Computational methods for physicists: compendium for students
Gespeichert in:
| Späterer Titel: | Širca, Simon Computational methods in physics |
|---|---|
| Hauptverfasser: | , |
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Heidelberg [u.a.]
Springer
2012
|
| Schriftenreihe: | Graduate texts in physics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XX, 715 S. Ill., graph. Darst. |
| ISBN: | 3642324770 9783642324772 |
Internformat
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| 245 | 1 | 0 | |a Computational methods for physicists |b compendium for students |c Simon Širca ; Martin Horvat |
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| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-642-32478-9 |
| 785 | 0 | 0 | |i Fortgesetzt durch |a Širca, Simon |t Computational methods in physics |d Cham : Springer, [2018] |z 978-3-319-78618-6 |
| 856 | 4 | 2 | |m X:MVB |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=4091875&prov=M&dok_var=1&dok_ext=htm |3 Inhaltstext |
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Datensatz im Suchindex
| _version_ | 1807955013163548672 |
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| adam_text |
IMAGE 1
CONTENTS
1 BASICS O F NUMERICAL ANALYSIS ]
1.1 INTRODUCTION 1
1.1.1 FINITE-PRECISION ARITHMETIC 1
1.2 APPROXIMATION O F EXPRESSIONS 6
1.2.1 OPTIMAL (MINIMAX) AND ALMOST OPTIMAL APPROXIMATIONS . 6 1.2.2
RATIONAL (PADE) APPROXIMATION 9
1.2.3 SUMMATION O F SERIES BY USING PADE APPROXIMATIONS (WYNN'S
E-ALGORITHM) 12
1.2.4 APPROXIMATION O F THE EVOLUTION OPERATOR FOR A HAMILTONIAN SYSTEM
14
1.3 POWER AND ASYMPTOTIC EXPANSION, ASYMPTOTIC ANALYSIS 16
1.3.1 POWER EXPANSION 17
1.3.2 ASYMPTOTIC EXPANSION 17
1.3.3 ASYMPTOTIC ANALYSIS OF INTEGRALS BY INTEGRATION BY PARTS . . 19
1.3.4 ASYMPTOTIC ANALYSIS OF INTEGRALS BY THE LAPLACE METHOD . 21 1.3.5
STATIONARY-PHASE APPROXIMATION 2 4
1.3.6 DIFFERENTIAL EQUATIONS WITH LARGE PARAMETERS 27
1.4 SUMMATION O F FINITE AND INFINITE SERIES 31
1.4.1 TESTS O F CONVERGENCE 32
1.4.2 SUMMATION OF SERIES IN FLOATING-POINT ARITHMETIC 33
1.4.3 ACCELERATION OF CONVERGENCE 36
1.4.4 ALTERNATING SERIES 38
1.4.5 LEVIN'S TRANSFORMATIONS 4 2
1.4.6 POISSON SUMMATION 44
1.4.7 BOREL SUMMATION 44
1.4.8 ABEL SUMMATION 45
1.5 PROBLEMS 46
1.5.1 INTEGRAL O F THE GAUSS DISTRIBUTION 4 6
1.5.2 AIRY FUNCTIONS 48
1.5.3 BESSEL FUNCTIONS 50
IX
HTTP://D-NB.INFO/1024548007
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X C O N T E N T S
1 .5.4 ALTERNATING SERIES 51
1.5.5 COULOMB SCATTERING AMPLITUDE AND BOREL RESUMMATION . . 52
REFERENCES 53
2 SOLVING NON-LINEAR EQUATIONS 57
2.1 SCALAR LIQUATIONS 59
2.1.1 BISECTION 59
2.1.2 THE FAMILY O F NEWTON'S METHODS AND THE NEWTONRAPHSON METHOD 6 0
2.1.3 THE SECANT METHOD AND ITS RELATIVES 64
2.1.4 MULLER'S METHOD 65
2.2 VECTOR EQUATIONS 67
2.2.1 NEWTON-RAPHSON'S METHOD 67
2.2.2 BROYDEN'S (SECANT) METHOD 69
2.3 CONVERGENCE ACCELERATION * 72
2.4 POLYNOMIAL EQUATIONS O F A SINGLE VARIABLE 73
