Table Of ContentAndreas Meister
Jens Struckmeier
Hyperbolic
Partial Differential Equations
Andreas Meister
Je ns Struckmeier
Hyperbolic
Partial Differential
Equations
Theory, Numerics and Applications
I I
vleweg
Hochschuldozent Dr. Andreas Meister
Medizinische Universitat zu Lubeck
Institut fur Mathematik
WallstraBe 40
D-23560 Lubeck, Germany
Professor Dr. Jens Struckmeier
Universitat Hamburg
Fachbereich Mathematik
BundesstraBe 55
D-20146 Hamburg, Germany
[email protected]
[email protected]
Die Deutsche Bibliothek - CIP-Cataloguing-in-Publication-Data
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Die Deutsche Bibliothek.
First edition, March 2002
All rights reserved
© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 2002
Softcover reprint of the hardcover 1st edition 2002
Vieweg is a company in the specialist publishing group BertelsmannSpringer.
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or transmitted, mechanical, photocopying or otherwise without prior per
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Cover design: Ulrike Weigel, www.CorporateDesignGroup.de
Printed on acid-free paper
ISBN -13: 978-3-322-80229·3 e-ISBN -13: 978-3-322-80227·9
DOl: 10.1007/978-3-322-80227·9
v
Preface
The following chapters summarize lectures given in March 2001 during the summerschool
on Hyperbolic Partial Differential Equations which took place at the Technical University of
Hamburg-Harburg in Germany. This type of meeting is originally funded by the Volkswa
genstiftung in Hannover (Germany) with the aim to bring together well-known leading
experts from special mathematical, physical and engineering fields of interest with PhD
students, members of Scientific Research Institutes as well as people from Industry, in
order to learn and discuss modern theoretical and numerical developments.
Hyperbolic partial differential equations play an important role in various applications
from natural sciences and engineering. Starting from the classical Euler equations in fluid
dynamics, several other hyperbolic equations arise in traffic flow problems, acoustics,
radiation transfer, crystal growth etc. The main interest is concerned with nonlinear
hyperbolic problems and the special structures, which are characteristic for solutions
of these equations, like shock and rarefaction waves as well as entropy solutions. As a
consequence, even numerical schemes for hyperbolic equations differ significantly from
methods for elliptic and parabolic equations: the transport of information runs along the
characteristic curves of a hyperbolic equation and consequently the direction of transport
is of constitutive importance. This property leads to the construction of upwind schemes
and the theory of Riemann solvers. Both concepts are combined with explicit or implicit
time stepping techniques whereby the chosen order of accuracy usually depends on the
expected dynamic of the underlying solution.
Numerical schemes for hyperbolic equations should always satisfy two contrary goals:
• smooth solutions should be approximated with a high spatial accuracy and
• discontinuities should be resolved as sharp as possible.
These two opposite purposes might even be found for scalar conservation laws in one
space dimension, like the inviscid Burgers equation. Hence, even on an elementary level,
it is possible to introduce to non-experts crucial concepts like stability, monotonicity or
the TVD-property (Total Variation Dimishing). A quite nice concept is the classification
of numerical schemes only using a numerical flux function, which approximates the flux
function defined by the given hyperbolic equation.
In the current book, all concepts mentioned above are discussed. Starting from basic
ideas on conservation principles to model various problems from natural and engineering
science, the lecturers develop the basic concepts from numerical analysis up to topics
which are actually treated by researchers working on numerical schemes for hyperbolic
equations.
Exactly this concept the editors had in mind when planning the summerschool. The aim
was to build a bridge from the elementary modeling of various phenomena using scalar
conservation laws up to the newest developments, like multi-scale analysis for low-Mach
number flows and the combination of computational fluid dynamics with accoustics.
In order to be more concrete, the five chapters of this book contain contributions from
(in alphabetical order)
• Michael Junk on Hyperbolic conservation laws and industrial applications
VI
• Claus-Dieter Munz on Computational fluid dynamics and aeroacoustics for low mach
number
• Milovan Peric. Ismet Demirdzic and Samir Muzaferija on Pressure-correction methods
for all flow speeds
• Giovanni Russo on Central schemes and systems of balance laws
• Thomas Sonar on Methods on unstructured grids including WENO and ENO re
covery techniques
The first chapter covers the theoretical aspects of conservation laws. Starting from initial
and boundary value problems for scalar equations, the concept of weak solutions and
entropy up to the theory of systems of equations are introduced. Additionally, the first
chapter contains a lot of physical and technical examples modeled by conservation laws.
