A theoretical introduction to numerical analysis:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, FL [u.a.]
Chapman & Hall/CRC
c2007
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 509-519) and index |
| Physical Description: | xiii, 537 p. ill. 25 cm |
| ISBN: | 9781584886075 1584886072 |
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| 100 | 1 | |a Rjabenʹkij, Viktor Solomonovič |d 1923- |e Verfasser |0 (DE-588)1028057067 |4 aut | |
| 245 | 1 | 0 | |a A theoretical introduction to numerical analysis |c Victor S. Ryaben'kii, Semyon V. Tsynkov |
| 264 | 1 | |a Boca Raton, FL [u.a.] |b Chapman & Hall/CRC |c c2007 | |
| 300 | |a xiii, 537 p. |b ill. |c 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references (p. 509-519) and index | ||
| 650 | 0 | 7 | |a Numerische Mathematik |0 (DE-588)4042805-9 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
| 689 | 0 | 0 | |a Numerische Mathematik |0 (DE-588)4042805-9 |D s |
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| 700 | 1 | |a Tsynkov, Semyon V. |e Verfasser |4 aut | |
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| adam_text | Contents
Preface xj
Acknowledgments xiii
1 Introduction 1
1.1 Discretization 4
Exercises 5
1.2 Conditioning 6
Exercises 7
1.3 Error 7
1.3.1 Unavoidable Error 8
1.3.2 Error oftheMethod 10
1.3.3 Round off Error 10
Exercises 11
1.4 On Methods of Computation 12
1.4.1 Accuracy 13
1.4.2 Operation Count 14
1.4.3 Stability 14
1.4.4 Loss of Significant Digits 15
1.4.5 Convergence 18
1.4.6 General Comments 18
Exercises 19
1 Interpolation of Functions. Quadratures 21
2 Algebraic Interpolation 25
2.1 Existence and Uniqueness of Interpolating Polynomial 25
2.1.1 The Lagrange Form of Interpolating Polynomial 25
2.1.2 The Newton Form of Interpolating Polynomial. Divided Dif
ferences 26
2.1.3 Comparison of the Lagrange and Newton Forms 31
2.1.4 Conditioning of the Interpolating Polynomial 32
2.1.5 On Poor Convergence of Interpolation with Equidistant
Nodes 33
Exercises 34
2.2 Classical Piecewise Polynomial Interpolation 35
2.2.1 Definition of Piecewise Polynomial Interpolation 35
iii
iv
2.2.2 Formula for the Interpolation Error 35
2.2.3 Approximation of Derivatives for a Grid Function 38
2.2.4 Estimate of the Unavoidable Error and the Choice of Degree
for Piecewise Polynomial Interpolation 40
2.2.5 Saturation of Piecewise Polynomial Interpolation 42
Exercises • 42
2.3 Smooth Piecewise Polynomial Interpolation (Splines) 43
2.3.1 Local Interpolation of Smoothness ä and Its Properties ... 43
2.3.2 Nonlocal Smooth Piecewise Polynomial Interpolation ... 48
2.3.3 Proofof Theorem 2.11 • 53
Exercises • 56
2.4 Interpolation of Functions of Two Variables 57
2.4.1 Structured Grids • 57
2.4.2 Unstructured Grids 59
Exercises • 60
3 Trigonometrie Interpolation 61
3.1 Interpolation of Periodic Functions 62
3.1.1 An Important Particular Choice of Interpolation Nodes ... 62
3.1.2 Sensitivity of the Interpolating Polynomial to Perturbations
of the Function Values 67
3.1.3 Estimate of Interpolation Error 68
3.1.4 An Alternative Choice of Interpolation Nodes ........ 72
3.2 Interpolation of Functions on an Interval. Relation between Alge
braic and Trigonometrie Interpolation 73
3.2.1 Periodization 73
3.2.2 Trigonometrie Interpolation 75
3.2.3 Chebyshev Polynomials. Relation between Algebraic and
Trigonometrie Interpolation 75
3.2.4 Properties of Algebraic Interpolation with poots of the
Chebyshev Polynomial Tn+ (x) as Nodes .......... 77
3.2.5 An Algorithm for Evaluating the Interpolating Polynomial . 78
3.2.6 Algebraic Interpolation with Extrema of the Chebyshev
Polynomial T„(x) as Nodes 79
3.2.7 More on the Lebesgue Constants and Convergence of Inter
polants 80
Exercises 89
4 Computation of Definite Integrals. Quadratures 91
4.1 Trapezoidal Rule, Simpson s Formula, and the Like ......... 91
4.1.1 General Construction of Quadrature Formulae . ¦ 92
4.1.2 Trapezoidal Rule 93
4.1.3 Simpson s Formula 98
Exercises 102
4.2 Quadrature Formulae with No Saturation. Gaussian Quadratures . 102
V
Exercises 107
4.3 Improper Integrals. Combination of Numerical and Analytical Meth
ods 108
Exercises HO
4.4 Multiple Integrals 110
4.4.1 Repeated Integrals and Quadrature Formulae 111
4.4.2 The Use of Coordinate Transformations 112
4.4.3 The Notion of Monte Carlo Methods 113
II Systems of Scalar Equations 115
5 Systems of Linear Algebraic Equations: Direct Methods 119
5.1 Different Forms of Consistent Linear Systems 120