2.4.1 LOCATING THE REGIONS CONTAINING ZEROS 75
2.4.2 DESCARTES' RULE AND THE STURM METHOD 77
2.4.3 NEWTON'S SUMS AND IN VIETO'S FORMULAS 79
2.4.4 ELIMINATING MULTIPLE ZEROS OF THE POLYNOMIAL 80
2.4.5 CONDITIONING O F THE COMPUTATION O F ZEROS 81
2.4.6 GENERAL HINTS FOR THE COMPUTATION O F ZEROS 81
2.4.7 BERNOULLI'S METHOD 82
2.4.8 HORNER'S LINEAR METHOD 83
2.4.9 BAIRSTOW'S (HORNER'S QUADRATIC) METHOD 84
2.4.10 LAGUERRE'S METHOD 87
2.4.11 MAEHLY-NEWTON-RAPHSON'S METHOD 88
2.5 ALGEBRAIC EQUATIONS O F SEVERAL VARIABLES * 89
2.6 PROBLEMS 94
2.6.1 WIEN'S LAW AND LAMBERT'S FUNCTION 9 4
2.6.2 HEISENBERG'S MODEL IN THE MEAN-FIELD APPROXIMATION . . . 96 2.6.3
ENERGY LEVELS O F SIMPLE ONE-DIMENSIONAL QUANTUM SYSTEMS 97
2.6.4 PROPANE COMBUSTION IN AIR 99
2.6.5 FLUID FLOW THROUGH SYSTEMS O F PIPES 100
2.6.6 AUTOMATED ASSEMBLY O F STRUCTURES 103
REFERENCES 106
3 MATRIX METHODS 109
3.1 BASIC OPERATIONS 109
3.1.1 MATRIX MULTIPLICATION 109
3.1.2 COMPUTING THE DETERMINANT I L L
3.2 SYSTEMS O F LINEAR EQUATIONS A X = B I L L
3.2.1 ANALYSIS O F ERRORS I L L
3.2.2 GAUSS ELIMINATION 113
3.2.3 SYSTEMS WITH BANDED MATRICES 115
IMAGE 3
CONTENTS XI
3.2.4 TOEPLITZ SYSTEMS 115
3.2.5 VANDERMONDE SYSTEMS 116
3.2.6 CONDITION ESTIMATES FOR MATRIX INVERSION 118
3.2.7 SPARSE MATRICES 118
3.3 LINEAR LEAST-SQUARE PROBLEM AND ORTHOGONALIZATION 119
3.3.1 THE Q R DECOMPOSITION 120
3.3.2 SINGULAR VALUE DECOMPOSITION (SVD) 122
3.3.3 THE MINIMAL SOLUTION O F THE LEAST-SQUARES PROBLEM . . . . 126 3.4
MATRIX EIGENVALUE PROBLEMS 127
3.4.1 NON-SYMMETRIC PROBLEMS 128
3.4.2 SYMMETRIC PROBLEMS 130
3.4.3 GENERALIZED EIGENVALUE PROBLEMS 133
3.4.4 CONVERTING A MATRIX TO ITS JORDAN FORM 134
3.4.5 EIGENVALUE PROBLEMS FOR SPARSE MATRICES 136
3.5 RANDOM MATRICES * 136
3.5.1 GENERAL RANDOM MATRICES 136
3.5.2 GAUSSIAN ORTHOGONAL OR UNITARY ENSEMBLE 139
3.5.3 CYCLIC ORTHOGONAL AND UNITARY ENSEMBLE 142
3.6 PROBLEMS 144
3.6.1 PERCOLATION IN A RANDOM-LATTICE MODEL 144
3.6.2 ELECTRIC CIRCUITS O F LINEAR ELEMENTS 146
3.6.3 SYSTEMS O F OSCILLATORS 147
3.6.4 IMAGE COMPRESSION BY SINGULAR VALUE DECOMPOSITION . . . 147 3.6.5
EIGENSTATES OF PARTICLES IN THE ANHARMONIC POTENTIAL . . . . 148 3.6.6
ANDERSON LOCALIZATION 150
3.6.7 SPECTRA O F RANDOM SYMMETRIC MATRICES 152
REFERENCES 154
4 TRANSFORMATIONS OF FUNCTIONS AND SIGNALS 159
4.1 FOURIER TRANSFORMATION 159
4.2 FOURIER SERIES 161
4.2.1 CONTINUOUS FOURIER EXPANSION 161
4.2.2 DISCRETE FOURIER EXPANSION 163
4.2.3 ALIASING 166
4.2.4 LEAKAGE 167
4.2.5 FAST DISCRETE FOURIER TRANSFORMATION (FFT) 168
4.2.6 MULTIPLICATION OF POLYNOMIALS BY USING THE FFT 170
4.2.7 POWER SPECTRAL DENSITY 171
4.3 TRANSFORMATIONS WITH ORTHOGONAL POLYNOMIALS 172
4.3.1 LEGENDRE POLYNOMIALS 174
4.3.2 CHEBYSHEV POLYNOMIALS 178
4.4 LAPLACE TRANSFORMATION 181
4.4.1 USE OF LAPLACE TRANSFORMATION WITH DIFFERENTIAL EQUATIONS 182
IMAGE 4
XII C O N T E N T S
4.5 HILBERT TRANSFORMATION * 184
4.5.1 ANALYTIC SIGNAL 186
4.5.2 KRAMERS-KRONIG RELATIONS 187
4.5.3 NUMERICAL COMPUTATION OF THE CONTINUOUS HILBERT TRANSFORM 190
