Table Of ContentZhu . Zhong . ehen· Zhang Flow Around Bodies
Zhu You-lan Zhong Xi-chang
ehen Bing-mu Zhang Zuo-min
Difference Methods for Initial
Boundary-Value Problems and
Flow Around Bodies
With 217 Figures and 40 Tables
Springer-Verlag Berlin Heidelberg GmbH
Zhu You-Ian Zhong Xi-chang
Chen Bing-mu Zhang Zuo-min
Computing Center
Chinese Academy of Sciences
Beijing
The People's Republic of China
Revised edition of the original Chinese edition published
by Science Press Beijing 1980 as the fourth volume in the
Series in Pure and Applied Mathematics.
Distribution rights throughout the world, exduding The People's
Republic of China, granted to Springer-Verlag Berlin Heidelberg
New York London Paris Tokyo
Mathematics Subject Classification (1980): 35-XX, 65-XX, 76-XX
Library of Congress Cataloging-in-Publication Data.
Ch'u pien chih wen t'i ch'a fen fang fa chi chiao liu. English. Difference methods for initial
boundary-value problems and flow around bodies / Zhu You-lan ... [et al.l. p. cm.
Translation of: Ch'u pien chih wen t'i ch'a fen fang fa chi chiao liu.
Half title: Initial-boundary-value problems and flow around bodies. "Revised edition of the
original Chinese edition published by Science Press Beijing 1980 as the fourth volume in the
Academia Sinica's series in pure andapplied mathematics"-T.p.verso.
Bibliography: p. Inciudes index.
ISBN 978-3-662-06709-3 ISBN 978-3-662-06707-9 (eBook)
DOI 10.1007/978-3-662-06707-9
1. Initial value problems. 2. Boundary value problems. 3. Difference equations. 4. Aerodyna
mies, Supersonic.l. Chu, Yu-lan. II. Title. 111. Title: lnitial-boundary-value problems andflow
around bodies. IV. Series: Ch'un ts'ui shu hsüeh yü ying yung shu hsüeh chuan chu ; ti 4
hao. QA378.C46813 1988 515.3'5-dc19 88-20115 CIP
This work is subject to copyright. All rights are reserved, whether the whole or part of the
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© Springer-Verlag Berlin Heidelberg 1988
Originally published by Springer-Verlag Berlin Heidelberg and Science Press Beijing in 1988.
Softcover reprint of the hardcover I st edition 1988
Typesetting: Science Press, Beijing, The People's Republic of China
2141/3140-543210
Preface to the English Edition
Sinoe the appearanoe of computers, numerical methodB for the
d.iscontinuous solutions of quasi-linear hyperbolio systems of partial
differon tial equations hav e been one of the most impt>rtant researoh su b joolis
in numürical analysis. 'rhe methods for this type of problems, genorally
.speaking, oan be olassified into two oategories. One category is the shook
oapturing method, where the shooks are obtained with a uniform scheme,
but are usually smeared. A typical representative of this category is. the
artiiicial visoosity method of von Neumann and Richtmyer. 'rhe other
category consists of methods whioh give definite looations of the shooks
and accurate flow fields, but are redundant to programming. The
..S ingularity-separating differenoe method developed by the authors belongs
to the latter category. 'l'his method has a high aocuraoy and asolid
theoretical basis, and has been suooessfully used for the solution of various
problems. It was sucoossful in the seventies, in computing supersonio
flow around combined bodies and in the early eighties, in solving
complicated unsteady flow, the Stefan problem, combustion problem and
hyperbolio equations with a nonconvex equation of state. In this book wo
shall introduce our work of the sevonties.
This monograph is divided into two parts. In Part I a numerioal
method for the initial-boundary-value problems of hyperbolio systems i8
discussed. In Part rr the application of the method to computation of
inviscid supersonio flow is desoribed. The authors' work on tho method of
lines, which has been used to compute subsonio-transonio flow around
blunt bodies and oonioal flow to provide the initial values of supersonic
flow field computation, is also briefly desoribed.
