Zhu . 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 material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provi sions of the German Copyright Law of September 9, 1965, in its version of June 24,1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © 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.