Microcomputer Modelling by Finite Differences

137 Pages·1986·8.436 MB·English

by  Gordon Reece (auth.)

Microcomputer Modelling by Finite Differences
Other Macmillan titles of related interest
Ian O. Angell Advanced Graphics with the IBM Personal Computer
Ian O. Angell and Brian J. Jones Advanced Graphics with the BBe Model B
Microcomputer
A. N. Barrett and A. L. Mackay Spatial Structures and the Microcomputer
G. P. McKeown and V. J. Rayward-Srnith Mathematics for Computing
Microcomputer Modelling
by Finite Differences
Gordon Reece
Department of Engineering Mathematics
University of Bristol
M
MACMILLAN
© Gordon Reece 1986
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with the provisions ofthe Copyright Act 1956 (as amended).
Any person who does any unauthorised act in relation to
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civil claims for damages.
First published 1986
Published by
MACMILLAN EDUCATION LTD
Houndmills, Basingstoke, Hampshire RG21 2XS
and London
Companies and representatives
throughout the world
Typeset by TecSet Ltd,
Wallington, Surrey
British Library Cataloguing in Publication Data
Reece, Gordon
Microcomputer modelling by finite
differences.
1. Differential equations - Data
processing 2. Mathematical physics -
Date processing
I. Title
530.1'5535 QC20.7.D5
ISBN 978-0-333-42692-0 ISBN 978-1-349-09051-8 (eBook)
DOI 10.1007/978-1-349-09051-8
Associated software diskette (see end of book)
ISBN 978-0-333-43937-1
For Nesta - who encouraged me
to start this book and who
would have loved to see it
finished - for Miriam, Helen
and David
Contents
Preface be
1 Introduction: What this Book is about 1
1.1 Why numerical methods? 1
1.2 The problems we shall solve 3
1.3 The computer program 4
2 The Finite-difference Method 7
2.1 Finite-difference methods 7
2.2 Second-order problems 13
2.3 The general second-order problem 18
2.4 Graphics 21
3 Convergence and Divergence 25
3.1 Convergence 25
3.2 Divergence 27
3.3 A cautionary tale 27
3.4 Solving systems of linear equations 29
4 Better Solution Methods 33
4.1 The general problem 33
4.2 The algorithm - a smart trick 34
4.3 The program revisited 36
4.4 A neater algorithm 40
4.5 Gradient boundary conditions 42
5 Further Improvements 45
5.1 Non-constant coefficients 45
5.2 Gradients at the boundaries 46
5.3 The non-uniform grid 47
5.4 The fmite-difference equations 48
6 More Ambitious Applications: The Heat Equation 51
6.1 Background 51
6.2 A simple form of the equation 54
6.3 The finite-difference form 54
Contents vii
7 Two-dimensional Phenomena - a Glimpse of Reality S9
7.1 Heat conduction in two dimensions 59
7.2 Deriving the heat equation 60
7.3 The fmite-difference form of the equation 62
7.4 The model of kpN etc. 62
7.5 Boundary conditions 64
Adiabatic boundary 64
Fixed-temperature boundaries 64
Finite heat transfer at the boundaries 65
Gradient-type boundary conditions 66
8 The THC (Transient Heat Conduction) Computer Program 68
8.1 Introduction 68
8.2 The MAIN control segment (1-999) 69
8.3 The START subroutine (1000-1999) 72
8.4 The PHYS subroutine (2000-2999) 75
8.5 The EDGE subroutine (3000-3999) 76
8.6 The WORK subroutine (4000-4999) 78
8.7 The PLOT subroutine (5000-5999) and the DRAW subroutine
(9000-9999) 82
9 Elementary Applications of the THC Program 84
9.1 Introduction 84
9.2 One-dimensional problems 84
9.2.1 Fixed temperatures at two ends of an insulated bar
(RUN 9.1) 84
9.2.2 Fully-developed laminar flow between parallel plates
(Couette flow) (RUN 9.2) 85
9.2.3 Flow in a circular pipe (poiseuille flow) (RUN 9.3) 86
9.2.4 The distribution of heat in an annular cylinder
(RUNS 9.4 and 9.5) 87
9.2.5 Cylindrical annulus with heat source (RUN 9.6) 88
9.3 Two-dimensional problems 88
9.3.1 The superposition principle (RUN 9.7) 88
9.3.2 Fully-developed laminar flow in a rectangular-sectioned
duct 89
9.3.3 The electric potential inside an infmite box 90
10 Further Applications of the THC Program 98
10.1 Update to allow for further modifications 98
10.2 Applications with standard-type boundary conditions 100
10.2.1 Rectangular intrusion 100
10.2.2 Hollow square 100
10.2.3 Non-uniform conductivity 101
10.3 Non-standard boundary conditions 101
viii Contents
10.3.1 Specified-gradient boundary conditions 101
10.3.2 Sine wave along an edge 101
10.4 Modifications to the contour plot: standardised contours 102
10.5 Epilogue 102
Appendix: Full Listing of the THC Program for the IBM PC
(Apple II modifications listed in REM statements) 115
Index 124
Details of associated software diskette 126
Preface
Mathematics should be fun. Too many maths teachers spend their time convincing
their pupils that maths is boring - and that it is difficult. I was lucky enough to
be taught by people who enjoyed maths and who showed me how to enjoy it.
One of the most exciting things that has happened in my life-time has been
the spread of knowledge and the removal of the mystique that used to surround
the possession of some items of knowledge. I do not mean just particle physics
or the structure of the earth's core - I mean things like how to mend a burst
pipe, service a car, buy a house or deal with the taxman. A new generation of
people has grown up that is not afraid to try things out for itself. Teachers and
publishers have a duty to provide people with books showing them how things
are done. 'Experts' have been reluctant to do this in the past - probably because
they were afraid that their expertise would be seen to be pretty trivial once any
one could use it. The usual excuse was that people might misuse the information
- well, that is our right.
Probably the greatest single liberator of the intellectual curiosity of ordinary
people will be the microcomputer. It is possible to buy a perfectly good computer
today for less than the cost of a new suit. With it there will usually come an
excellent manual on BASIC. Such computers can be used for playing games, but
far more excitingly they can be used to illustrate - even to solve - some prob
lems in maths that would have been virtually impossible to undertake when I
was born.
We have not even begun to explore the real capabilities of the smallest com
puters - in maths or in the physical and social sciences. Some attempts have
been made to use computers in teaching, and a whole area of study (CAL:
Computer Assisted Learning) exists, but CAL is probably only in its babyhood
and has yet to reach its infancy.
This book takes subjects that have never been tackled in schools and are
hardly covered in undergraduate courses. Using the simplest (or anyway the
most popular) computer language, BASIC, and a knowledge of maths little
beyond O-level, it illustrates most of the ideas with computer programs that can
be run on most microcomputers. You will obviously have to make occasional
slight modifications to handle the variations in BASIC dialects.
The spirit of this book is, I hope, entirely in keeping with the ideal of making
the maths in it be fun. I enjoy it and hope you will too. The subjects covered are
fmite-difference methods and how they can be used to solve real problems in
physics and engineering.
be