Advanced Numerical Methods with Matlab 2: Resolution of Nonlinear, Differential and Partial Differential Equations

211 Pages·2018·4.24 MB·English

by  Bouchaib Radi

Advanced Numerical Methods with Matlab® 2
Mathematical and Mechanical Engineering Set
coordinated by
Abdelkhalak El Hami
Volume 7
Advanced Numerical
®
Methods with Matlab 2

Resolution of Nonlinear, Differential and
Partial Differential Equations

Bouchaib Radi
Abdelkhalak El Hami
First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
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© ISTE Ltd 2018
The rights of Bouchaib Radi and Abdelkhalak El Hami to be identified as the authors of this work have
been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2018934991

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A CIP record for this book is available from the British Library
ISBN 978-1-78630-293-9
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Part1.SolvingEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter1.SolvingNonlinearEquations . . . . . . . . . . . . . . . . . . 3
1.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.Separatingtheroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.Approximatingaseparatedroot . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1.Bisectionmethod(ordichotomymethod). . . . . . . . . . . . . . . 4
1.3.2.Fixed-pointmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.3.Firstconvergencecriterion . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.4.Iterativestoppingcriteria . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.5.Secondconvergencecriterion(localcriterion) . . . . . . . . . . . . 9
1.3.6.Newton’smethod(orthemethodoftangents) . . . . . . . . . . . . 10
1.3.7.Secantmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.8.Regulafalsimethod(orfalsepositionmethod) . . . . . . . . . . . . 17
1.4.Orderofaniterativeprocess . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.UsingMatlab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.1.Findingtherootsofpolynomials. . . . . . . . . . . . . . . . . . . . 19
1.5.2.Bisectionmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5.3.Newton’smethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter2.NumericallySolvingDifferentialEquations . . . . . . . . . 25
2.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.Cauchyproblemanddiscretization . . . . . . . . . . . . . . . . . . . . 27
2.3.Euler’smethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1.Interpretation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2.Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
vi AdvancedNumericalMethodswithMatlab2
2.4.One-stepRunge–Kuttamethod . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1.Second-orderRunge–Kuttamethod . . . . . . . . . . . . . . . . . . 32
2.4.2.Fourth-orderRunge–Kuttamethod . . . . . . . . . . . . . . . . . . 33
2.5.Multi-stepAdamsmethods . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.1.OpenAdamsmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.2.ClosedAdamsformulas. . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.Predictor–Correctormethod . . . . . . . . . . . . . . . . . . . . . . . . 41
2.7.UsingMatlab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Part2.SolvingPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter3.FiniteDifferenceMethods . . . . . . . . . . . . . . . . . . . . 49
3.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.Presentationofthefinitedifferencemethod. . . . . . . . . . . . . . . . 51
3.2.1.Convergence,consistencyandstability . . . . . . . . . . . . . . . . 53
3.2.2.Courant–Friedrichs–Lewycondition . . . . . . . . . . . . . . . . . 56
3.2.3.VonNeumannstabilityanalysis . . . . . . . . . . . . . . . . . . . . 57
3.3.Hyperbolicequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.1.Keyresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.2.Numericalschemesforsolvingthetransport
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.3.Waveequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.4.Burgersequation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.Ellipticequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4.1.Poissonequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.Parabolicequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5.1.Heatequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6.UsingMatlab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Chapter4.FiniteElementMethod . . . . . . . . . . . . . . . . . . . . . . 83
4.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2.One-dimensionalfiniteelementmethods . . . . . . . . . . . . . . . . . 83
4.3.Two-dimensionalfiniteelementmethods . . . . . . . . . . . . . . . . . 88
4.4.Generalprocedureofthemethod . . . . . . . . . . . . . . . . . . . . . 93
4.5.Finiteelementmethodforcomputingelasticstructures . . . . . . . . . 93
4.5.1.Linearelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.5.2.Variationalformulationofthelinearelasticity
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.5.3.Planarlinearelasticityproblems . . . . . . . . . . . . . . . . . . . . 99
4.5.4.Applyingthefiniteelementmethodtoplanar
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.5.5.Axisymmetricproblems . . . . . . . . . . . . . . . . . . . . . . . . 105
4.5.6.Three-dimensionalproblems . . . . . . . . . . . . . . . . . . . . . . 107
Contents vii
4.6.UsingMatlab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.6.1.SolvingPoisson’sequation . . . . . . . . . . . . . . . . . . . . . . . 108
4.6.2.Solvingtheheatequation . . . . . . . . . . . . . . . . . . . . . . . . 111
4.6.3.Computingstructures . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Chapter5.FiniteVolumeMethods . . . . . . . . . . . . . . . . . . . . . . 117
5.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.Finitevolumemethod(FVM) . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.1.Conservationpropertiesofthemethod . . . . . . . . . . . . . . . . 118
