Nonlinear Hyperbolic Equations - Theory. Computation .
Josef BallmannRolf Jeltsch (Eds.)Nonlinear Hyperbolic Equations Theory. Computation Methodsand Applications
Notes on Numerical Fluid MechanicsVolume 24Series Editors: Ernst Heinrich Hirschel, MUnchenKozo Fujii, TokyoKeith William Morton, OxfordEarll M. Murman, M.I.T., CambridgeMaurizio Pandolfi, TorinoArthur Rizzi, StockholmBernard Roux, Marseille(Addresses of the Editors: see inner back cover)Volume 1Boundary Algorithms for Multidimensional Inviscid Hyperbolic Flows(K. Forster, Ed.)Volume 2Proceedings of the Third GAMM-Conference on Numerical Methods in FluidMechanics (E. H. Hirschel, Ed.) (out of print)Volume 3Numerical Methods for the Computation of Inviscid Transonic Flows with ShockWaves (A. Rizzi I H. Viviand, Eds.)Shear Flow in Surface-Oriented Coordinates (E. H. Hirschel/W. Kordulla)Proceedings of the Fourth GAMM-Conference on Numerical Methods in FluidMechanics (H. Viviand, Ed.) (out of print)Volume 4Volume 5Volume 6Volume 7Volume 8Volume 9Numerical Methods in Laminar Flame Propagation (N.Petersl J.Warnatz, Eds.)Proceedings of the Fifth GAMM-Conference on Numerical Methods in FluidMechanics (M.Pandolfi/R.Piva, Eds.)Vectorization of Computer Programs with Applications to Computational FluidDynamics (W.Gentzsch)Analysis of Laminar Flow over a Backward FaCing Step (K.Morganl J.PeriauxlF. Thomasset, Eds.)Volume 10 Efficient Solutions of Elliptic Systems (W. Hackbusch, Ed.)Volume 11 Advances in Multi-Grid Methods (D. Braess/W. Hackbusch I U. Trottenberg, Eds.)Volume 12 The Efficient Use of Vector Computers with Emphasis on Computational FluidDynamics (W.Schonauer/W.Gentzsch, Eds.)Volume 13 Proceedings of the Sixth GAMM-Conference on Nurner;ical Methods in FluidMechanics (Q. Rues/W. Kordulla, -Eds.Hout of pri"nt)' Volume 14 Finite Approximations in Fluid Mech nics.(E:H:Hir'sChel, Ed.)Volume 15 Direct and Large Eddy Simulation of "t:urbulence{U. Schumannl R. Friedrich, Eds.)Volume 16 Numerical Techniques in Continuum Mechanics IW.HaGkbusch/K.Witsch, Eds.)Volume 17 Research in NumeriCal Fluid DynamieslP.Wesseling, Ed.)Volume 18 Numerical Simulation of Compressible Navier-Stokes Flows IM.O.BristeaulR.Glowinskil J.Periauxl H. Viviand, Eds.)Volume 19 Three-Dimensional Turbulent Boundary Layers - Calculations and Experiments(B. van den Berg/D.A. Humphreys/E. Krausel J.P. F. Lindhout)Volume 20 Proceedings of the Seventh GAMM-Conference on Numerical Methods in FluidMechanics (M.Deville, Ed.)Volume 21 Panel Methods in Fluid Mechanics with Emphasis on Aerodynamics (J.BalimannlR.Eppler/W.Hackbusch, Eds.)Volume 22 Numerical Simulation of the Transonic DFVLR-F5 Wing Experiment(W.Kordulla, Ed.)Volume 23 Robust Multi-Grid Methods (W.Hackbusch, Ed.)Volume 24 Nonlinear Hyperbolic Equations - Theory, Computation Methods, andApplications (J. Ballmannl R. Jeltsch, Eds.)Volume 25 Finite Approximations in Fluid Mechanics II IE. H. Hirschel, Ed.)