4.5.4 DISCRETE HILBERT TRANSFORMATION 192
4.6 WAVELET TRANSFORMATION * 195
4.6.1 NUMERICAL COMPUTATION O F THE WAVELET TRANSFORM 197
4.6.2 DISCRETE WAVELET TRANSFORM 199
4.7 PROBLEMS 200
4.7.1 FOURIER SPECTRUM OF SIGNALS 200
4.7.2 FOURIER ANALYSIS O F THE DOPPLER EFFECT 200
4.7.3 USE O F LAPLACE TRANSFORMATION AND ITS INVERSE 201
4.7.4 USE O F THE WAVELET TRANSFORMATION 202
REFERENCES 203
5 STATISTICAL ANALYSIS AND MODELING O F DATA 207
5.1 BASIC DATA ANALYSIS 207
5.1.1 PROBABILITY DISTRIBUTIONS 207
5.1.2 MOMENTS OF DISTRIBUTIONS 208
5.1.3 UNCERTAINTIES OF MOMENTS OF DISTRIBUTIONS 209
5.2 ROBUST STATISTICS 210
5.2.1 HUNTING FOR OUTLIERS 212
5.2.2 M-ESTIMATES OF LOCATION 213
5.2.3 M-ESTIMATES O F SCALE 216
5.3 STATISTICAL TESTS 217
5.3.1 COMPUTING THE CONFIDENCE INTERVAL FOR THE SAMPLE MEAN . 217 5.3.2
COMPARING THE MEANS O F TWO SAMPLES WITH EQUAL VARIANCES 218
5.3.3 COMPARING THE MEANS OF TWO SAMPLES WITH DIFFERENT VARIANCES 219
5.3.4 DETERMINING THE CONFIDENCE INTERVAL FOR THE SAMPLE VARIANCE 220
5.3.5 COMPARING TWO SAMPLE VARIANCES 221
5.3.6 COMPARING HISTOGRAMMED DATA TO A KNOWN DISTRIBUTION . 223 5.3.7
COMPARING TWO SETS OF HISTOGRAMMED DATA 224
5.3.8 COMPARING NON-HISTOGRAMMED DATA TO A CONTINUOUS DISTRIBUTION 224
5.4 CORRELATION 225
5.4.1 LINEAR CORRELATION 225
5.4.2 NON-PARAMETRIC CORRELATION 226
5.5 LINEAR AND NON-LINEAR REGRESSION 227
5.5.1 LINEAR REGRESSION 228
5.5.2 REGRESSION WITH ORTHOGONAL POLYNOMIALS 229
5.5.3 LINEAR REGRESSION (FITTING A STRAIGHT LINE) 230
IMAGE 5
C O N T E N T S XIII
5.5.4 LINEAR REGRESSION (FITTING A STRAIGHT LINE) WITH ERRORS IN
BOTH COORDINATES 232
5.5.5 FITTING A CONSTANT 233
5.5.6 GENERALIZED LINEAR REGRESSION BY USING SVD 236
5.5.7 ROBUST METHODS FOR ONE-DIMENSIONAL REGRESSION 237
5.5.8 NON-LINEAR REGRESSION 239
5.6 MULTIPLE LINEAR REGRESSION 240
5.6.1 THE BASIC METHOD 240
5.6.2 PRINCIPAL-COMPONENT MULTIPLE REGRESSION 242
5.7 PRINCIPAL-COMPONENT ANALYSIS 244
5.7.1 PRINCIPAL COMPONENTS BY DIAGONALIZING THE COVARIANCE MATRIX 246
5.7.2 STANDARDIZATION O F DATA FOR PCA 248
5.7.3 PRINCIPAL COMPONENTS FROM THE SVD O F THE DATA MATRIX . . 249
5.7.4 IMPROVEMENTS O F PCA: NON-LINEARITY, ROBUSTNESS 249
5.8 CLUSTER ANALYSIS * 249
5.8.1 HIERARCHICAL CLUSTERING 250
5.8.2 PARTITIONING METHODS: ^-MEANS 253
5.8.3 GAUSSIAN MIXTURE CLUSTERING AND THE EM ALGORITHM . . . . 256 5.8.4
SPECTRAL METHODS 258
5.9 LINEAR DISCRIMINANT ANALYSIS * 259
5.9.1 BINARY CLASSIFICATION 259
5.9.2 LOGISTIC DISCRIMINANT ANALYSIS 261
5.9.3 ASSIGNMENT TO MULTIPLE CLASSES 262
5.10 CANONICAL CORRELATION ANALYSIS * 263
5.11 FACTOR ANALYSIS * 265
5.11.1 DETERMINING THE FACTORS AND WEIGHTS FROM THE COVARIANCE MATRIX
266
5.11.2 STANDARDIZATION O F DATA AND ROBUST FACTOR ANALYSIS . . . . 269
5.12 PROBLEMS 270
5.12.1 MULTIPLE REGRESSION 270
5.12.2 NUTRITIONAL VALUE O F FOOD 270
5.12.3 DISCRIMINATION OF RADAR SIGNALS FROM IONOSPHERIC REFLECTIONS 271
5.12.4 CANONICAL CORRELATION ANALYSIS O F OBJECTS IN THE CDFS AREA 271
