Numerical mathematics:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2000
|
| Series: | Texts in applied mathematics
37 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 654 S. Ill., graph. Darst. |
| ISBN: | 0387989595 |
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| 240 | 1 | 0 | |a Mathematica numerica |
| 245 | 1 | 0 | |a Numerical mathematics |c Alfio Quarteroni ; Riccardo Sacco ; Fausto Saleri |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2000 | |
| 300 | |a XX, 654 S. |b Ill., graph. Darst. | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Texts in applied mathematics |v 37 | |
| 650 | 4 | |a Numerische Mathematik - Lehrbuch | |
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| 700 | 1 | |a Sacco, Riccardo |e Verfasser |4 aut | |
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I
ALFIO QUARTERONI RICCARDO SACCO
FAUSTO
SALERI
NUMERICAL MATHEMATICS
WITB 134 LUUSTRATIONS
SPRINGER
CONTENTS
SERIES PREFACE V
PREFACE VII
PART I: GETTING STARTED
1.
FOUNDATIONS OF MATRIX ANALYSIS 1
1.1 VECTOR SPACES 1
1.2 MATRICES 3
1.3 OPERATIONS WITH MATRICES 5
1.3.1 INVERSE OF A MATRIX 6
1.3.2 MATRICES AND LINEAR MAPPINGS 7
1.3.3 OPERATIONS WITH BLOCK-PARTITIONED MATRICES .
.
.
. 7
1.4
TRACE AND DETERMINANT OF A MATRIX 8
1.5 RANK AND KERNEL OF A MATRIX 9
1.6 SPECIAL MATRICES 10
1.6.1 BLOCK DIAGONAL MATRICES 10
1.6.2 TRAPEZOIDAL AND TRIANGULAER MATRICES 11
1.6.3 BANDED MATRICES 11
1.7 EIGENVALUES AND EIGENVECTORS 12
1.8 SIMILARITY TRANSFORMATIONS 14
1.9 THE SINGULAR VALUE DECOMPOSITION (SVD) 16
1.10 SCALAR PRODUCT AND NORMS IN VECTOR SPACES 17
1.11 MATRIX NORMS 21
XII CONTENTS
1.11.1
RELATION BETWEEN NORMS AND THE
SPECTRAL
RADIUS OF A MATRIX 25
1.11.2
SEQUENCES AND SERIES OF MATRICES 26
1.12 POSITIVE DEFINITE, DIAGONALLY DOMINANT AND M-MATRICES . 27
1.13
EXERCISES 30
2.
PRINCIPLES OF NUMERICAL MATHEMATICS 33
2.1 WELL-POSEDNESS AND CONDITION NUMBER OF A PROBLEM .
.
. 33
2.2
STABILITY OF NUMERICAL METHODS 37
2.2.1 RELATIONS BETWEEN STABILITY AND CONVERGENCE .
.
. 40
2.3
A PRIORI
AND
A POSTERIORI
ANALYSIS 41
2.4 SOURCES OF ERROR IN COMPUTATIONAL MODELS 43
2.5 MACHINE REPRESENTATION OF NUMBERS 45
2.5.1 THE POSITIONAL SYSTEM 45
2.5.2 THE FLOATING-POINT NUMBER SYSTEM 46
2.5.3 DISTRIBUTION OF FLOATING-POINT NUMBERS .
.
.
.
.
. 49
2.5.4 IEC/IEEE ARITHMETIC 49
2.5.5 ROUNDING OF A REAL NUMBER IN ITS
MACHINE
REPRESENTATION 50
2.5.6 MACHINE FLOATING-POINT OPERATIONS 52
2.6 EXERCISES 54
PART II: NUMERICAL LINEAR ALGEBRA
3.
