Lecture Notes inMathematicsEdited by A. Dold and B. Eckmann

1270

C. Carasso P-A. Raviart D. Serre (Eds.)

Nonlinear Hyperbolic ProblemsProceedings of an Advanced Research Workshopheld in St. Etienne, FranceJanuary 13-17, 1986

Springer-VerlagBerlin Heidelberg NewYork London Paris Tokyo

Editors

Claude CarassoDenis SerreAnalyse Numerique,23 rue du Dr. Paul Michelon

42023 Saint Etienne Cedex 2, France

Pierre-Arnaud RaviartUniversite Pierre et Marie Curie

Analyse Nurnerique4, place Jussieu75230 Paris Cedex 05, France

Mathematics Subject Classification (1980): 35L, 65M, 76N

ISBN 3-540-18200-4 Springer-Verlag Berlin Heidelberg New YorkISBN 0-387-18200-4 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the materialis concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation,broedcastinqreproduction on microfilms or in other ways, and storage in data banks. Duplicationof this publication or parts thereof is only permitted under the provisions of the German CopyrightLaw of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always bepaid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1987Printed in Germany

Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr.2146/3140-543210

CONTENTS

I Numerical Analysis

a -General theory.

A.Bourgeade, Ph.Le Floch, P.A.Raviart: Approximate solution of the generalized

Riemann problem and applications.

B.Engquist: Computation of oscillatory solutions to partial differential equations.

A.Harten: Preliminary results on the extension of ENO schemes to two-dimensional

problems.

P.L.Roe: Upwind differencing schemes for hyperbolic conservation laws with source

terms. -

E.Tadmor: The entropy dissipation by numerical viscosity in nonlinear conservative

difference schemes.

b Main applications.

V.Billey, J.Periaux, P.Perrier, B.Stoumet: 2-D and 3-D Euler computations

with [mite element methods in aerodynamics.

J-J.Chattot, S.Malet: A "box-scheme" for the Euler equations.

J-F.Colombeau, A-Y.Leroux: Numerical techniques in elastoplasticity.

V.Daru, A.Lerat: An implicit centered scheme which gives non-oscillatory steady

shocks.

y.I.Shokin: A computer aided system for investigation and construction of difference

schemes of gas dynamics.

10

23

41

52

64

82

103

115

128

135

163

150

152

II fuperbolic eD E theory

a - Surveys, open questions.

S.K.Godunov: Lois de conservation et integrales d'energie des equations

hyperboliques.

P.D.Lax: On symmetrizing hyperbolic differential equations.

B-L.Keyfitz: A sUIVey of nonstrictly hyperbolic conservation laws.

M.Slemrod: Admissibility criteria for phase boundaries.

D.H.Wagner: The transformation from Eulerian to Lagrangian coordinates for solutions

with discontinuities. 172

b Existence theory.

D.Hoff: Two existence theorems for systems of conservation laws with dissipation. 181

IV

210

221

195

286

238

253

265

Li Ta Tsien: Global solutions for some free boundary value problems for quasilinear

hyperbolic systems and applications.

M.Rascle: Un theoreme d'existence globale en elasticite non lineaire

mono-dimensionnelle.

Wang Jing Una: A nonhomogeneous system of equations of nonisentropic gas

dynamics.

c -Qualitative propertiesofsolutions.

B.Gnstafsson: Far field boundary conditions for steady state solutions to hyperbolic

systems.

M.Schatzman: Can hyperbolic systems of conservation laws be well-posed in BV?

D.Serre: Propagation des oscillations dans les systemes hyperboliques non lineaires,

B.Temple: Stability and decay in systems of conservation laws.

Ill- MisceUaneousproblems

C.Bardos: Different approaches for the relations between the kinetic and the macroscopic

equations. 308

J.M.Greenberg: Integral transport processes. 324

PREFACE

The International Conference on Non-linear Hyperbolic Problemswas held in Saint-Etienne on January 13-17, 1986. Our aim was to get anoverview of a branch of applied mathematics which has been expanding veryfast over the last few years. As we all know applications are possible inmany fields (aerodynamics, multifluid flows, combustion, detonics...), butthis involves difficulties of a theoretical as well as a numerical nature. It istherefore not surprising that, in the end, papers dealing with theoreticalproblems raised by the resolution of non-linear hyperbolic systemsoutnumbered those which concentrated on numerical methods.