REFERENCES 273
6 MODELING AND ANALYSIS O F TIME SERIES 277
6.1 RANDOM VARIABLES 278
6.1.1 BASIC DEFINITIONS 278
6.1.2 GENERATION OF RANDOM NUMBERS 279
6.2 RANDOM PROCESSES 280
6.2.1 BASIC DEFINITIONS 280
6.3 STABLE DISTRIBUTIONS AND RANDOM WALKS 283
6.3.1 CENTRAL LIMIT THEOREM 283
IMAGE 6
X I V C O N T E N T S
6.3.2 STABLE DISTRIBUTIONS 284
6.3.3 GENERALIZED CENTRAL LIMIT THEOREM 287
6.3.4 DISCRETE-TIME RANDOM WALKS 287
6.3.5 CONTINUOUS-TIME RANDOM WALKS 290
6.4 MARKOV CHAINS * 292
6.4.1 DISCRETE-TIME OR CLASSICAL MARKOV CHAINS 292
6.4.2 CONTINUOUS-TIME MARKOV CHAINS 297
6.5 NOISE 299
6.5.1 TYPES OF NOISE 300
6.5.2 GENERATION O F NOISE 302
6.6 TIME CORRELATION AND AUTO-CORRELATION 304
6.6.1 SAMPLE CORRELATIONS O F SIGNALS 306
6.6.2 REPRESENTATION O F TIME CORRELATIONS 308
6.6.3 FAST COMPUTATION OF DISCRETE SAMPLE CORRELATIONS 308
6.7 AUTO-REGRESSION ANALYSIS O F DISCRETE-TIME SIGNALS * 310
6.7.1 AUTO-REGRESSION (AR) MODEL 311
6.7.2 USE OF A R MODELS 314
6.7.3 ESTIMATE OF THE FOURIER SPECTRUM 316
6.8 INDEPENDENT COMPONENT ANALYSIS * 319
6.8.1 ESTIMATE O F THE SEPARATION MATRIX AND THE FASTICA ALGORITHM 321
6.8.2 THE FASTICA ALGORITHM 322
6.8.3 STABILIZATION O F THE FASTICA ALGORITHM 323
6.9 PROBLEMS 324
6.9.1 LOGISTIC MAP 324
6.9.2 DIFFUSION AND CHAOS IN THE STANDARD MAP 326
6.9.3 PHASE TRANSITIONS IN THE TWO-DIMENSIONAL ISING MODEL . . 328 6.9.4
INDEPENDENT COMPONENT ANALYSIS 329
REFERENCES 331
7 INITIAL-VALUE PROBLEMS FOR ODE 335
7.1 EVOLUTION EQUATIONS 335
7.2 EXPLICIT EULER'S METHODS 337
7.3 EXPLICIT METHODS O F THE RUNGE-KUTTA TYPE 339
7.4 ERRORS O F EXPLICIT METHODS 340
7.4.1 DISCRETIZATION AND ROUND-OFF ERRORS 341
7.4.2 CONSISTENCY, CONVERGENCE, STABILITY 342
7.4.3 RICHARDSON EXTRAPOLATION 343
7.4.4 EMBEDDED METHODS 344
7.4.5 AUTOMATIC STEP-SIZE CONTROL 346
7.5 STABILITY OF ONE-STEP METHODS 347
7.6 EXTRAPOLATION METHODS * 349
7.7 MULTI-STEP METHODS * 351
7.7.1 PREDICTOR-CORRECTOR METHODS 353
7.7.2 STABILITY OF MULTI-STEP METHODS 354
7.7.3 BACKWARD DIFFERENTIATION METHODS 356
IMAGE 7
CONTENTS X V
7.8 CONSERVATIVE SECOND-ORDER EQUATIONS 357
7.8.1 RUNGE-KUTTA-NYSTROM METHODS 358
7.8.2 MULTI-STEP METHODS 359
7.9 IMPLICIT SINGLE-STEP METHODS 359
7.9.1 SOLUTION BY NEWTON'S ITERATION 362
7.9.2 ROSENBROCK LINEARIZATION 363
7.10 STIFF PROBLEMS 365
7.11 IMPLICIT MULTI-STEP METHODS * 367
7.12 GEOMETRIC INTEGRATION * 368
7.12.1 PRESERVATION O F INVARIANTS 368
7.12.2 PRESERVATION O F THE SYMPLECTIC STRUCTURE 372
7.12.3 REVERSIBILITY AND SYMMETRY 373
7.12.4 MODIFIED HAMILTONIANS AND EQUATIONS OF MOTION . . . . 374
7.13 LIE-SERIES INTEGRATION * 375
7.13.1 TAYLOR EXPANSION OF THE TRAJECTORY 376
7.14 PROBLEMS 380
7.14.1 TIME DEPENDENCE O F FILAMENT TEMPERATURE 380
7.14.2 OBLIQUE PROJECTILE MOTION WITH DRAG FORCE AND WIND . . 380 7.14.3
INFLUENCE O F FOSSIL FUELS ON ATMOSPHERIC CO2 CONTENT . 381 7.14.4
SYNCHRONIZATION O F GLOBALLY COUPLED OSCILLATORS . . . . 383 7.14.5
EXCITATION OF MUSCLE FIBERS 384