DIRECT METHODS FOR THE SOLUTION OF LINEAR SYSTEMS 57
3.1 STABILITY ANALYSIS OF LINEAR SYSTEMS 58
3.1.1 THE CONDITION NUMBER OF A MATRIX 58
3.1.2 FORWARD
A PRIORI
ANALYSIS 60
3.1.3 BACKWARD
A PRIORI
ANALYSIS 63
3.1.4
A POSTERIORI
ANALYSIS 64
3.2 SOLUTION OF TRIANGULAER SYSTEMS 65
3.2.1 IMPLEMENTATION OF SUBSTITUTION METHODS 65
3.2.2 ROUNDING ERROR ANALYSIS 67
3.2.3 INVERSE OF A TRIANGULAER MATRIX 67
3.3 THE GAUSSIAN ELIMINATION METHOD (GEM) AND
LU
FACTORIZATION 68
3.3.1 GEM AS A FACTORIZATION METHOD 72
3.3.2 THE EFFECT OF ROUNDING ERRORS 76
3.3.3 IMPLEMENTATION OF LU FACTORIZATION 77
3.3.4 COMPACT FORMS OF FACTORIZATION 78
3.4 OTHER TYPES OF FACTORIZATION 79
3.4.1 LDM
T
FACTORIZATION 79
3.4.2 SYMMETRIE AND POSITIVE DEFINITE MATRICES:
THE
CHOLESKY FACTORIZATION 80
3.4.3 RECTANGULAR MATRICES: THE QR FACTORIZATION .
.
. 82
CONTENTS XIII
3.5 PIVOTING . 85
3.6 COMPUTING THE INVERSE OF A MATRIX 89
3.7 BANDED SYSTEMS 90
3.7.1 TRIDIAGONAL MATRICES 91
3.7.2 IMPLEMENTATION ISSUES 92
3.8 BLOCK SYSTEMS 93
3.8.1 BLOCK LU FACTORIZATION 94
3.8.2 INVERSE OF A BLOCK-PARTITIONED MATRIX 95
3.8.3 BLOCK TRIDIAGONAL SYSTEMS 95
3.9 SPARSE MATRICES 97
3.9.1 THE CUTHILL-MCKEE ALGORITHM 98
3.9.2 DECOMPOSITION INTO SUBSTRUCTURES 100
3.9.3 NESTED DISSECTION 103
3.10 ACCURACY OF THE SOLUTION ACHIEVED USING GEM 103
3.11 AN APPROXIMATE COMPUTATION OF
K
(A)
106
3.12 IMPROVING THE ACCURACY OF GEM 109
3.12.1 SCALING 110
3.12.2 ITERATIVE REFINEMENT 111
3.13 UNDETERMINED SYSTEMS 112
3.14 APPLICATIONS 115
3.14.1 NODAL ANALYSIS OF A STRUCTURED FRAME 115
3.14.2 REGULARIZATION OF A TRIANGULAER GRID 118
3.15 EXERCISES 121
4.
ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS 123
4.1 ON THE CONVERGENCE OF ITERATIVE METHODS 123
4.2 LINEAR ITERATIVE METHODS 126
4.2.1 JACOBI, GAUSS-SEIDEL AND RELAXATION METHODS . . . 127
4.2.2
CONVERGENCE RESULTS FOR JACOBI AND
GAUSS-SEIDEL
METHODS 129
4.2.3 CONVERGENCE RESULTS FOR THE RELAXATION METHOD . 131
4.2.4
A PRIORI
FORWARD ANALYSIS 132
4.2.5 BLOCK MATRICES 133
4.2.6 SYMMETRIE FORM OF THE GAUSS-SEIDEL AND
SOR
METHODS 133
4.2.7 IMPLEMENTATION ISSUES 135
4.3 STATIONARY AND NONSTATIONARY ITERATIVE METHODS 136
4.3.1 CONVERGENCE ANALYSIS OF THE RICHARDSON METHOD . 137
4.3.2
PRECONDITIONING MATRICES 139
4.3.3 THE GRADIENT METHOD 146
4.3.4 THE CONJUGATE GRADIENT METHOD 150
4.3.5 THE PRECONDITIONED CONJUGATE GRADIENT METHOD . 156
4.3.6
THE ALTERNATING-DIRECTION METHOD 158
4.4 METHODS BASED ON KRYLOV SUBSPACE ITERATIONS 159
4.4.1 THE ARNOLDI METHOD FOR LINEAR SYSTEMS 162
XIV CONTENTS
4.4.2 THE GMRES METHOD 165
4.4.3 THE LANCZOS METHOD FOR SYMMETRIE SYSTEMS . . . 167
4.5
THE LANCZOS METHOD FOR UNSYMMETRIC SYSTEMS 168
4.6 STOPPING CRITERIA 171
4.6.1 A STOPPING TEST BASED ON THE INCREMENT 172
4.6.2 A STOPPING TEST BASED ON THE RESIDUAL 174
4.7 APPLICATIONS . 174
4.7.1 ANALYSIS OF AN ELECTRIC NETWORK \ 174
4.7.2 FINITE DIFFERENCE ANALYSIS OF BEAM BENDING .
.