Josef BallmannRolf Jeltsch (Eds.)NonlinearHyperbolic Equations Theory, Computation Methods,and ApplicationsProceedings of the Second International Conferenceon Nonlinear Hyperbolic Problems,Aachen, FRG, March 14 to 18, 1988Friedr. Vieweg & SohnBraunschweig I Wiesbaden
CIP-Titelaufnahme der Deutschen BibliothekNonlinear hyperbolic equations: theory,computation methods, and applications;proceedings of the 2nd InternationalConference on Nonlinear HyperbolicProblems, Aachen, FRG, March 14 to18, 1988/ Josef Ballmann; Rolf Jeltsch(eds.). - Braunschweig; Wiesbaden:Vieweg, 1989(Notes on numerical fluid mechanics;Vol. 24)NE: Ballmann, Josef (Hrsg.]; InternationalConference on Nonlinear HyperbolicProblems(02,1988, Aachen); GTManuscripts should have well over 100 pages. As they will be reproduced photomechanicallythey should be typed with utmost care on special stationary which will be supplied on request.In print, the size will be reduced linearly to approximately 75 per cent. Figures and diagrammsshould be lettered accordingly so as to produce letters not smaller than 2 mm in print. The sameis valid for handwritten formulae. Manuscripts (in English) or proposals should be sent to thegeneral editor Prof. Dr. E. H. Hirschel, Herzog-Heinrich-Weg 6, 0-8011 Zorneding.Vieweg is a subsidiary company of the Bertelsmann Publishing Group.All rights reserved Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1989No part of this publication may be reproduced, stored in a retrievalsystem or transmitted, mechanical, photocopying or otherwise, withoutprior permission of the copyright holder.Produced by W. Langeliiddecke, BraunschweigISBN-13: 978-3-528-08098-3e-ISBN-13: 978-3-322-87869-4DOl: 10.1007/978-3-322-87869-4
FOREWORDOn the occasion of the International Conference on Nonlinear Hyperbolic Problemsheld in St. Etienne, France, 1986 it was decided to start a two years cycle ofconferences on this very rapidly expanding branch of mathematics and it·sapplications in Continuum Mechanics and Aerodynamics. The second conferencetoolc place in Aachen, FRG, March 14-18, 1988. The number of more than 200participants from more than 20 countries all over the world and about 100invited and contributed papers, well balanced between theory, numerical analysisand applications, do not leave any doubt that it was the right decision to startthis cycle of conferences, of which the third will be organized in Sweden in 1990.ThiS volume contains sixty eight original papers presented at the conference,twenty two cif them dealing with the mathematical theory, e.g. existence,uniqueness, stability, behaviour of solutions, physical modelling by evolutionequations. Twenty two articles in numerical analysis are concerned with stabilityand convergence to the physically relevant solutions such as schemes especiallydeviced for treating shoclcs, contact discontinuities and artificial boundaries.Twenty four papers contain multidimensional computational applications tononlinear waves in solids, flow through porous media and compressible fluidflow including shoclcs, real gas effects, multiphase phenomena, chemicalreactions etc.The editors and organizers of the Second International Conference on HyperbolicProblems would lilce to thanlc the Scientific Committee for the generous supportof recommending invited lectures and selecting the contributed papers of theconference.The meeting was made possible by the efforts of many people to whom theorganizers are most grateful. It is a particular pleasure to aclcnowledge thehelp of Riikka Tuominen for preparing the abstract book and Bert Pohl for hisdedicated help organizing the conference. It is also a pleasure to thank SylvieWiertz, Angela Schneider, Gabriele