The number of participants, the diversity of the teams,researchers and countries represented do not leave any doubt as to theopportuneness of such a conference. It was particularly useful in as much asit produced the original papers contained in this volume and also because ithelped establish conditions of scientific cooperation between researchteams. We are confident that the wish expressed by many participants to seethe conference become a regular fixture will be fulfilled.

The success of such an undertaking requires the help of manypeople and organizations. We are particularly gratefUl to :

Ie Centre National de la Recherche Scientifique (CNRS),the National Science Foundation (NSF),la Direction des Recherches et Etudes Techniques (DRET, contratn085/1392),the European Research Office of the United States Army,l'Acadernie des Sciences de Chine,Ie Commissariat a l'Energie Atomique (CEA),Ie Centre National d'Etudes Spatiales (CNES),la Societe de MathSmatiques Appliquees et Industrielles (SMAI),la Societe MathSmatique de France (SMF),les collectlvites territoriales (Mairie and Conseil general),

for their precious support.We also wish to thank Nohra RAHMANI and HelEme CHABREUIL who

greatly contributed to the success of the conference by attending to itspractical organization.

Claude CARASSO, Pierre-Arnaud RAVIART, Denis SERRE

PREFACE

Le Conqres International sur les Problemes Hyperboliques Nonl.ineaires s'est deroule a Saint-Etienne du 13 au 17 janvier 1986. Noussouhaitions, par cette rencontre, faire Ie point sur un secteur desMathernatiques Appllquses qui s'est considerabtement devetoppe cesdernleres annees. Les domaines d'application sont en eftet multiples(aerodynarnique, ecoulements multiphases, combustion, detonique...) maisegalement difficiles, tant du point de vue theorique que nurnerique. II n'estd'ailleurs pas etonnant que les communications sur les problernes theoriquessoulevss par la resolution des systsmes hyperboliques non lineaires alent, endefinitive, ete plus nombreuses que celles portant sur les methodesnumerlques.

Le nombre de participants, la variate des equipes, des chercheurset des pays representee montrent a l'evidence qu' une telle rencontre etaitnecessaire. Elle a ete utile d'une part par les contributions originalescontenues dans ce volume, et d'autre part par les liens de cooperationscientifique qui ont pu s'etablir entre les equipes de chercheurs. Noussommes certains que Ie souhait des participants d'une rencontre periodiquesur Ie rnsme theme pourra etre exauce.

La realisation d'une telle rencontre necesslte des aides multiples,qu'elles soient rnatertelles ou humaines. Nous remercions particulierernent:

Ie Centre National de la Recherche Scientifique (CNRS),the National Science Foundation (NSF),la Direction des Recherches et Etudes Techniques (DRET, contratn085/1392),the European Research Office of the United States Army,l'Acadernie des Sciences de Chine,Ie Commissariat a l'Energie Atomique (CEA),Ie Centre National d'Etudes Spatiales (CNES),la Societe de MatMmatiques Appliquees et Industrielles (SMAI),la Societe Mathernatique de France (SMF),les collectivites territoriales (Mairie et Conseil general),

pour leur aide, ainsi que Nohra RAHMANI et Helene CHABREUIL qui ont assureIe succes materiel du Conqres.

Claude CARASSO, Pierre-Arnaud RAVIART, Denis SERRE

P.S. Ce Workshop etait classs U.S.-French conference, ainsi queJournees SMF. Le present document a ete etabli en execution du contratn°85/1392 passe par la direction des recherches et etudes techniques,direction scientifique, section soutien a la recherche (Delegation generalepour I'armement).

Liste des participants

AFIF Mohamed Universite Kadiayad, faculte des sciences,Marrakech, Maroc

ALVES A.S Lab. de rnecanlque theorique, Univ. P. et M.Curie, 4 place Jussieu, 75230 Paris Cedex

AMIRAT Youcef INRIA, Domaine de Rocquencourt, BP 105,78135 Le Chesnay

AROSIO Alberto Dipart. Matematica, V. Buonaroti 2, 56100Pisa,ltalie

ARRIGO Jean-Luc E. Polytechnique FeQerale de Lausanne,DMACite Universitaire, CH-1015 Lausanne

AUBRY D. E.Centrale des art et manufactures, 92295Chatenay Malabry Cedex

AUDOUNET Jacques Universite P.Sabatier, 118 route de Narbonne,31062 Toulouse Cedex