.
. 177
4.8
EXERCISES
179
5. APPROXIMATION OF EIGENVALUES AND EIGENVECTORS 183
5.1 GEOMETRICAL LOCATION OF THE EIGENVALUES 183
5.2 STABILITY AND CONDITIONING ANALYSIS 186
5.2.1
A PRIORI
ESTIMATES 186
5.2.2
A POSTERIORI
ESTIMATES 190
5.3 THE POWER METHOD 192
5.3.1 APPROXIMATION OF THE EIGENVALUE OF
LARGEST
MODULE 192
5.3.2 INVERSE ITERATION 195
5.3.3 IMPLEMENTATION ISSUES 196
5.4 THE QR ITERATION 200
5.5 THE BASIC QR ITERATION 201
5.6 THE QR METHOD FOR MATRICES IN HESSENBERG FORM 203
5.6.1 HOUSEHOLDER AND GIVENS TRANSFORMATION MATRICES 204
5.6.2
REDUCING A MATRIX IN HESSENBERG FORM 207
5.6.3 QR FACTORIZATION OF A MATRIX IN HESSENBERG FORM 209
5.6.4
THE BASIC QR ITERATION STARTING FROM
UPPER
HESSENBERG FORM 210
5.6.5 IMPLEMENTATION OF TRANSFORMATION MATRICES .
.
.
. 212
5.7
THE QR ITERATION WITH SHIFTING TECHNIQUES 215
5.7.1 THE QR METHOD WITH SINGLE SHIFT 215
5.7.2 THE QR METHOD WITH DOUBLE SHIFT 218
5.8 COMPUTING THE EIGENVECTORS AND THE SVD OF A MATRIX . . 221
5.8.1
THE HESSENBERG INVERSE ITERATION 221
5.8.2 COMPUTING THE EIGENVECTORS FROM THE
SCHUR
FORM OF A MATRIX 221
5.8.3 APPROXIMATE COMPUTATION OF THE SVD OF A MATRIX 222
5.9
THE GENERALIZED EIGENVALUE PROBLEM 224
5.9.1 COMPUTING THE GENERALIZED REAL SCHUR FORM . . . 225
5.9.2
GENERALIZED REAL SCHUR FORM OF
SYMMETRIC-DEFINITE
PENCILS 226
5.10 METHODS FOR EIGENVALUES OF SYMMETRIE MATRICES 227
5.10.1 THE JACOBI METHOD 227
5.10.2 THE METHOD OF STURM SEQUENCES 230
CONTENTS XV
5.11 THE LANCZOS METHOD 233
5.12 APPLICATIONS 235
5.12.1 ANALYSIS OF THE BUECKLING OF A BEAM 236
5.12.2 FREE DYNAMIC VIBRATION OF A BRIDGE 238
5.13 EXERCISES 240
PART III: AROUND FUNCTIONS AND FUNCTIONALS
6. ROOTFLNDING FOR NONLINEAR EQUATIONS 245
6.1 CONDITIONING OF A NONLINEAR EQUATION 246
6.2 A GEOMETRIE APPROACH TO ROOTFLNDING 248
6.2.1 THE BISECTION METHOD 248
6.2.2 THE METHODS OF CHORD, SECANT AND REGULA FALSI
AND
NEWTON'S METHOD 251
6.2.3 THE DEKKER-BRENT METHOD 256
6.3 FIXED-POINT ITERATIONS FOR NONLINEAR EQUATIONS 257
6.3.1 CONVERGENCE RESULTS FOR
SOME
FIXED-POINT METHODS 260
6.4 ZEROS OF ALGEBRAIC EQUATIONS 261
6.4.1 THE HOMER METHOD AND DEFLATION 262
6.4.2 THE NEWTON-HOMER METHOD 263
6.4.3 THE MULLER METHOD 267
6.5 STOPPING CRITERIA 269
6.6 POST-PROCESSING TECHNIQUES FOR ITERATIVE METHODS 272
6.6.1
AITKEN'S ACCELERATION 272