7.14.6 RESTRICTED THREE-BODY PROBLEM (ARENSTORF ORBITS) . . . 386 7.14.7
LORENZ SYSTEM 388
7.14.8 SINE PENDULUM 389
7.14.9 CHARGED PARTICLES IN ELECTRIC AND MAGNETIC FIELDS . . . . 390
7.14.10 CHAOTIC SCATTERING 391
7.14.11 HYDROGEN BURNING IN THE PP I CHAIN 392
7.14.12 OREGONATOR 394
7.14.13 KEPLER'S PROBLEM 395
7.14.14 NORTHERN LIGHTS 396
7.14.15 GALACTIC DYNAMICS 397
REFERENCES 398
8 BOUNDARY-VALUE PROBLEMS FOR ODE 401
8.1 DIFFERENCE METHODS FOR SCALAR BOUNDARY-VALUE PROBLEMS 402 8.1.1
CONSISTENCY, STABILITY, AND CONVERGENCE 404
. 8.1.2 NON-LINEAR SCALAR BOUNDARY-VALUE PROBLEMS 405
8.2 DIFFERENCE METHODS FOR SYSTEMS O F BOUNDARY-VALUE PROBLEMS . . . 408
8.2.1 LINEAR SYSTEMS 411
8.2.2 SCHEMES O F HIGHER ORDERS 411
8.3 SHOOTING METHODS 413
8.3.1 SECOND-ORDER LINEAR EQUATIONS 414
8.3.2 SYSTEMS O F LINEAR SECOND-ORDER EQUATIONS 416
8.3.3 NON-LINEAR SECOND-ORDER EQUATIONS 418
8.3.4 SYSTEMS O F NON-LINEAR EQUATIONS 419
IMAGE 8
XVI C O N T E N T S
8.3.5 MULTIPLE (PARALLEL) SHOOTING 421
8.4 ASYMPTOTIC DISCRETIZATION SCHEMES * 424
8.4.1 DISCRETIZATION 426
8.5 COLLOCATION METHODS * 429
8.5.1 SCALAR LINEAR SECOND-ORDER BOUNDARY-VALUE PROBLEMS . . 430 8.5.2
SCALAR LINEAR BOUNDARY-VALUE PROBLEMS O F HIGHER ORDERS . 432 8.5.3
SCALAR NON-LINEAR BOUNDARY-VALUE PROBLEMS O F HIGHER ORDERS 436
8.5.4 SYSTEMS OF BOUNDARY-VALUE PROBLEMS 438
8.6 WEIGHTED-RESIDUAL METHODS * 439
8.7 BOUNDARY-VALUE PROBLEMS WITH EIGENVALUES 441
8.7.1 DIFFERENCE METHODS 443
8.7.2 SHOOTING METHODS WITH PRIIFER TRANSFORMATION 446
8.7.3 PRUESS METHOD 449
8.7.4 SINGULAR STURM-LIOUVILLE PROBLEMS 452
8.7.5 EIGENVALUE-DEPENDENT BOUNDARY CONDITIONS 453
8.8 ISOSPECTRAL PROBLEMS * 454
8.9 PROBLEMS 455
8.9.1 GELFAND-BRATU EQUATION 455
8.9.2 MEASLES EPIDEMIC 456
8.9.3 DIFFUSION-REACTION KINETICS IN A CATALYTIC PELLET 457
8.9.4 DEFLECTION O F A BEAM WITH INHOMOGENEOUS ELASTIC MODULUS 459
8.9.5 A BOUNDARY-LAYER PROBLEM 459
8.9.6 SMALL OSCILLATIONS OF AN INHOMOGENEOUS STRING 460
8.9.7 ONE-DIMENSIONAL SCHRODINGER EQUATION 462
8.9.8 A FOURTH-ORDER EIGENVALUE PROBLEM 463
REFERENCES 464
9 DIFFERENCE METHODS FOR ONE-DIMENSIONAL PDE 467
9.1 DISCRETIZATION OF THE DIFFERENTIAL EQUATION 469
9.2 DISCRETIZATION O F INITIAL AND BOUNDARY CONDITIONS 471
9.3 CONSISTENCY + 473
9.4 IMPLICIT SCHEMES 475
9.5 STABILITY AND CONVERGENCE * 476
9.5.1 INITIAL-VALUE PROBLEMS 476
9.5.2 INITIAL-BOUNDARY-VALUE PROBLEMS 479
9.6 ENERGY ESTIMATES AND THEOREMS ON MAXIMA * 481
9.6.1 ENERGY ESTIMATES 481
9.6.2 THEOREMS ON MAXIMA 482
9.7 HIGHER-ORDER SCHEMES 484
9.8 HYPERBOLIC EQUATIONS 485
9.8.1 EXPLICIT SCHEMES 486
9.8.2 IMPLICIT SCHEMES 489
9.8.3 WAVE EQUATION 490
IMAGE 9
CONTENTS X V I I
9.9 NON-LINEAR EQUATIONS AND EQUATIONS OF MIXED TYPE * 491
9.10 DISPERSION AND DISSIPATION * 494
9.11 SYSTEMS O F HYPERBOLIC AND PARABOLIC PDE * 497
9.12 CONSERVATION LAWS AND HIGH-RESOLUTION SCHEMES * 500
9.12.1 HIGH-RESOLUTION SCHEMES 502
9.12.2 LINEAR PROBLEM V, + C V X = 0 504
9.12.3 NON-LINEAR CONSERVATION LAWS OF THE FORM V, + = 0 505