AYADI Abdelhamid Inst. de Math., Univ. de Constantine, Algerie

BALABANE Mikhael CMA, Ecole Normale Superieure, 45 rue d'Ulm,75230 Paris Cedex

BALLMAN Joseph RWTH Aachen, Templergraben 64 0-5100Aachen

BARAS Pierre TIM 3, BP 68, 38402 St-Martin d'Heres

BARDOS Claude CMA, Ecole Normale Superieure, 45 rue d'Ulm75005 Paris

BATE-AYE Hans Universite de Yaounde, Dept. de Maths., Fac.des Sciences, BP 812, Yaounde Cameroun

BENABDALLAH Assia CMA, Ecole Normale Sup., 45 rue d'Ulm, 75230Paris Cedex 05

BENBOUCHTA Adellatif Univ. Scientifique et Medicale de Grenoble,Lab. de Physique des Plasmas, 38402St-Martin d'Heres

BERNARD Michele CEAlCadarache DRELSTRE 13108 St-Palu lezDurane

BERNARDI Christine Lab. d'Analyse Numerique, Tour 55-65, 5eetage, 4 place Jussieu, 75252 Paris

BESNAULT DRET, 26 Bd Victor 75996, Paris ArmeesBIGOT Jean-Yves Lab. de Spectroscopie et d'optique du corps

solide, 5 r. de l'Universite, 67084 StrasbourgCedex

BONNEFILLE Max Analyse Nurnerique, 23 rue Dr Paul Michelon,42023 St-Etienne Cedex 2

I3OlJCH()\J Helene CEA, BP 7, 77181 Courtry

IX

BOlJRDEL ONERA-CERT, 2 Av. Edouard Belin Toulouse

BOURGEAT Alain Analyse Numerique, 23 rue du Dr PaulMichelon, 42023 St-Etienne Cedex

BOUZIT Mohamed Univ. de Constantine, Inst. de Maths Algerie

BRAUNER CI-Michel Departement M.I.S., Ecole Centrale de Lyon,BP 163, 69131 ECULLY Cedex

BRENIER Yann INRIA, Domaine de Rocquencourt, BP 105,78135 Le Chesnay

BRUNEAU Ch-Henri Universite Paris Sud, Bat.425, 91405 OrsayCedex

CABANNES Henri Universite de Paris 6, Mecanique Theorique,4 place Jussieu 75005 Paris

CALLEJA Pierre C.E.A, BP 7,77181 Courtry

CARASSO Claude Analyse Numerlque, 23 rue Dr Paul Michelon42023 St-Etienne Cedex 2

CHALABI Abdallah TIM 3, BP 68,38402 St-Martin d'Heres

CHAMBAT Michele UER Math de Lyon 1, 69622 VilleurbanneCedex

CHARNAY Michel INSA Lyon, Centre de Maths., 20 Av. A.Einstein,69621 Villeurbanne Cedex

CHARRIER Pierre UER Maths. Info., Univ. Bordeaux 1, Talence33405

CHATTOT J-Jacques MATRA BP 1 78146 Velizy-VillacoublayCedex

DARU Virginie ENSAM, 151 Bd de l'h6pital, 75013 Paris

DEGOND Pierre CMAP, Ecole Polytechnique, 91128 PalaiseauCedex

DELANNOY Guy SNPE/Centre de Recherches du Bouchet, BP 2,91710 Vert Ie Petit

DELORME Philippe Direction de l'Energetique (DE) ONERA BP 7292322 Chatillon Cedex

CO;1INGUEZ Jesus HISPANOIL C/PEZ Volador, 28007 MadridDE LA BASILLA EspagneDOUGLASJr Jim Univ. of Chicago Mathematics 5734 University

Av., Chicago IL 60637 USADUBOIS Francols CMAP, Ecole Polytechnique, 91118 Palaiseau

CedexDUBROCA B. CEA-CESTA, BP 2 Le Barp, 33830

Belin-BelietDURAND Gilbert DRET/SDRlG61-26 Bd Victor 75996 Paris

Armees

x

EINFELT Bernd RWTH Aachen, Templergraben 55,D-5100Aachen

ELDABAGHJ Fadi INRJA, Bat 16, 78153 Le Chesnay

ENDRE Suli Oxford Univ. Computing Lab. 8-11 Keble RoadOxford OX1 30D Angleterre

ERIKSSON KENNETH Department of Maths. Chalmers Univ. of Tech.S-41296 Goteborg Suede

FORESTIER Alain CEA, Saclay 911191 Gif sur Yvette Cedex

FREISTHULER H. RWTH Aachen, Templergraben 55, D-5100Aachen

FRESNR Christophe Univ. de Bordeaux 1, UER de Maths. 351 coursde La Liberation 33405 Talence