'l'he prerequisites for reading this book are a knowledge of caloulos
and of numerical analysis and familiarity with the basic methods of
mathematical physios. For the theoretical proofs, some funotional analysis
and matrix theory is required.
'rhe authors wiSh to pay partioular tribute to Professors Feng Kang
and Zhuang Fenggan, who reviewed the Chinese manuscript of this
book and gavo a number of valuable suggestions during the course of
this projoct. Sincere than~ are owed to Professor Huang Dun and
Professor '~J..1ranslator-editor Sun Xianrou for reading the ~~ngliHh
manusoript. Gratitude is owed to Wang Ruquan, Li Yinfan, Wu Huamo,
VI Preface to the English Edition
Liu Xuezong, Fu Dexun and Cai Dayong for thcir oomments and also to
Bai Degin, Ma Dehui, Su Anjie and Wang Hui for the:ir assiStanoe in the
diffioult mathematical typing.
Also the authors are grateful to Wang Rnquan, Zhang Guanqnan,
Q.in Bailiang and others who worked with us for a short period in the
computation of flow around bodies.
Zhn Youlan
Zhong Xichang
Chen Bingmu
Zhang Zuomin
Beijing
JUfIß,1987.
CONTENTS
PART I NUMERIOAL METHODS
Ohapter 1 Numerical Methods for lnitial-Botm.dary-Value Problems
for First Order Quasilinea.r Hyperbolio Systems in Two
Independent Variables
Introduction .............................................•..•.................................... a
§ 1 Formulation of Problems········································.· .. ·.· ................. 3
§ 2 Four Model Problems ................................................................... 7
§ 3 Some Difference Schemes ....•........ '" ..•..•.............. , ..... , . ....•. ..... . .. . ... 18
§ 4 The Stability of Difference Schemes for Initial-Boundary-Value
Problems and the "Condition" of Systems of Difference Equations ....... 40
§ 5 Solution of Systems of Difference Equations •.•................. .•............... 88
§ 6 The Stability of the Prooedure of Elimination and the Procedure of
Calculation of the Unknowns, and the Convergence of Iteration .......... 99
Appendix 1 Stability of Difference Schemes for Pure-Initial-Value
Problems with Varia.ble Coefficients ...•..•....................... , .... '" .,. .. . .. .. 111
Appendix 2 A Block-Double-Sweep Method for "Incomplete" Linear
Aigebraic Systems and Its Stability ...•....••...................•.................. 124
Appendix 3 Stability and Convergence of Difference Schemes for Linear
Initial-Boundary-Value Problems ................................................ 160
Ohapter 2 Numerical Methods for a Oert&in 018S!l of Initial
Boundary-Value Problems for the First Order Quasi linear
Hyperbolic Systems in Three Independent Variables
Introduction .................................................................................. 171
§ 1 Formulation of Problems. ... ·········································•·· ............. 171
§ 2 Numerical Methods ................................................................... 176
Ohapter 3 Numerical Schemes for Oertain Boundary-Value
Problems of Mixed-Type and Elliptioal Equations
§ 1 Formulation of Problems·· .. ··········································· .............. 194
§ 2 Numerical Schemes ...........................................................•....... 197
§ 3 Iteration Methods ............................................................ .......... 199
§ 4 Interpolation Polynomials .......................................................... 202
§ 5 Remarks on Improperly Posed Problems········································ 204
PARr.r II INVISOID SUPERSONIO FWW AROUND BODIES
Introduction to Part II ........................................................... 210
§ 1 Outline of Part 11· .. ············ .. ··················· .. ·.········· ...................... 210
§ 2 Literature Review························ '.' .. . .. . ... .. . ... .. . ... .. . .. . .. . ...... ... ...... 212
VIIl Contents
Ohapter 4 Inviseid Steady Flow
§ 1 The System of Fundamental Differential Equations and Its
Characteristics· ........................................................ , ................... ·235
§ 2 Discontinuities, Singularities, and the Intersection and Refiection of
Strong Discontinuities ................................................................. ·265
§ 3 Boundary Conditions and Internal Boundary Conditions .................... ·301
§ 4 Calculation of Thermodynamic Properties of Equilibrium Air ............ 311
§ 5 A Non-equilibrium Model of Air .................................................. ·326
Ohapter 5 Oaloulation of Supersonio Flow around Blunt Bodies