The second chapter is devoted to the description of higher order central schemes on rect
angular grids using CWENO (Central-Weighted Essentially Non-Oscillatory) reconstruction
techniques. Beside the scalar case also systems of conservation laws are considered and
the treatment of stiff source terms is discussed. The objective of the third chapter is
the development of finite volume approximations of hyperbolic conservation laws on un
structured grids. Thereby, primary as well as secondary grid methods are presented and
different time stepping schemes are explained. Special attention is laid on ENO- and
WENO-recovery techniques and grid adaptivity. The main principles of various pres
sure correction methods like SIMPLE, SIMPLEC and PISO are outlined in chapter 4.
Therein, a particular finite volume scheme for the discretization of the Navier-Stokes
equations is described. The applicability of the method for the simulation of flow fields
at all speeds is demonstrated by means of different laminar and turbulent test cases.
Numerical schemes for low Mach number flows based on an asymptotic single as well
as multiple scale analysis are considered in the concluding fifth chapter. Thereby, the
compressible pressure correction approach and the multiple pressure variable method
are outlined. Furthermore, a comprehensive description concerning the coupling of me
thods for fluid calculation with schemes for sound propagation due to acoustic waves is
presented.
First of all, the editors would like to thank the Volkswagen-Stiftung in Hannover, which
financially supported the summerschool in the context of the DMV-GAMM-GI Fach
gruppe Numerische Software. Next this book would be impossible without the contri
butions of the lecturers of the school. Hence, we would like to thank Michael Junk
(University of Kaiserslautern), Claus-Dieter Munz (University of Stuttgart), Milovan
Peric (Technical University of Hamburg-Harburg), Giovanni Russo (University of Cata
nia, Italy) and Thomas Sonar (Technical University of Braunschweig) for their impres
sive talks as well as their fascinating and well-written contributions. Moreover, we are
also grateful to Wolfgang Mackens, Professor at the Technical University of Hamburg
Harburg for initiating a large number of summerschools, for giving us the possibility to
realize this course and for his collaboration and help during the organization and execu
tion. Finally, we would like to thank Frau Schmickler-Hirzebruch from Vieweg publishers,
who was the responsible person to edit the book.
Hamburg, November 2001 Andreas Meister and Jens Struckmeier.
Vll
Contents
1 Hyperbolic Conservation Laws and Industrial Applications 1
1.1 Transport theorem and balance laws 1
1.1.1 The flux vector . . . . . 1
1.1.2 The transport theorem . 3
1.1.3 Continuum hypothesis . 5
1.1.4 Examples of balance laws 6
Population model. 6
Traffic model ... 6
Kinetic equations . 8
Euler equation · . 10
1.2 Linear initial and boundary value problems 12
1.2.1 The method of characteristics 12
Advection equation. . . . . . 14
Stationary radiative transport. 16
Vlasov-equation with constant force 19
Crystal precipitation 19
1.2.2 Linear systems · .. 21
Symmetric hyperbolic systems 21
A problem with a moving boundary 23
1.3 Weak solutions and entropy 27
1.3.1 Weak solutions · . 30
1.3.2 Entropy condition 36
1.3.3 Ion etching in semiconductor production . 39
1.4 Systems of conservation laws 44
1.4.1 Characteristic surfaces . . 44
1.4.2 The Saint-Venant system 45
1.4.3 Rarefaction waves 49
1.4.4 Shock waves. . . . 51
Vlll Contents
Bibliography 57
2 Central Schemes and Systems of Balance Laws 59
2.1 Second order central schemes 60
2.1.1 Hyperbolic systems .. 60
2.1.2 Conservative schemes 62
2.1.3 Godunov scheme ... 64
2.1.4 The Nessyahu-Tadmor scheme 68
2.2 High order central schemes . . . . 71
2.2.1 Time evolution: Runge-Kutta methods with Natural Continuous
Extension . . . . . . . . . . . . . . . . . . . . . . . 72
2.2.2 The Central-CWENO (CWENO) Reconstruction. 74
2.2.3 Numerical results. 79
Scalar equation . . 79
Systems of equations 80
2.2.4 Improvements.... 83
2.2.5 Semidiscrete central schemes 85
2.3 Multidimensional central schemes. 86