9.13 PROBLEMS 505
9.13.1 DIFFUSION EQUATION 505
9.13.2 INITIAL-BOUNDARY VALUE PROBLEM FOR V, + C V X - 0 . . . . 506
9.13.3 DIRICHLET PROBLEM FOR A SYSTEM OF NON-LINEAR HYPERBOLIC PDE 507
9.13.4 SECOND-ORDER AND FOURTH-ORDER WAVE EQUATIONS . . . . 508 9.13.5
BURGERS EQUATION 509
9.13.6 THE SHOCK-TUBE PROBLEM 511
9.13.7 KORTEWEG-DE VRIES EQUATION 512
9.13.8 NON-STATIONARY SCHRODINGER EQUATION 513
9.13.9 NON-STATIONARY CUBIC SCHRODINGER EQUATION 515
REFERENCES 517
10 DIFFERENCE METHODS FOR PDE IN SEVERAL DIMENSIONS 519
10.1 PARABOLIC AND HYPERBOLIC PDE 519
10.1.1 PARABOLIC EQUATIONS 519
10.1.2 EXPLICIT SCHEME 520
10.1.3 CRANK-NICOLSON SCHEME 522
10.1.4 ALTERNATING DIRECTION IMPLICIT SCHEMES 523
10.1.5 THREE SPACE DIMENSIONS 526
10.1.6 HYPERBOLIC EQUATIONS 527
10.1.7 EXPLICIT SCHEMES 527
10.1.8 SCHEMES FOR EQUATIONS IN THE FORM O F CONSERVATION LAWS 528
10.1.9 IMPLICIT AND ADI SCHEMES 529
10.2 ELLIPTIC PDE 530
10.2.1 DIRICHLET BOUNDARY CONDITIONS 530
10.2.2 NEUMANN BOUNDARY CONDITIONS 532
10.2.3 MIXED BOUNDARY CONDITIONS 532
10.2.4 RELAXATION METHODS 532
10.2.5 CONJUGATE GRADIENT METHODS 537
10.3 HIGH-RESOLUTION SCHEMES * 537
10.4 PHYSICALLY MOTIVATED DISCRETIZATIONS 540
10.4.1 TWO-DIMENSIONAL DIFFUSION EQUATION IN POLAR COORDINATES 542
10.4.2 TWO-DIMENSIONAL POISSON EQUATION IN POLAR COORDINATES 544
IMAGE 10
X V I I I C O N T E N T S
10.5 BOUNDARY ELEMENT METHOD * 545
10.6 FINITE-ELEMENT METHOD * 549
10.6.1 ONE SPACE DIMENSION 549
10.6.2 TWO SPACE DIMENSIONS 553
10.7 MIMETIC DISCRETIZATIONS * 557
10.8 MULTI-GRID AND MESH-FREE METHODS * 557
10.8.1 A MESH-FREE METHOD BASED ON RADIAL BASIS FUNCTIONS . 559 10.9
PROBLEMS 560
10.9.1 TWO-DIMENSIONAL DIFFUSION EQUATION 560
10.9.2 NON-LINEAR DIFFUSION EQUATION 563
10.9.3 TWO-DIMENSIONAL POISSON EQUATION 565
10.9.4 HIGH-RESOLUTION SCHEMES FOR THE ADVECTION EQUATION . 567 10.9.5
TWO-DIMENSIONAL DIFFUSION EQUATION IN POLAR COORDINATES 568
10.9.6 TWO-DIMENSIONAL POISSON EQUATION IN POLAR COORDINATES 568
10.9.7 FINITE-ELEMENT METHOD 569
10.9.8 BOUNDARY ELEMENT METHOD FOR THE TWO-DIMENSIONAL LAPLACE EQUATION
570
REFERENCES 571
11 SPECTRAL METHODS FOR PDE 575
11.1 SPECTRAL REPRESENTATION OF SPATIAL DERIVATIVES 577
11.1.1 FOURIER SPECTRAL DERIVATIVES 577
11.1.2 LEGENDRE SPECTRAL DERIVATIVES 580
11.1.3 CHEBYSHEV SPECTRAL DERIVATIVES 581
11.1.4 COMPUTING THE CHEBYSHEV SPECTRAL DERIVATIVE BY FOURIER
TRANSFORMATION 583
11.2 GALERKIN METHODS 586
11.2.1 FOURIER-GALERKIN 586
11.2.2 LEGENDRE-GALERKIN 587
11.2.3 CHEBYSHEV-GALERKIN 589
11.2.4 TWO SPACE DIMENSIONS 591
11.2.5 NON-STATIONARY PROBLEMS 591
11.3 TAU METHODS 594
11.3.1 STATIONARY PROBLEMS 594
11.3.2 NON-STATIONARY PROBLEMS 596
11.4 COLLOCATION METHODS 597
11.4.1 STATIONARY PROBLEMS 598
11.4.2 NON-STATIONARY PROBLEMS 599
11.4.3 SPECTRAL ELEMENTS: COLLOCATION WITH FI-SPLINES 600
11.5 NON-LINEAR EQUATIONS 601
11.6 TIME INTEGRATION * 605
11.7 SEMI-INFINITE AND INFINITE DEFINITION DOMAINS * 606
11.8 COMPLEX GEOMETRIES * 607
IMAGE 11
CONTENTS X I X