GALEONE Luciano Universita, Dipartimento de Matematica ViaNicolai 2 70100 Bari Italie

GALLOUET Thierry Maths. Univ., 73000 Oharnbery

GAUDY Catherine CEA, Saclay 91191 Gif sur Yvette

GILOUIN Herve Analyse Nurnerique, 23 rue du Dr PaulMichelon, 42023 St Etienne Cedex 2

GODOUI\OV SK Inst. of Math., Siberian Div. ofthe Academy ofSciences, 630090 Novossibirsk 90 USSR

GOUTAL Nicole. 6 quai Watier 78400 Chatou

GREENBERG James M. University of Maryland, Baltimore CountryCatonsville, Maryland 21228 USA

GRUNPAHN Klaus RWTH Aachen, Templergraben 64, D-5100Aachen

GUSTAFSSON Bertil Univ. of Uppsala, Compo Dept. , Sturegatan 4B,Uppsala, Suede

HALPERN Laurence CMAP, Ecole Polytechnique, 91128 PalaiseauCedex

HAMDACHE Kamel CNRS et ENSTA-GHN, Centre de I'Yvette,Chemin de la Huniere, 91120 Palaiseau

HANOU2ET Bernard Univ. de Bordeaux 1, UER de Math. et Info.,351cours de la Liberation, 33405 Talence Cedex

HARTEN Ami University ot California, Dpt of Mathematics,Los Angeles 90024 USA

HENRY Jean-Joseph DOWELL-SCHLUMBERGER 2.1.Molina laChazotte 42000 Saint Etienne

HERVOUET Jean-Michel EDF/AEE/LNH, 6 quai Watier, 78400 Chatou

HIRSCH C. Wrije Univ. Bruxelles, Depart. FluidMechanics, Plein land 2, Bruxelles, Belgique

XI

HOFF David Indiana Univ. Dept of Math.,Bloomington,lndiana 47405 USA

JETSCH RWTH Aachen, Templergraben 55, D-5100

JOLY Jean-Luc Univ. de Bordeaux 1, UER de Math. et Info., 351Cours de la Liberation, 33405 Talence Cedex

JOHNSON Claes Maths. Dept., Chalmers Univ. of Technology,41296 Goteborq, Suede

JOSTEIN Alvestad STATOIL, Varlikroken 25N-4 000 StavangerNorveqe

KEYFITZ Barbara Maths. ,Univ. of Houston, Univ. Park, Houston,TX77004 USA

KLEIN Rupert Inst. fUr Allgemeine Mechanik, Templergraben55, D-5100 Aachen

KLINGENBERG Christian Dept. of Applied Maths., Univ. of Heidelberg,1m Neuenheimer feld 294, Heidelberg, FRG

LACAS Francols Labo. EM2C, Ecole Centrale de Paris, Gde voledes Vignes, 92295 Chatenay-Malabry Cedex

LAMARQUE CI-Henri DRET, Section Formation, Bureau du Personnel26 Bd Victor 75996 Paris Armees

LARGILLIER Alain Analyse Nurnerique, 23 rue du Dr PaulMichelon, 42023 Saint Etienne Cedex 2

LATROBE Andre Info. Intern, 57 rue pierre Semard, 38000Grenoble

LAYEC Yves Dept. G. M. P. I I. U. T. de Brest, rue de laGrandiere, 29287 Brest Cedex

LECOQ Gerard EDF/DER/PHR, 1 avenue du General De Gaulle,91141 Clamart Cedex

LEFLOCH Philippe Ecole Polytechnique, CMAP, 91128 Palaiseau

LEQUESNE Paul 6 Quai Watier, 78400 Chatou

LENNART Egnesund Uppsala Universitat, Formansgatan 17,S-72466 Vasteras Suede

LERAT Alain Ecole Nationale Sup. des Arts et Metiers, 151Bd. de I'hopital, 75640 Paris Cedex

LEROUX Alain-Yves Univ. de Bordeaux 1, Maths et Infor, 351 Coursde la Liberation 33400 Talence

LESAINT Pierre Lab. d'Analyse Nurnerique, Faculte desSciences, Route de Gray, 25030 Besancon

LEVEQUE Randal Dpt of Maths., Univ. of Washington, Seattle,WHA 98195 USA

LINDBERG Hans Dept. of Computer Science, Sturegatan 4B,S-75234 Uppsala Suede

L1U Tai Ping Univ. of Maryland, Dept. of Maths., CollegePark, MD 20742 USA

LOCHAK Pierre

LOESENER Charles

MADAY Yvon

MAGGI-VIDELAINE Pascale

MALET Sylvie

MARX Yves

MAS-GALLIC Sylvie

MASLOV V.P.

