Computational Physics

331 Pages·2018·31.617 MB·English

by  Michael Bestehorn

MichaelBestehorn
ComputationalPhysics
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Michael Bestehorn
Computational
Physics
|
With Worked Out Examples in FORTRAN® and MATLAB®
PhysicsandAstronomyClassificationScheme2010
05.10.Ln,05.45.-a,05.65.+b,07.05.Mh,44.05.+e,45.10.-b,47.11.-j,47.85.-g
Author
Prof.Dr.MichaelBestehorn
BTUCottbus-Senftenberg
Dept.ofStatisticalPhysicsandNonlinearDynamics
Erich-Weinert-Str.1
03046Cottbus
Germany
[email protected]
ISBN978-3-11-051513-8
e-ISBN(PDF)978-3-11-051514-5
e-ISBN(EPUB)978-3-11-051521-3
LibraryofCongressControlNumber:2018934247
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2018WalterdeGruyterGmbH,Berlin/Boston
Coverimage:MichaelBestehorn
Typesetting:le-texpublishingservicesGmbH,Leipzig
Printingandbinding:CPIbooksGmbH,Leck
♾Printedonacid-freepaper
PrintedinGermany
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Contents
1 Introduction|1
1.1 Goal,contents,andoutline|1
1.2 Theenvironmentrequiredforprogramdevelopment|4
1.2.1 Operatingsystem|4
1.2.2 Softwarepackages|5
1.2.3 Graphics|5
1.2.4 Programdevelopmentandasimplescript|6
1.3 Afirstexample–thelogisticmap|7
1.3.1 Map|7
1.3.2 FORTRAN|8
1.3.3 Problems|11
2 Nonlinearmaps|12
2.1 Frenkel–Kotorovamodel|12
2.1.1 Classicalformulation|12
2.1.2 Equilibriumsolutions|13
2.1.3 Thestandardmap|13
2.1.4 Problems|14
2.2 ChaosandLyapunovexponents|15
2.2.1 Stability,butterflyeffect,andchaos|15
2.2.2 Lyapunovexponentofthelogisticmap|16
2.2.3 Lyapunovexponentsformultidimensionalmaps|17
2.3 Affinemapsandfractals|20
2.3.1 Sierpinskitriangle|22
2.3.2 Aboutfernsandotherplants|23
2.3.3 Problems|24
2.4 Fractaldimension|25
2.4.1 Box-counting|25
2.4.2 Application:Sierpinskitriangle|26
2.4.3 Problem|27
2.5 Neuralnetworks|27
2.5.1 Perceptron|28
2.5.2 Self-organizedmaps:Kohonen’smodel|34
2.5.3 Problems|39
3 Dynamicalsystems|40
3.1 Quasilineardifferentialequations|40
3.1.1 Example:logisticmapandlogisticODE|41
3.1.2 Problems|42
VI | Contents
3.2 Fixedpointsandinstabilities|43
3.2.1 Fixedpoints|43
3.2.2 Stability|43
3.2.3 Trajectories|44
3.2.4 Gradientdynamics|45
3.2.5 SpecialcaseN=1|45
3.2.6 SpecialcaseN=2|45
3.2.7 SpecialcaseN=3|47
3.3 Hamiltoniansystems|50
3.3.1 Hamiltonfunctionandcanonicalequations|50
3.3.2 Symplecticintegrators|51
3.3.3 Poincarésection|56
4 OrdinarydifferentialequationsI|58
4.1 Newton’smechanics|58
4.1.1 Equationsofmotion|58
4.1.2 Themathematicalpendulum|59
4.2 Numericalmethodsofthelowestorder|60
4.2.1 Eulermethod|60
4.2.2 NumericalstabilityoftheEulermethod|60
4.2.3 Implicitandexplicitmethods|63
4.3 Higherordermethods|64
4.3.1 Heun’smethod|64
4.3.2 Problems|67
4.3.3 Runge–Kuttamethod|67
4.4 RK4applications:celestialmechanics|73
4.4.1 Keplerproblem:closedorbits|73
4.4.2 Quasiperiodicorbitsandapsidalprecession|76
4.4.3 Multipleplanets:isoursolarsystemstable?|76
4.4.4 Thereducedthree-bodyproblem|79
4.4.5 Problems|86
4.5 Moleculardynamics(MD)|86
4.5.1 Classicalformulation|86
4.5.2 Boundaryconditions|87
4.5.3 Microcanonicalandcanonicalensemble|88
4.5.4 Asymplecticalgorithm|89
4.5.5 Evaluation|90
4.5.6 Problems|94
4.6 Chaos|95
4.6.1 Harmonicallydrivenpendulum|95
4.6.2 Poincarésectionandbifurcationdiagrams|97
Contents | VII
4.6.3 Lyapunovexponents|98
4.6.4 Fractaldimension|106
4.6.5 Reconstructionofattractors|109
4.7 ODEswithperiodiccoefficients|111
4.7.1 Floquettheorem|111
4.7.2 Stabilityoflimitcycles|112
4.7.3 Parametricinstability:pendulumwithanoscillatingsupport|112
4.7.4 Mathieuequation|114
4.7.5 Problems|116