§ 1 Introduction ............................................................................. ·337
§ 2 Formulation of Problems ............................................. · .. · .. · ........ 338
§ 3 Methods of Solution .............................................................. ·· .... ·342
§ 4 Calculation of the Axisymmetric Flow ............................................ ·346
§ 5 Calculation of the Three-dimensional Flow .................................... · .. 353
§ 6 Results· .................. · .. · ............................................................... ·364
Appendix Application of the Method of Lines to Supersonic Regions of
Flow ......................................................................... ···· .. · .. · .. ··388
Ohapter 6 Oaloulation of Supersonjo Conioal Flow
§ 1 Introduction ............................................................................. ·395
§ 2 Formulation of Problems .. · ................................................... · ........ 396
§ 3 Methods of Solution · .................. · ....... · ......... · .. · .. · .. · .... ·· .. · .. · .. · .. · .... 399
§ 4 Results ...................................................................................... ·404
Ohapter 7 Solution of Supersonio Regions of Flow around
Oorn bined Bodies
§ 1 Introduction ............................................................................. ·423
§ 2 Formulation of Problems ..................................................... · ........ ·425
§ 3 Numerical Methods ........................................................ · .... · .. · .. ··441
§ 4 Computed Results ....................................................................... ·468
Appendix A Numerical Method with High Accuracy for Calculating
the Interactions between Discontinuities in Three Independent
Variables ........................................ · .. · .................................... 563
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 573
General References .......................................................................... ·573
Special References A: Numerical Calculation of Flow in Subsonic and
Transonic Regions .... · .. · .. · .. · ........................................................ 578
Special References B: N umerical Calculation of Conical Flow .................... ·589
Special References C: Numerical Calculation of Flow in Supersonic
Regions·· .. · ........ · .. · .. · .. · .. · ......................................................... ·592
Su bject Index ......................................................................................... 597
PART I
NUMERICA.L METHODS
Chapter 1
Numerical Methods for Initial-Boundary-Value
Problems for First Order Quasilinear Hyperbolic
Systems in Two Independent Variables
lntroduction
When discussing numerical methods for hyperbolic systems, it is
usual to construct difference schemes and do theoretical analysis only for
pure initial-value problems. However, most of the problems which exist
in practice are initial-boundary-value problems. When applying the
results from the pure initial-value problems (PIVP) to the initial
boundary--value problems (IBVP), diffioulties are encountered since we
usually do not know how to calculate the bounday points and how to
.ascertain whether an algorithm for boundary points is reasonable.
There have been several works written on initial-boundary
value problemsU-14l, but they have been imperfect. Therefore, further
development of numerical methods for initial-boundary-value problems
is urgently needed. In this chapter, we shall discuss this problem
thoroughly and systematically in the case of two independent variables.
That is, we shall carefully describe a difference method for initial
boundary-value problems of the first order quasi-linear hyperbolic
systems in two independent variables. Firstly, a way of constructing
schemes for initial-boundary-value problems is given and several schemes
.are presented. Then, the stability of several classes of schemes for initial
boundary-value problems with variables coefficients is discussed. Finally,
.a method of solving difference equations--a block-double-sweep
method for "segmental", incomplete linear algebraic systems--is
described, and the stability of this direct method for the systems of
difference equations with variable coefficients is discussed. In passing,
three appendices which are extensions of the text are given.
The difference method described in this chapter has the following
features.