Goblet and Thomas Hoerkens for preparingthese proceedings. Finally the organizers are indebted to the host organizationsRheinisch Westflilische Technische Hochschule Aachen and the city of Aachenand to those organizations which provided the needed financial support for theconference: Control Data GmbH, Cray Research GmbH, Deutsche Forschungsgemeinschaft, Diehl GmbH & Co., Digital Equipment GmbH, FAHO Gesellschaftvon Freunden der Aachener Hochschule, IBM Deutschland GmbH, MathematischNaturWissenschaftliche Fakultlit der RWTH, Ministerium fUr Wissenschaft undForschung des Landes Nordrhein-Westfalen, Office of Naval Research Branchof London, Rheinmetall GmbH, US Air Force EOARD, US Army European ResearchOffice of London, Wegmann GmbH & Co.Aachen, September 1988Josef BallmannRolf Jeltsch
ContentsPageArminjon, P., Dervieux, A., Fezoui, 1., Steve, H., Stouffiet, B.: Non-oscillatoryschemes for multidimensional Euler calculations with unstructured grids . 1Billet, G.: Finite-difference schemes with dissipation control joined to a generalization of van Leer flux splitting . 11Binninger, B., Jeschke, M., Henke, H., Hanel, D.: Computation of inviscid vortical flows in Piston engines . 21Bourdel, F., Delorme, Ph., Mazet, P.A.: Convexity in hyperbolic problems. Application to a discontinuous Galerkin method for the resolution of the polydimensional Euler equations . 31Brio, M.: Admissibility conditions for weak solutions of nonstrictly hyperbolicsystems . 43Cahouet, J., Coquel, F.: Uniformly second order convergent schemes for hyperbolic conservation laws including Leonard's approach . 51Causon, D.M.: High resolution finite volume schemes and computational aerodynamics . 63Christiansen, S.: A stability analysis of a Eulerian method for some surface gravity wave problems . 75Degond, P., Mustieles, F.J., Niclot, B.: A quadrature approximation ofthe Boltzmann collision operator in axisymmetric geometry and its application to particle methods . 85Dubois, F., Le Floch, P.: Boundary conditions for nonlinear hyperbolic systemsof conservation laws . 96Eliasson, P., Rizzi, A., Andersson, H.I.: Time-marching method to solve steadyincompressible Navier-Stokes equations for laminar and turbulent flow . 105Engelbrecht, J.: On the finite velocity of wave motion modelled by nonlinear evolution equations . 115Fernandez, G., Larrouturou, B.: Hyperbolic schemes for multi-component Eulerequations . 128VI
PageFogwell, T.W., Brakhagen, F.: Multigrid methods for solution of porous mediamulti phase flow equations . 139Freistiihler, H.: A standard model of generic rotational degeneracy . 149Gimse, T.: A numerical method for a system of equations modelling one-dimensional three-phase flow in a porous medium . 159Glimm, J.: Nonuniqueness of solutions for Riemann problems . 169Goldberg, M., Tadmor, E.: Simple stability criteria for difference approximationsof hyperbolic initial-boundary value problems . 179Greenberg, J.M.: Hyperbolic heat transfer problems with phase transitions. 186Gustafsson, B.: Unsymmetric hyperbolic systems and almost incompressibleflow . 193Hackbusch, W., Hagemann S.: Frequency decomposition multi-grid methods forhyperbolic problems . 209Hanyga, A., Fabrizio, M.: Existence and uniqueness for linear hyperbolic systemswith unbounded coefficients . 218Harabetian, E.: A numerical method for computing viscous shock layers . 220Henke, H.H.: Solution of the Euler equations for unsteady, two-dimensional transonic flow . 230Holden, H., Holden, L.: On some recent results for an explicit conservation lawof mixed type in one dimension . 238Hsiao, L.: Qualitative behaviour of solutions for Riemann problems of conservation laws of mixed type . 246Hunter, J.K.: Strongly nonlinear hyperbolic waves . 257Isaacson, E., Marchesin, D., Plohr, B.J.: The structure of the Riemann solutionfor non-strictly hyperbolic conservation laws . 269Klein, R.: Detonation initiation due to shock wave-boundary interactions . 279VII