5 OrdinarydifferentialequationsII|117
5.1 Preliminaryremarks|117
5.1.1 Boundaryconditions|117
5.1.2 Example:ballisticflight|118
5.2 Finitedifferences|118
5.2.1 Discretization|119
5.2.2 Example:Schrödingerequation|122
5.3 Weightedresidualmethods|128
5.3.1 Weightandbasefunctions|128
5.3.2 Example:Starkeffect|130
5.4 Nonlinearboundaryvalueproblems|132
5.4.1 Nonlinearsystems|132
5.4.2 Newton–Raphson|133
5.4.3 Example:thenonlinearSchrödingerequation|134
5.4.4 Example:amoonshot|136
5.5 Shooting|140
5.5.1 Themethod|140
5.5.2 Example:freefallwithquadraticfriction|141
5.5.3 Systemsofequations|142
5.6 Problems|143
6 PartialdifferentialequationsI,basics|144
6.1 Classification|144
6.1.1 PDEsofthefirstorder|144
6.1.2 PDEsofthesecondorder|147
6.1.3 Boundaryandinitialconditions|149
6.2 Finitedifferences|153
6.2.1 Discretization|153
6.2.2 EllipticPDEs,example:Poissonequation|156
6.2.3 ParabolicPDEs,example:heatequation|162
6.2.4 HyperbolicPDEs,example:convectionequation,waveequation|167
VIII | Contents
6.3 Alternativediscretizationmethods|173
6.3.1 Chebyshevspectralmethod|173
6.3.2 SpectralmethodbyFouriertransformation|178
6.3.3 Finite-elementmethod|182
6.4 NonlinearPDEs|186
6.4.1 RealGinzburg–Landauequation|186
6.4.2 Numericalsolution,explicitmethod|188
6.4.3 Numericalsolution,semi-implicitmethod|189
6.4.4 Problems|190
7 PartialdifferentialequationsII,applications|193
7.1 Quantummechanicsinonedimension|193
7.1.1 Stationarytwo-particleequation|193
7.1.2 Time-dependentSchrödingerequation|196
7.2 Quantummechanicsintwodimensions|202
7.2.1 Schrödingerequation|202
7.2.2 Algorithm|202
7.2.3 Evaluation|203
7.3 Fluidmechanics:flowofanincompressibleliquid|204
7.3.1 Hydrodynamicbasicequations |204
7.3.2 Example:drivencavity|206
7.3.3 Thermalconvection:(A)squaregeometry|211
7.3.4 Thermalconvection:(B)Rayleigh–Bénardconvection|218
7.4 Patternformationoutofequilibrium|225
7.4.1 Reaction-diffusionsystems|225
7.4.2 Swift–Hohenbergequation|234
7.4.3 Problems|238
8 MonteCarlomethods(MC)|240
8.1 Randomnumbersanddistributions|240
8.1.1 Randomnumbergenerator|240
8.1.2 Distributionfunction,probabilitydensity,meanvalues|241
8.1.3 Otherdistributionfunctions|242
8.2 MonteCarlointegration|245
8.2.1 Integralsinonedimension|245
8.2.2 Integralsinhigherdimensions|247
8.3 Applicationsfromstatisticalphysics|250
8.3.1 Two-dimensionalclassicalgas|250
8.3.2 TheIsingmodel|255
8.4 Differentialequationsderivedfromvariationalproblems|265
8.4.1 Diffusionequation|265
8.4.2 Swift–Hohenbergequation|267
Contents | IX
A Matricesandsystemsoflinearequations|270
A.1 Realmatrices|270
A.1.1 Eigenvaluesandeigenvectors|270
A.1.2 Characteristicpolynomial|270
A.1.3 Notations|271
A.1.4 Normalmatrices|271
A.2 Complexmatrices|272
A.2.1 Notations|272
A.2.2 Jordancanonicalform|273
A.3 Inhomogeneoussystemsoflinearequations|274
A.3.1 Regularandsingularsystemmatrices|274
A.3.2 Fredholmalternative|275
A.3.3 Regularmatrices|275
A.3.4 LUdecomposition|276
A.3.5 Thomasalgorithm|278
A.4 Homogeneoussystemsoflinearequations|279
A.4.1 Eigenvalueproblems|279
A.4.2 Diagonalization|279
A.4.3 Application:zerosofapolynomial|282
B Programlibrary|284
B.1 Routines|284
B.2 Graphics|285
B.2.1 init|285
B.2.2 contur|285
B.2.3 contur1|285
B.2.4 ccontu|286
B.2.5 image|286
B.2.6 ccircl|286
B.3 Runge–Kutta|286
B.3.1 rkg|286
B.3.2 drkg|287
B.3.3 drkadt|287
B.4 Miscellaneous|287
B.4.1 tridag–Thomasalgorithm|287
B.4.2 ctrida|287
B.4.3 dlyap_exp–Lyapunovexponents|288
B.4.4 schmid–orthogonalization|288
B.4.5 FUNCTIONvolum–volumeinndimensions|288
B.4.6 FUNCTIONdeter–determinant|289
B.4.7 random_init–randomnumbers|289