5.2.2.Thestagesofthemethod . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2.3.Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.4.Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.5.Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3.Advectionschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.3.1.Two-dimensionalFVM . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3.2.Convection-diffusionequation . . . . . . . . . . . . . . . . . . . . . 129
5.3.3.Centraldifferencingscheme . . . . . . . . . . . . . . . . . . . . . . 131
5.3.4.Upwind(decentered)scheme. . . . . . . . . . . . . . . . . . . . . . 133
5.3.5.Hybridscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.6.Power-lawscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.7.QUICKscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3.8.Higher-orderschemes . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3.9.Unsteadyone-dimensionalconvection-diffusion
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3.10.Explicitscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.3.11.Crank–Nicolsonscheme. . . . . . . . . . . . . . . . . . . . . . . . 142
5.3.12.Implicitscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4.UsingMatlab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Chapter6.MeshlessMethods . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.2.LimitationsoftheFEMandmotivationofmeshless
methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.3.Examplesofmeshlessmethods . . . . . . . . . . . . . . . . . . . . . . 148
6.3.1.Advantagesofmeshlessmethods . . . . . . . . . . . . . . . . . . . 149
6.3.2.Disadvantagesofmeshlessmethods . . . . . . . . . . . . . . . . . . 150
6.3.3.Comparisonofthefiniteelementmethod
andmeshlessmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.4.Basisofmeshlessmethods . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.4.1.Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.4.2.Kernel(weight)functions. . . . . . . . . . . . . . . . . . . . . . . . 152
6.4.3.Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.4.4.Partitionofunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
viii AdvancedNumericalMethodswithMatlab2
6.5.Meshlessmethod(EFG) . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.5.1.Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.5.2.MovingLeast-SquaresApproximation . . . . . . . . . . . . . . . . 153
6.6.Applicationofthemeshlessmethodtoelasticity . . . . . . . . . . . . . 163
6.6.1.Formulationofstaticlinearelasticity . . . . . . . . . . . . . . . . . 163
6.6.2.Imposingessentialboundaryconditions. . . . . . . . . . . . . . . . 165
6.7.Numericalexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.7.1.Fixed-freebeam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.7.2.Compressedblock . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.8.UsingMatlab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Part3.Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Appendix1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Appendix2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Preface
Most physical problems can be expressed in the form of mathematical equations
(e.g. differential equations, integral equations). Historically, mathematicians had to
findanalyticsolutionstotheequationsencounteredinengineeringandrelatedfields
(e.g. mechanics, physics, biology). These equations are sometimes highly complex,
requiring significant work to be simplified. However, in the mid-20th Century, the
introduction of the first computers gave rise to new methods for solving equations:
numerical methods. This new approach allows us to solve the equations that we
encounter(whenconstructingmodels)asaccuratelyaspossible,therebyenablingus
to approximate the solutions of the problems that we are studying. These
approximate solutions are typically calculated by computers using suitable
algorithms.
Practicalexperiencehasshownthat,comparedtostandardnumericalapproaches,
a carefully planned and optimized methodology can improve the speed of
computation by a factor of 100 or even higher. This can transform a completely
unreasonable calculation into a perfectly routine computation, hence our great
interest in numerical methods! Clearly, it is important for researchers and engineers
to understand the methods that they are using and, in particular, the limitations and
advantages associated with each approach. The computations needed by most
scientific fields require techniques to represent functions as well as algorithms to
calculatederivativesandintegrals, solvedifferentialequations, locatezeros, findthe
eigenvectorsandeigenvaluesofamatrix,andmuchmore.
The objective of this book is to present and study the fundamental numerical
methods that allow scientific computations to be executed. This involves
implementing a suitable methodology for the scientific problem at hand, whether
derived from physics (e.g. meteorology, pollution) or engineering (e.g. structural
mechanics,fluidmechanics,signalprocessing).
x AdvancedNumericalMethodswithMatlab2
This book is divided into two parts, with two appendices. The first part contains
twochaptersdedicatedtosolvingnonlinearequationsanddifferentialequations.The
secondpartconsistsoffourchaptersonthevariousnumericalmethodsthatareusedto
solve partial differential equations: finite differences, finite elements, finite volumes
andmeshlessmethods.
Each chapter starts with a brief overview of relevant theoretical concepts and
definitions, with a range of illustrative numerical examples and graphics. At the end
of each chapter, we introduce the reader to the various Matlab commands for
implementing the methods that have been discussed. As is often the case, practical
applications play an essential role in understanding and mastering these methods.
Thereislittlehopeofbeingabletoassimilatethemwithouttheopportunitytoapply
themtoarangeofconcreteexamples.Accordingly,wewillpresentvariousexamples
and explore them with Matlab. These examples can be used as a starting point for
practicalexploration.
Matlabiscurrentlywidelyusedinteaching,industryandresearch.Ithasbecome
astandardtool invariousfields thanksto itsintegratedtoolboxes(e.g.optimization,
statistics, control, image processing). Graphical interfaces have been improved
considerably in recent versions. One of our appendices is dedicated to introducing
readerstoMatlab.
BouchaibRADI
AbdelkhalakELHAMI
March2018