MAZET Pierre-Alain

MILANI Albert

MORICE Phlippe

MUNZ CI-Dieter

NATALINI Roberto

NAPOLITANO M.

NEDELEC Jean-Claude

NEGRO Angelo

ORAN Elaine

OSHER Stanley

PART Eva

PAUMIER Jean-Claude

PERIAUX Jacques

PERTHAME Benoit

XII

CMA, Ecole Normale Sup., 45 rue d'Ulm, 75230Paris Cedex 05BRGM, BP 6009,45060 Orleans Cedex 2

Lab. d'Analyse Nurnerique, Tour 55-65,5e etage, 4 pI. Jussieu, 75252 Paris Cedex 05USMG, Lab. de physique des Plasmas, DomaineUniversitaire, BP 68, 38402 St-Martind'HeresMATRA, BP 1, 78146 Velizy-VillacoublayCedexE.N.S.M. Nantes, 1 rue de la Noe, 44072 Nantes

CNRS Lab. d'Analyse Numerique, Tour 55-65,4 place Jussieu, 75252 Paris Cedex 2Moscow Institute of Electronic, Machinebuilding B. Vusovsky 3/12 Moscou USSRONERA-CERT, 2 avenue Edouard Belin, 32000ToulouseDipartimento di Matematica, Universita diTorino, V. Principe Amadeo, 8 10123 TorinoOff. Nat. d'Etudes et de RecherchesAerospateles, BP 72, 92322 Chatillon CedexMathematisches Institut II, unlversttatKarlsruhe, Englerstr. 2, D-7500 Karlsruhe 1Math. et Info. , 351 Cours de la Liberation,Univ. de Bordeaux 1 ,33405 Talence CedexInst. Macchine, Univ. di Bari, Via re DavidAdo, 0125 Bari, ItalieCMAP, Ecole Polytechnique, 91128 PalaiseauCedexDipar. di Mat. Univ. dl Torino, V. PrincipeAmadeo 8, 10123 Torino ItalieLabo. for Computer Physics Naval ResearchLabo code 4040 D.C. 20375 USAMaths., UCLA Los Angeles, CA 90024 USA

Dept. of Computer Sci., Uppsala Univ.,Sturegatan 4B, 2 TR S-752 23 Uppsala SuedeTIM 3, BP 68, 38402 Saint Martin d'Heres

Avions M. DassauIVBAlDEA, 78 Quai Carnot,92214 Saint CloudCMA, Ecole Normale Superieure, 45 rue d'Ulm,75230 Paris Cedex 05

XIII

PFERTZEL Agnes 40 rue de la Sabliere, 75014 Paris

PHAM NGOC-DINH 19 rue Erard, 75012 Paris

PIRONNEAU Olivier Univ. Paris 6, Analyse Numerique, 4 placeJussieu, 75252 Paris

PIROZZI Antonietta Instituto di Maths., Fac Di Scienze Nautiche,Via Acton 38, Napoli Italie

PCX3U Marc I.N.SA de Rennes, 20 avenue des Buttes deCoesmes, 35043 Rennes Cedex

POINSARD Thierry CNET, 38-40 rue de General Leclerc, 92131Issy les Moulineaux

POINSOT Thierry CNRS Labo de thermique, Ecole Centrale deParis 92290 Chatenay Malabry

RASCLE Michel Analyse Numerique, 23 rue du Dr PaulMichelon, 42023 Saint Etienne

RAUCH Jeffrey Mathematics, Univ. of Michigan, Ann Arbor,MI48109 USA

RAVIART P-Arnaud Univ. P. et M. Curie, Analyse Numerique, 4place Jussieu, 75230 Paris Cedex 05

REBOURCET Bernard CEA-V, BP 27,94190 Villeneuve St-Georges

REGGIO Marcelo Ecole Polytechnique de Montreal, Maths. Appl.,Case Postale 6079, Succursale A, MontrealCanada H3C 3A7

RIGAL Alain Analyse Nurnerique, Univ. P. Sabatier, 118route de Narbonne, 31062 Toulouse Oedex

RIOUX Francoise CNRS Lab. d'Elec. , Bat 214,91405 Orsay

ROCHE Jean Univ. Nancy 1, BP 239, 54506 Vandoeuvre lesNancy

ROE P.L. College of Aeronautics, Cranfield Inst. ofTechnology, Cranfield, Bedford, MK 43 OAL