PageKlingenberg, C., Osher, S.: Nonconvex scalar conservation laws in one and twospace dimensions . 289Koren, B.: Upwind schemes for the Navier-Stokes equations . 300Kosinski, S.: Normal reflection transmission of shock waves on a plane interfacebetween two rubber-like media . 310Kosinski, W.: On the concept of weak solutions in the BV-space . 320Kozel, K., Vavi'incova, M., Nhac, N.: Numerical solution of the Euler equationsused for simulation of 2D and 3D steady transonic flows . 329Kroner, D.: Numerical schemes for the Euler equations in two dimensions withoutdimensional splitting . 342Lar'kin, N.A.: Initial-boundary value problems for transonic equations in the unbounded domain . 353Le Floch, P.: Entropy weak solutions to nonlinear hyperbolic systems in nonconservation form . 362Le Roux, A.Y., De Luca, P.: A velocity-pressure model for elastodynamics .374MandaI, J.C., Deshpande, S.M.: Higher order accurate kinetic flux vector splitting method for Euler equations . 384Marshall, G.: Monte Carlo finite difference methods for the solution of hyperbolicequations . 393Mertens, J., Becker, K.: Numerical solution of flow equations. An aircraft designer's view . 403Montagne, J.-L., Yee, H.C., Klopfer, G.H., Vinokur, M.: Hypersonic blunt bodycomputations including real gas effects . 413Moretti, G., Dadone, A.: Airfoil calculations in Cartesian grids . .423Morton, K.W., Childs, P.N.: Characteristic Galerkin methods for hyperbolic systems . 435Munz, C.D., Schmidt, L.: Numerical simulations of compressible hydrodynamicinstabilities with high resolution schemes . .456VIII
PagePandolfi, M.: On the "Flux-difference splitting" formulation . .466Peradzynski, Z.: On overdetermined hyperbolic systems . 482Pfitzner, M.: Runge-Kutta split-matrix method for the simulation of real gas hypersonic flows . . 489Pham Ngoc Dinh, A., Dang Dinh Ang: On some viscoelastic strongly dampednonlinear wave equations . 499Rostand, P., Stouffiet, B.: TVD schemes to compute compressible viscous flowson unstructured meshes . , . 510Schick, P., Hornung, K.: Nonstationary shock wave generation in droplet vapourmixtures . 521Schaffel, St.U.: Nonlinear resonance phenomena for the Euler-equations coupledwith chemical reaction-kinetics . 530Sethian, J., Osher, S.: The design of algorithms for hypersurfaces moving withcurvature-dependent speed . 544Shapiro, R.A.: Prediction of dispersive errors in numerical solutions of the Eulerequations . 552Sommerfeld, M.: Numerical prediction of shock wave focusing phenomena in airwith experimental verification . 562Staat, M., Ballmann, J.: Fundamental aspects of numerical methods for the propagation of multi-dimensional nonlinear waves in solids . 574Stra.skraba, I.: On a nonlinear telegraph equation with a free boundary . " .589Sweby, P.K.: "TVD" schemes for inhomogenous conservations laws . 599Temple, B.: The L1-Norm distinguishes the strictly hyperbolic from a non-strictlyhyperbolic theory of the initial value problem for systems of conservationlaws . 608Ting, T.e.T.: The Riemann problem with umbilic lines for wave propagation inisotropic elastic solids . 617IX
