, M2AN, submitted.