11.9 PROBLEMS 607
11.9.1 GALERKIN METHODS FOR THE HELMHOLTZ EQUATION 607
11.9.2 GALERKIN METHODS FOR THE ADVECTION EQUATION 608
11.9.3 GALERKIN METHOD FOR THE DIFFUSION EQUATION 609
11.9.4 GALERKIN METHOD FOR THE POISSON EQUATION: POISEUILLE LAW 611
11.9.5 LEGENDRE TAU METHOD FOR THE POISSON EQUATION 613
11.9.6 COLLOCATION METHODS FOR THE DIFFUSION EQUATION I . . . . 614
11.9.7 COLLOCATION METHODS FOR THE DIFFUSION EQUATION II . . . 616
11.9.8 BURGERS EQUATION 617
REFERENCES 619
APPENDIX A MATHEMATICAL TOOLS 62 1
A.L ASYMPTOTIC NOTATION 621
A.2 THE NORMS IN SPACES L P ( Q ) AND 1 P OO 622
A.3 DISCRETE VECTOR NORMS 623
A.4 MATRIX AND OPERATOR NORMS 625
A.5 EIGENVALUES O F TRIDIAGONAL MATRICES 626
A.6 SINGULAR VALUES O F X AND EIGENVALUES O F X T X AND X X R 627
A.7 THE "SQUARE ROOT" O F A MATRIX 628
REFERENCES 628
APPENDIX B STANDARD NUMERICAL DATA TYPES 629
B.L REAL NUMBERS IN FLOATING-POINT ARITHMETIC 629
B . L . L COMBINING TYPES WITH DIFFERENT PRECISIONS 632
B.2 INTEGER NUMBERS 633
B.3 (ALMOST) ARBITRARY PRECISION 634
REFERENCES 635
APPENDIX C GENERATION O F PSEUDORANDOM NUMBERS 637
C . L UNIFORM GENERATORS: FROM INTEGERS TO REALS 637
C.2 TRANSFORMATIONS BETWEEN DISTRIBUTIONS 638
C.2.1 DISCRETE DISTRIBUTION 639
C.2.2 CONTINUOUS DISTRIBUTION 640
C.3 RANDOM NUMBER GENERATORS AND TESTS OF THEIR RELIABILITY . . . . 646
C.3.1 LINEAR GENERATORS 646
C.3.2 NON-LINEAR GENERATORS 648
C.3.3 USING AND TESTING GENERATORS 648
REFERENCES 649
APPENDIX D CONVERGENCE THEOREMS FOR ITERATIVE METHODS 65 1
D. 1 GENERAL THEOREMS 651
D.2 THEOREMS FOR THE NEWTON-RAPHSON METHOD 653
REFERENCES 654
IMAGE 12
X X
C O N T E N T S
APPENDIX E NUMERICAL INTEGRATION 655
E. 1 GAUSS QUADRATURE 657
E . L . L GAUSS-KRONROD QUADRATURE 658
E.1.2 QUADRATURE IN TWO DIMENSIONS 659
E.2 INTEGRATION OF RAPIDLY OSCILLATING FUNCTIONS 660
E.2.1 ASYMPTOTIC METHOD 660
E.2.2 FILON'S METHOD 662
E.3 INTEGRATION O F SINGULAR FUNCTIONS 664
REFERENCES 665
APPENDIX F FIXED POINTS AND STABILITY * 667
F. 1 LINEAR STABILITY 667
F.2 SPURIOUS FIXED POINTS 669
F.3 NON-LINEAR STABILITY 671
REFERENCES 673
APPENDIX G CONSTRUCTION O F SYMPLECTIC INTEGRATORS * 675
REFERENCES 680
APPENDIX H TRANSFORMING PDE TO SYSTEMS O F ODE: TWO WARNINGS . . 68 1 H
. L DIFFUSION EQUATION 681
H.2 ADVECTION EQUATION 684
REFERENCES 686
APPENDIX I NUMERICAL LIBRARIES, AUXILIARY TOOLS, AND LANGUAGES . . . 687
I.1 IMPORTANT NUMERICAL LIBRARIES 687
1.2 BASICS OF PROGRAM COMPILATION 690
1.3 USING LIBRARIES IN C/C++AND FORTRAN 691
1.3.1 SOLVING SYSTEMS O F EQUATIONS A X = B BY USING THE G S L LIBRARY
691
1.3.2 SOLVING THE SYSTEM A X = B IN C/C++ LANGUAGE AND FORTRAN LIBRARIES
692
1.3.3 SOLVING THE SYSTEM A X = B IN FORTRAN95 BY USING A FORTRAN77
LIBRARY 694
1.4 AUXILIARY TOOLS 694
1.5 CHOOSING THE PROGRAMMING LANGUAGE 696
REFERENCES 697
APPENDIX J MEASURING PROGRAM EXECUTION TIMES ON LINUX SYSTEMS . . 699
REFERENCES 702
INDEX 703 |
| any_adam_object | 1 |
| author | Širca, Simon 1969- Horvat, Martin 1977- |