PageToro, E.F.: Random-choice based hybrid methods for one and two dimensionalgas dynamics . 630Voskresensky, G.P.: Some features of numerical algorithms for supersonic flowcomputation around wings of lifting vehicles . 640Wada, Y., Kubota, H., Ishiguro, T., Ogawa, S.: Fully implicit high-resolutionscheme for chemically reacting compressible flows . 648Warming, R.F., Beam, R.M.: Stability of semi-discrete approximations for hyperbolic initial-boundary-value problems: Stationary modes . 660Wendroff, B., White, A.B.: Some supraconvergent schemes for hyperbolic equations on irregular grids . 671Westenberger, H., Ballmann, J.: The homogeneous homentropic compression orexpansion - A test case for analyzing Sod's operator-splitting . 678Zajaczkowski, W.M.: Global existence of solutions for noncharacteristic mixedproblems to nonlinear symmetric dissipative systems of the first order . 688Zi61ko, M.: Stability of initial - boundary value problems for hyperbolic systems . 698List of Participants and Authors . 708Support and Sponsorship Acknowledgements . 718x
Volume 15 Direct and Large Eddy Simulation of "t:urbulence{U. Schumannl R. Friedrich, Eds.) Volume 16 Numerical Techniques in Continuum Mechanics IW.HaGkbusch/K.Witsch, Eds.) Volume 17 Research in NumeriCal Fluid DynamieslP.Wesseling, Ed.) Volume 18 Numerical Simulation of Compressible Navier-Stokes Flows IM.O.Bristeaul
equal to ˇ. In hyperbolic geometry, 0 ˇ. In spherical geometry, ˇ . Figure 1. L to R, Triangles in Euclidean, Hyperbolic, and Spherical Geometries 1.1. The Hyperbolic Plane H. The majority of 3-manifolds admit a hyperbolic struc-ture [Thurston], so we shall focus primarily on the hyperbolic geometry, starting with the hyperbolic plane, H.
Volume in hyperbolic geometry H n - the hyperbolic n-space (e.g. the upper half space with the hyperbolic metric ds2 dw2 y2). Isom(H n) - the group of isometries of H n. G Isom(H n), a discrete subgroup )M H n G is a hyperbolic n-orbifold. M is a manifold ()G is torsion free. We will discuss finite volume hyperbolic n-manifolds and .
The angle between hyperbolic rays is that between their (Euclidean) tangent lines: angles are congruent if they have the same measure. q Lemma 5.10. The hyperbolic distancea of a point P from the origin is d(O, P) cosh 1 1 jPj2 1 jPj2 ln 1 jPj 1 jPj aIt should seem reasonable for hyperbolic functions to play some role in hyperbolic .
1 Hyperbolic space and its isometries 1 1.1 Möbius transformations 1 1.2 Hyperbolic geometry 6 1.2.1 The hyperbolic plane 8 1.2.2 Hyperbolic space 8 1.3 The circle or sphere at infinity 12 1.4 Gaussian curvature 16 1.5 Further properties of Möbius transformations 19 1.5.1 Commutativity 19 1.5.2 Isometric circles and planes 20 1.5.3 Trace .
shortest paths connecting two point in the hyperbolic plane. After a brief introduction to the Poincar e disk model, we will talk about geodesic triangleand we will give a classi cation of the hyperbolic isome-tries. In the end, we will explain why the hyperbolic geometry is an example of a non-Euclidean geometry.
metrical properties of the hyperbolic space are very differ-ent. It is known that hyperbolic space cannot be isomet-rically embedded into Euclidean space [18, 24], but there exist several well-studied models of hyperbolic geometry. In every model, a certain subset of Euclidean space is en-dowed with a hyperbolic metric; however, all these models
One difficulty for defining the hyperbolic CVT is how to well de-fine the centroid of a given region in 2D hyperbolic space. In this paper, we extend the model centroid [Galperin 1993] to define the centroid of a Voronoi cell in 2D hyperbolic space. Another chal-lenge is how to effectively and efficiently compute the hyperbolic CVT.
Unfortunately, each model of the Hyperbolic geometry depicts a warped version of the Hyperbolic space, just as a two-dimensional map represents the Earth in a distorted way. To describe the Hyperbolic space, all we need to know is the amount of distortion introduced . Hyperbolic space of curvature k is: d(x,y) arccosh s (1 Xn i 1 x2