6.6.2 TECHNIQUES FOR MULTIPLE ROOTS 275
6.7 APPLICATIONS 276
6.7.1 ANALYSIS OF THE STATE EQUATION FOR A REAL GAS . . 276
6.7.2
ANALYSIS OF A NONLINEAR ELECTRICAL CIRCUIT 277
6.8 EXERCISES 279
7. NONLINEAR SYSTEMS AND NUMERICAL OPTIMIZATION 281
7.1 SOLUTION OF SYSTEMS OF NONLINEAR EQUATIONS 282
7.1.1 NEWTON'S METHOD AND ITS VARIANTS 283
7.1.2 MODIFIED NEWTON'S METHODS 284
7.1.3 QUASI-NEWTON METHODS 288
7.1.4 SECANT-LIKE METHODS 288
7.1.5 FIXED-POINT METHODS 290
7.2 UNCONSTRAINED OPTIMIZATION 294
7.2.1 DIRECT SEARCH METHODS 295
7.2.2 DESCENT METHODS 300
7.2.3 LINE SEARCH TECHNIQUES 302
7.2.4 DESCENT METHODS FOR QUADRATIC FUNCTIONS 304
7.2.5 NEWTON-LIKE METHODS FOR FUNCTION MINIMIZATION . 307
7.2.6
QUASI-NEWTON METHODS 308
XVI CONTENTS
7.2.7 SECANT-LIKE METHODS 309
7.3 CONSTRAINED OPTIMIZATION 311
7.3.1 KUHN-TUCKER NECESSARY CONDITIONS FOR
NONLINEAR
PROGRAMMING 313
7.3.2 THE PENALTY METHOD 315
7.3.3 THE METHOD OF LAGRANGE MULTIPLIERS 317
7.4 APPLICATIONS 319
7.4.1 SOLUTION OF A NONLINEAR SYSTEM ARISING FROM
SEMICONDUCTOR
DEVICE SIMULATION 320
7.4.2 NONLINEAR REGULARIZATION OF A DISCRETIZATION GRID . 323
7.5
EXERCISES 325
8. POLYNOMIAL INTERPOLATION 327
8.1 POLYNOMIAL INTERPOLATION 328
8.1.1 THE INTERPOLATION ERROR 329
8.1.2 DRAWBACKS OF POLYNOMIAL INTERPOLATION ON EQUALLY
SPACED
NODES AND RUNGE'S COUNTEREXAMPLE .
.
.
. 330
8.1.3
STABILITY OF POLYNOMIAL INTERPOLATION 332
8.2 NEWTON FORM OF THE INTERPOLATING POLYNOMIAL 333
8.2.1 SOME PROPERTIES OF NEWTON DIVIDED DIFFERENCES . . 335
8.2.2
THE INTERPOLATION ERROR USING DIVIDED DIFFERENCES 337
8.3
PIECEWISE LAGRANGE INTERPOLATION 338
8.4 HERMITE-BIRKOFF INTERPOLATION 341
8.5 EXTENSION TO THE TWO-DIMENSIONAL CASE 343
8.5.1 POLYNOMIAL INTERPOLATION 343
8.5.2 PIECEWISE POLYNOMIAL INTERPOLATION 344
8.6 APPROXIMATION BY SPLINES 348
8.6.1 INTERPOLATORY CUBIC SPLINES 349
8.6.2 B-SPLINES 353
8.7 SPLINES IN PARAMETRIC FORM 357
8.7.1 BEZIER CURVES AND PARAMETRIC B-SPLINES 359
8.8 APPLICATIONS 362
8.8.1 FINITE ELEMENT ANALYSIS OF A CLAMPED BEAM . . . 363
8.8.2
GEOMETRIE RECONSTRUCTION BASED ON
COMPUTER
TOMOGRAPHIES 366
8.9 EXERCISES 368
9. NUMERICAL INTEGRATION 371
9.1 QUADRATURE FORMULAE 371
9.2 INTERPOLATORY QUADRATURES 373
9.2.1 THE MIDPOINT OR RECTANGLE FORMULA 373
9.2.2 THE TRAPEZOIDAL FORMULA 375
9.2.3 THE CAVALIERI-SIMPSON FORMULA 377