| author_GND | (DE-588)1030054010 (DE-588)1030054118 |
| author_facet | Širca, Simon 1969- Horvat, Martin 1977- |
| author_role | aut aut |
| author_sort | Širca, Simon 1969- |
| author_variant | s š sš m h mh |
| building | Verbundindex |
| bvnumber | BV040717013 |
| classification_rvk | SK 950 SK 955 |
| classification_tum | PHY 016f |
| ctrlnum | (OCoLC)827409712 (DE-599)DNB1024548007 |
| dewey-full | 530.15 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.15 |
| dewey-search | 530.15 |
| dewey-sort | 3530.15 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| genre_facet | Lehrbuch |
| id | DE-604.BV040717013 |
| illustrated | Illustrated |
| indexdate | 2024-08-21T00:31:00Z |
| institution | BVB |
| isbn | 3642324770 9783642324772 |
| language | English |
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| oclc_num | 827409712 |
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| physical | XX, 715 S. Ill., graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Springer |
| record_format | marc |
| series2 | Graduate texts in physics |
| spelling | Širca, Simon 1969- Verfasser (DE-588)1030054010 aut Computational methods for physicists compendium for students Simon Širca ; Martin Horvat Heidelberg [u.a.] Springer 2012 XX, 715 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in physics Computerphysik (DE-588)4273564-6 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Computerphysik (DE-588)4273564-6 s DE-604 Horvat, Martin 1977- Verfasser (DE-588)1030054118 aut Erscheint auch als Online-Ausgabe 978-3-642-32478-9 Fortgesetzt durch Širca, Simon Computational methods in physics Cham : Springer, [2018] 978-3-319-78618-6 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4091875&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025697302&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Širca, Simon 1969- Horvat, Martin 1977- Computational methods for physicists compendium for students Computerphysik (DE-588)4273564-6 gnd |
| subject_GND | (DE-588)4273564-6 (DE-588)4123623-3 |
| title | Computational methods for physicists compendium for students |
| title_auth | Computational methods for physicists compendium for students |
| title_exact_search | Computational methods for physicists compendium for students |
| title_full | Computational methods for physicists compendium for students Simon Širca ; Martin Horvat |
| title_fullStr | Computational methods for physicists compendium for students Simon Širca ; Martin Horvat |
| title_full_unstemmed | Computational methods for physicists compendium for students Simon Širca ; Martin Horvat |
| title_new | Širca, Simon Computational methods in physics |
| title_short | Computational methods for physicists |
| title_sort | computational methods for physicists compendium for students |
| title_sub | compendium for students |
| topic | Computerphysik (DE-588)4273564-6 gnd |
| topic_facet | Computerphysik Lehrbuch |
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