2.3.1 Multidimensional CWENO reconstruction 89
2.3.2 The Reconstruction from Cell Averages 89
2.3.3 The Bi-Quadratic Polynomials 92
2.3.4 The Weights ..... ..... 92
2.3.5 The Reconstruction of Flux Derivatives 94
2.3.6 The Algorithm . . .. 96
2.3.7 Systems of Equations 96
2.3.8 Numerical Examples . 97
Two-Dimensional Gas Dynamics Equations 98
2.3.9 Improvements .... 100
2.4 Treatment of the source 102
Further developments 108
Bibliography 110
ix
3 Methods on unstructured grids, WENO and ENO Recovery techniquesl15
3.1 Introduction to finite volume approximations 115
3.2 Governing equations . . . . . . 115
3.3 Finite volume approximations . 118
3.3.1 Conforming triangulations . 119
3.3.2 Primary grid methods for the Euler equations . 121
3.3.3 Box methods for the Euler equations . 123
3.3.4 Remarks on the spatial accuracy 125
3.3.5 Numerical examples ....... 127
3.3.6 Discretisation of the viscous fluxes 133
3.4 Time stepping schemes 139
3.4.1 The CFL condition . 139
3.4.2 Explicit discretisations . 140
3.4.3 Implicit time discretisations . 141
3.5 Remarks on the philosophy of ENO schemes 144
3.6 Polynomial recovery ...... 145
3.6.1 Stencil selection algorithms 146
3.6.2 Computation of the recovery polynomial . 148
3.6.3 Recovery on primary grids . 152
3.6.4 Recovery for box methods . 160
3.7 WEN 0 approximations .... 165
3.8 The theory of optimal recovery 166
3.8.1 Basic notions 168
3.8.2 Splines .... 173
3.9 Grid adaptivity for box methods 179
3.9.1 A refinement procedure 179
3.9.2 Red-Green Refinement . 180
Algorithm 1 . 181
3.9.3 History ... 181
3.9.4 Recoarsening 182
Algorithm 2 . 183
Algorithm 3 . 184
3.9.5 Conservati vity 186
x Contents
3.10 Error and residual 187
3.11 Experience with L2 189
3.11.1 On the necessity of proper weighting 189
3.11.2 The weighted L2 -norm 195
3.11.3 Unsteady Problems. 199
3.12 The dual graph-norm 203
3.12.1 Friedrichs systems 203
3.12.2 Symmetrising the Euler equations 204
3.12.3 Cell error and transported error 208
3.12.4 A weak a posteriori error estimate 209
3.12.5 A discrete dual graph-norm indicator. 213
3.12.6 The error indicator ... 215
3.12.7 Numerical experiments. 219
3.12.8 Problems with the dual graph-norm 224
3.13 Closing of the circle: L2 meets dual graph norm 228
Bibliography 229
4 Pressure-Correction Methods for all Flow Speeds 233
4.1 Introduction ..... . 233
4.2 Conservation Equations 233
4.3 Pressure-Correction Equation for Incompressible Flows 236
4.3.1 SIMPLE-Method. 237
4.3.2 SIMPLEC-Method 238
4.3.3 PISO-Method ... 239
4.3.4 SIMPLER-Method 240
4.3.5 Under-Relaxation Strategies. 241
4.3.6 Fractional-Step Methods . . . 242
4.4 Pressure-Correction Equation for Compressible Flows 243
4.5 Solution Algorithm for all Flow Speeds .. 245
4.6 FV-Method for Arbitrary Control Volumes 246
4.7 Pressure-Correction Algorithm for FV-Methods . 249
4.8 Implementation of Boundary Conditions . 252
4.9 Examples of Application. . . . . . . . . . 254
Xl
4.9.1 Subsonic Laminar Flow Around Airfoil .. 255
4.9.2 Subsonic Turbulent Flow Around Airfoil . 256
4.9.3 Transonic Flow Around Airfoil . 259
4.9.4 Supersonic Flow Around Airfoil. 259
4.9.5 Flow Through an Orifice. 261
4.10 Conclusions 262
Bibliography 267
5 Computational Fluid Dynamics and Aeroacoustics for Low Mach Num-
ber Flow 269
5.1 Introduction 269
5.2 N on-Dimensionalisation of the Governing Equations 272
5.3 The Incompressible Limit of a Compressible Fluid Flow 273
5.4 Numerical Methods for Low Mach Number Fluid Flow. 275
5.4.1 Compressible Pressure Correction Schemes .. 276
5.4.2 Multiple Pressure Variables (MPV-)Approach . 279
5.4.3 Preconditioning ............. . 283
5.5 Sound Generation and Sound Propagation . 284
5.5.1 The Equations of Linear Acoustics 284
5.5.2 Lighthill's Acoustic Analogy ... 285
5.5.3 The Janzen-Rayleigh Expansion 286
5.5.4 Expansion around Incompressible Flow 287
5.5.5 Numerical Example: Co-Rotating Vortex Pair. 289
5.6 Multiple Scale Considerations .. 290
5.6.1 Outer Asymptotic Expansion 291
5.6.2 Multiple Scale Asymptotic Expansion 292
5.6.3 MPV-Approach for Thermo-Acoustic Applications 295
5.6.4 Multiple Scale Considerations of Noise Generation 298
5.7 Numerical Aeroacoustics . 301
5.8 Conclusions 314
Bibliography 317