9.3 NEWTON-COTES FORMULAE 378
9.4 COMPOSITE NEWTON-COTES FORMULAE 383
CONTENTS XVII
9.5 HERMITE QUADRATURE FORMULAE 386
9.6 RICHARDSON EXTRAPOLATION 387
9.6.1 ROMBERG INTEGRATION 389
9.7 AUTOMATIC INTEGRATION 391
9.7.1 NON ADAPTIVE INTEGRATION ALGORITHMS 392
9.7.2 ADAPTIVE INTEGRATION ALGORITHMS 394
9.8 SINGULAR INTEGRALS 398
9.8.1 INTEGRALS OF FUNCTIONS WITH FINITE
JUMP
DISCONTINUITIES 398
9.8.2 INTEGRALS OF INFINITE FUNCTIONS 398
9.8.3 INTEGRALS OVER UNBOUNDED INTERVALS 401
9.9 MULTIDIMENSIONAL NUMERICAL INTEGRATION 402
9.9.1 THE METHOD OF REDUCTION FORMULA 403
9.9.2 TWO-DIMENSIONAL COMPOSITE QUADRATURES 404
9.9.3
MONTE CARLO METHODS FOR
NUMERICAL
INTEGRATION 407
9.10 APPLICATIONS 408
9.10.1 COMPUTATION OF AN ELLIPSOID SURFACE 408
9.10.2 COMPUTATION OF THE WIND ACTION ON A
SAILBOAT
MAST 410
9.11 EXERCISES 412
PART IV: TRANSFORMS, DIFFERENTIATION
AND
PROBLEM DISCRETIZATION
10.
ORTHOGONAL POLYNOMIALS IN APPROXIMATION THEORY 415
10.1
APPROXIMATION OF FUNCTIONS BY GENERALIZED FOURIER SERIES 415
10.1.1
THE CHEBYSHEV POLYNOMIALS 417
10.1.2 THE LEGENDRE POLYNOMIALS 419
10.2 GAUSSIAN INTEGRATION AND INTERPOLATION 419
10.3 CHEBYSHEV INTEGRATION AND INTERPOLATION 424
10.4 LEGENDRE INTEGRATION AND INTERPOLATION 426
10.5 GAUSSIAN INTEGRATION OVER UNBOUNDED INTERVALS 428
10.6 PROGRAMS FOR THE IMPLEMENTATION OF GAUSSIAN QUADRATURES 429
10.7
APPROXIMATION OF A FUNCTION IN THE LEAST-SQUARES SENSE . 431
10.7.1
DISCRETE LEAST-SQUARES APPROXIMATION 431
10.8 THE POLYNOMIAL OF BEST APPROXIMATION 433
10.9 FOURIER TRIGONOMETRIE POLYNOMIALS 435
10.9.1 THE GIBBS PHENOMENON 439
10.9.2 THE FAST FOURIER TRANSFORM 440
10.10 APPROXIMATION OF FUNCTION DERIVATIVES 442
10.10.1 CLASSICAL FINITE DIFFERENCE METHODS 442
10.10.2 COMPACT FINITE DIFFERENCES 444
10.10.3 PSEUDO-SPECTRAL DERIVATIVE 448
10.11 TRANSFORMS AND THEIR APPLICATIONS 450
XVIII CONTENTS
10.11.1 THE FOURIER TRANSFORM 450
10.11.2 (PHYSICAL) LINEAR SYSTEMS AND FOURIER TRANSFORM . 453
10.11.3
THE LAPLACE TRANSFORM 455
10.11.4 THE Z-TRANSFORM 457
10.12 THE WAVELET TRANSFORM 458
10.12.1 THE CONTINUOUS WAVELET TRANSFORM 458
10.12.2 DISCRETE AND ORTHONORMAL WAVELETS 461
10.13 APPLICATIONS 463
10.13.1 NUMERICAL COMPUTATION OF BLACKBODY RADIATION . 463
10.13.2
NUMERICAL SOLUTION OF SCHROEDINGER EQUATION .
.
.
. 464
10.14
EXERCISES 467
11.
NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 469
11.1
THE CAUCHY PROBLEM 469
11.2 ONE-STEP NUMERICAL METHODS 472
11.3 ANALYSIS OF ONE-STEP METHODS 473
11.3.1 THE ZERO-STABILITY 475
11.3.2 CONVERGENCE ANALYSIS 477
11.3.3 THE ABSOLUTE STABILITY 479
11.4 DIFFERENCE EQUATIONS . 482
11.5 MULTISTEP METHODS 487
11.5.1 ADAMS METHODS 490
11.5.2 BDF METHODS 492
11.6 ANALYSIS OF MULTISTEP METHODS 492
11.6.1 CONSISTENCY 493
11.6.2 THE ROOT CONDITIONS 494
11.6.3 STABILITY AND CONVERGENCE ANALYSIS FOR
MULTISTEP
METHODS 495
11.6.4 ABSOLUTE STABILITY OF MULTISTEP METHODS 499
11.7 PREDICTOR-CORRECTOR METHODS 502
11.8 RUNGE-KUTTA METHODS 508
11.8.1 DERIVATION OF AN EXPLICIT RK METHOD .
.
.
.
.
.
. 511
11.8.2 STEPSIZE ADAPTIVITY FOR RK METHODS 512
11.8.3 IMPLICIT RK METHODS 514
11.8.4 REGIONS OF ABSOLUTE STABILITY FOR RK METHODS .
. 516
11.9
SYSTEMS OF ODES 517
11.10 STIFF PROBLEMS 519
11.11 APPLICATIONS 521
11.11.1 ANALYSIS OF THE MOTION OF A FRICTIONLESS PENDULUM 522
11.11.2
COMPLIANCE OF ARTERIAL WALLS 523
11.12 EXERCISES 527
12.
TWO-POINT BOUNDARY VALUE PROBLEMS 531
12.1 A MODEL PROBLEM 531
12.2 FINITE DIFFERENCE APPROXIMATION 533
CONTENTS XIX
12.2.1 STABILITY ANALYSIS BY THE ENERGY METHOD 534
12.2.2 CONVERGENCE ANALYSIS .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 538
12.2.3 FINITE DIFFERENCES FOR TWO-POINT BOUNDARY
VALUE
PROBLEMS WITH VARIABLE COEFFICIENTS 540
12.3
THE SPECTRAL COLLOCATION METHOD 542
12.4 THE GALERKIN METHOD 544
12.4.1 INTEGRAL FORMULATION OF BOUNDARY-VALUE PROBLEMS 544
12.4.2
A QUICK INTRODUCTION TO DISTRIBUTIONS 546
12.4.3 FORMULATION AND PROPERTIES OF THE
GALERKIN
METHOD . 547
12.4.4 ANALYSIS OF THE GALERKIN METHOD 548
12.4.5 THE FINITE ELEMENT METHOD . 550
12.4.6 IMPLEMENTATION ISSUES 556
12.4.7 SPECTRAL METHODS 559
12.5 ADVECTION-DIFFUSION EQUATIONS 560
12.5.1 GALERKIN FINITE ELEMENT APPROXIMATION 561
12.5.2 THE RELATIONSHIP BETWEEN FINITE ELEMENTS AND
FINITE
DIFFERENCES; THE NUMERICAL VISCOSITY .
.
.
. 563
12.5.3
STABILIZED FINITE ELEMENT METHODS 567
12.6 A QUICK GLANCE TO THE TWO-DIMENSIONAL CASE .
.
.
.
.
.
. 572
12.7 APPLICATIONS 575
12.7.1 LUBRICATION OF A SLIDER 575
12.7.2 VERTICAL DISTRIBUTION OF SPORE
CONCENTRATION
OVER WIDE REGIONS 576
12.8 EXERCISES 578
13.
PARABOLIC AND HYPERBOLIC INITIAL BOUNDARY
VALUE
PROBLEMS 581
13.1 THE HEAT EQUATION 581
13.2 FINITE DIFFERENCE APPROXIMATION OF THE HEAT EQUATION . . 584
13.3
FINITE ELEMENT APPROXIMATION OF THE HEAT EQUATION . . . 586
13.3.1
STABILITY ANALYSIS OF THE OE-METHOD 588
13.4 SPACE-TIME FINITE ELEMENT METHODS FOR THE
HEAT
EQUATION 593
13.5 HYPERBOLIC EQUATIONS: A SCALAR TRANSPORT PROBLEM .
.
.