ROMSTEDT Peter Gesellschaft fUr Reaktorsicherheit 0-8046Garching

ROSS Catherine Ecole Normale Sup., C.M.A. ,45 rue d'Ulm75005 Paris

SAIAC Herve C.N.A.M., 117 Bd Jourdan, 75014 Paris

SAMADEN Guy C.EA, BP 12, 91680 Bruyeres Ie Chatel

SANTOS Raphael Paris XIII, Dept. Math, Av. J.B. Clement, 93430Villetaneuse

SAUVETERRE Maryline Analyse Numerique, 23 rue du Dr PaulMichelon, 42023 Saint Etienne Cedsx 2

SCHATZMAN Michelle Univ. Lyon 1, 43 Bel du 11 Novembre 191869622 Villeurbanne Cedex

SCHMIDT-LAINE Claudine

SCHOENAUER M

SHERWOOD J.D.

SERRE Denis

SHOKIN Youri

SIBE Evelyne

SIDERIS Thomas

SLEMROD Marshall

SOLER Juan

SJOGREEN Bjorn

STEPHAN Yann

SURRY Claude

SWEBY PK

TADMOR Eitan

TAPIERO Roland

TARTAR Luc

TEIPEL Ing.1

TEMAM Roger

TEMPLE Blake

TOURE HAMIDOU

TRUC Joel

VAN HARTEN Aart

VILA Jean-Paul

XIV

Ecole Centrale de Lyon BP 163 69131 EcullyCedexCMAP Ecole Polytechnique 91128 Palaiseau

Etudes et fabrication Dowell-SchlumbergerBP 90, 42003 Saint Etienne Cedex 1Analyse Numerique , 23 rue du Dr PaulMichelon, 42023 Saint Etienne Cedex 2Computing Center. USSR Academy of Science,660036 Krasnoyarsk 36C.EA , BP 27, 94190 Villeneuve St-Georges

Courant Institute, 251 Mercer Steet,New-York N.Y. 10012 USADepar. of Math. Science, Rensselear Poly.Institute Troy. New York 12181 USALab. d'Analyse Numerique.Tour 55-65,5emeetage, Univ. P. et M. Curie,75005 ParisDept. of Computer SCience, Sturegatan 4B,S-752 23 Uppsala SuedeInformatique Internationale , BP 24, 91190Gif sur YvetteEcole Nationale d'lnqenieurs de Saint Etienne,rue Jean Parot, 42100 Saint EtienneDepartment of Mathematics Univ. of ReadingP.O. Box 220 Reading, RG 6 2AX , UKTel-Aviv Univ. Tel Aviv Israel

Analyse Nurnerique, Univ. Lyon 1 ,43 Bd du 11novembre 1918, 69622 Villeurbanne OedexCEA, Centre de Limeil-Valenton, BP 27,94190 Villeneuve Saint GeorgesUniv. of Hannover, Appelstr. 11, D-3000Hannover 12Analyse Numerique, Bat 425, Univ. Paris Sud914050rsayMath. Res. Center, 610 Walnut Street, MadisonWis. 53705 USAInst. de Maths. BP 7021 Ouagadoudou BurkinaFasoCNIN, BP 161, 83507 La Seyne sur Mer

Maths. Inst., Rijksuniversiteit UtrechtPostbus 80.010 3508 Ta Utrecht NederlandCEMAGREF Div. Nivologie, CentreUniversitaire, BP 76, Saint Martin d'Heres

VINCENT

WAGNER

Alain

Alain

xv

CNES, 18 avenue E. Belin 31055 ToulouseCedexCNES, Rond - Point des Pcetes, 91000 Evry

WAGNER David Dept. of Mathematics, University of Houston,University Park, Houston, Texas 77004

WANG JHING HUA Inst. of Systeme Science, Academia SINICABeijing 0 100080 China

WAZNER Alain TIM3 BP 68 38402 Saint Martin d'HeresCedex

WERNER Klaus RWTH Aachen, Templergraben 55,D-5100Aachen

WITOMSKI Patrick Univ. Grenoble 1 Labo. IMAG-TIM3 BP 6838402 Saint Martin d'Heres Cedex

WOODWARD Paul Dep. of Astronomy, Univ. of Minnesota, 116Church Street SO, Minneapolis, Minnesota55455

ZITI Cherif Analyse Nurnerique , 23 rue du Dr PaulMichelon , 42023 Saint Etienne Cedex 2