. 597
13.6
SYSTEMS OF LINEAR HYPERBOLIC EQUATIONS 599
13.6.1 THE WAVE EQUATION 601
13.7 THE FINITE DIFFERENCE METHOD FOR HYPERBOLIC EQUATIONS . . 602
13.7.1
DISCRETIZATION OF THE SCALAR EQUATION 602
13.8 ANALYSIS OF FINITE DIFFERENCE METHODS 605
13.8.1 CONSISTENCY 605
13.8.2 STABILITY 605
13.8.3 THE CFL CONDITION 606
13.8.4 VON NEUMANN STABILITY ANALYSIS 608
13.9 DISSIPATION AND DISPERSION 611
XX CONTENTS
13.9.1 EQUIVALENT EQUATIONS 614
13.10 FINITE ELEMENT APPROXIMATION OF HYPERBOLIC EQUATIONS . . 618
13.10.1
SPACE DISCRETIZATION WITH CONTINUOUS AND
DISCONTINUOUS
FINITE ELEMENTS 618
13.10.2 TIME DISCRETIZATION 620
13.11 APPLICATIONS 623
13.11.1 HEAT CONDUCTION IN A BAR 623
13.11.2 A HYPERBOLIC MODEL FOR BLOOD FLOW
INTERACTION
WITH ARTERIAL WALLS 623
13.12 EXERCISES 625
REFERENCES 627
INDEX OF MATLAB PROGRAMS 643
INDEX 647 |
| any_adam_object | 1 |
| author | Quarteroni, Alfio 1952- Sacco, Riccardo Saleri, Fausto 1965-2007 |
| author_GND | (DE-588)120370158 (DE-588)120434571 |
| author_facet | Quarteroni, Alfio 1952- Sacco, Riccardo Saleri, Fausto 1965-2007 |
| author_role | aut aut aut |
| author_sort | Quarteroni, Alfio 1952- |
| author_variant | a q aq r s rs f s fs |
| building | Verbundindex |
| bvnumber | BV013049292 |
| callnumber-first | Q - Science |
| callnumber-label | QA297 |
| callnumber-raw | QA297 |
| callnumber-search | QA297 |
| callnumber-sort | QA 3297 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 900 |
| classification_tum | MAT 650f |
| ctrlnum | (OCoLC)247799952 (DE-599)BVBBV013049292 |
| dewey-full | 519.4 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.4 |
| dewey-search | 519.4 |
| dewey-sort | 3519.4 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre_facet | Bibliografie Aufgabensammlung Lehrbuch |
| id | DE-604.BV013049292 |
| illustrated | Illustrated |
| indexdate | 2025-03-10T15:06:16Z |
| institution | BVB |
| isbn | 0387989595 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008891288 |
| oclc_num | 247799952 |
| open_access_boolean | |
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| physical | XX, 654 S. Ill., graph. Darst. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer |
| record_format | marc |
| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spelling | Quarteroni, Alfio 1952- Verfasser (DE-588)120370158 aut Mathematica numerica Numerical mathematics Alfio Quarteroni ; Riccardo Sacco ; Fausto Saleri New York [u.a.] Springer 2000 XX, 654 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 37 Numerische Mathematik - Lehrbuch Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf 1\p (DE-588)4006432-3 Bibliografie gnd-content 2\p (DE-588)4143389-0 Aufgabensammlung gnd-content 3\p (DE-588)4123623-3 Lehrbuch gnd-content Numerische Mathematik (DE-588)4042805-9 s DE-604 Sacco, Riccardo Verfasser aut Saleri, Fausto 1965-2007 Verfasser (DE-588)120434571 aut Texts in applied mathematics 37 (DE-604)BV002476038 37 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008891288&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Quarteroni, Alfio 1952- Sacco, Riccardo Saleri, Fausto 1965-2007 Numerical mathematics Texts in applied mathematics Numerische Mathematik - Lehrbuch Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4006432-3 (DE-588)4143389-0 (DE-588)4123623-3 |
| title | Numerical mathematics |
| title_alt | Mathematica numerica |
| title_auth | Numerical mathematics |
| title_exact_search | Numerical mathematics |
| title_full | Numerical mathematics Alfio Quarteroni ; Riccardo Sacco ; Fausto Saleri |
| title_fullStr | Numerical mathematics Alfio Quarteroni ; Riccardo Sacco ; Fausto Saleri |
| title_full_unstemmed | Numerical mathematics Alfio Quarteroni ; Riccardo Sacco ; Fausto Saleri |
| title_short | Numerical mathematics |
| title_sort | numerical mathematics |
| topic | Numerische Mathematik - Lehrbuch Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerische Mathematik - Lehrbuch Numerische Mathematik Bibliografie Aufgabensammlung Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008891288&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
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