5.1.1 Canonical Form of a Linear System 120
5.1.2 Operator Form 121
5.1.3 Finite Difference Dirichlet Problem for the Poisson Equa
tion 121
Exercises 124
5.2 Linear Spaces, Norms, and Operators 124
5.2.1 Normed Spaces 126
5.2.2 Norm of a Linear Operator 129
Exercises 131
5.3 Conditioning of Linear Systems 133
5.3.1 Condition Number 134
5.3.2 Characterization of a Linear System by Means of Its Condi¬
tion Number 136
Exercises 139
5.4 Gaussian Elimination and Its Tri Diagonal Version 140
5.4.1 Standard Gaussian Elimination 141
5.4.2 Tri Diagonal Elimination 145
5.4.3 Cyclic Tri Diagonal Elimination 148
5.4.4 Matrix Interpretation of the Gaussian Elimination. LU Fac
torization 149
5.4.5 Cholesky Factorization 153
5.4.6 Gaussian Elimination with Pivoting 154
5.4.7 An Algorithm with a Guaranteed Error Estimate 155
Exercises 156
5.5 Minimization of Quadratic Functions and Its Relation to Linear Sys¬
tems I57
Exercises 159
5.6 TheMethodofConjugateGradients 159
5.6.1 Construction of the Method 159
5.6.2 Flexibility in Specifying the Operator A 163
5.6.3 Computational Complexity 163
Exercises 163
vi
5.7 Finite Fourier Series 164
5.7.1 Fourier Series for Grid Functions 165
5.7.2 Representation of Solution as a Finite Fourier Series .... 168
5.7.3 Fast Fourier Transform 169
Exercises 171
6 Iterative Methods for Solving Linear Systems 173
6.1 Richardson Iterations and the Like 174
6.1.1 Generallteration Scheme 174
6.1.2 A Necessary and Sufficient Condition for Convergence . . . 178
6.1.3 The Richardson Method for A =A* 0 181
6.1.4 Preconditioning 188
6.1.5 Scaling 192
Exercises 193
6.2 Chebyshev Iterations and Conjugate Gradients 194
6.2.1 Chebyshev Iterations 194
6.2.2 Conjugate Gradients 196
Exercises 197
6.3 Krylov Subspace Iterations 198
6.3.1 Definition of Krylov Subspaces 199
6.3.2 GMRES 201
Exercises 204
6.4 Multigrid Iterations 204
6.4.1 Ideaof the Method 205
6.4.2 Description ofthe Algorithm 208
6.4.3 Bibliography Comments 210
Exercises 210
¦ 7 Overdetermined Linear Systems. The Method of Least Squares 211
I 7.1 Examples of Problems that Result in Overdetermined Systems ... 211
1 7.1.1 Processing of Experimental Data. Empirical Formulae ... 211
7.1.2 Improving the Accuracy of Experimental Results by Increas
ing the Number of Measurements 213
7.2 Weak Solutions of Füll Rank Systems. 0Ä Factorization 214
7.2.1 Existence and Uniquenessof Weak Solutions 214
7.2.2 Computation of Weak Solutions. QR Factorization 217
7.2.3 Geometrie Interpretation of the Method of Least Squares . . 220
7.2.4 Overdetermined Systems in the Operator Form 221
Exercises 222
7.3 Rank Deficient Systems. Singular Value Decomposition 225
7.3.1 Singular Value Decomposition and Moore Penrose Pseu
doinverse 225
7.3.2 Minimum Norm Weak Solution 227
Exercises 229
vii
8 Numerical Solution of Nonlinear Equations and Systems 231
8.1 Commonly Used Methods of Rootfinding 233
8.1.1 The Bisection Method 233
8.1.2 The Chord Method 234
8.1.3 The Secant Method 235
8.1.4 Newton s Method 236
8.2 Fixed Point Iterations 237
8.2.1 TheCaseofOneScalarEquation 237
8.2.2 The Case ofa System of Equations 240
Exercises 242
8.3 Newton s Method 242
8.3.1 Newton s Linearization for One Scalar Equation 242
8.3.2 Newton s Linearization for Systems 244
8.3.3 Modified Newton s Methods 246
Exercises 247
III The Method of Finite Differences for the Numerical Solu¬
tion of Differential Equations 249
9 Numerical Solution of Ordinary Differential Equations 253
9.1 Examples of Finite Difference Schemes. Convergence 253
9.1.1 Examples of Difference Schemes 254
9.1.2 Convergent Difference Schemes 256
9.1.3 Verification of Convergence for a Difference Scheine .... 259
9.2 Approximation of Continuous Problem by a Difference Scheme.
Consistency 260
9.2.1 Truncation Error öfw 261
9.2.2 Evaluation of the Truncation Error 8fw 262
9.2.3 Accuracyof Order/** 264
9.2.4 Examples 265
9.2.5 Replacement of Derivatives by Difference Quotients .... 269
9.2.6 Other Approaches to Constructing Difference Schemes . . . 269
Exercises 271
9.3 Stability of Finite Difference Schemes 271
9.3.1 Definition of Stability 272
9.3.2 The Relation between Consistency, Stability, and Conver¬
gence 273
9.3.3 Convergent Scheme for an Integral Equation 277
9.3.4 The Effect of Rounding 278
9.3.5 General Comments. A stability 280
Exercises 283
9.4 The Runge Kutta Methods 284
9.4.1 The Runge Kutta Schemes 284
9.4.2 Extension to Systems 286
Exercises 288
viii
9.5 SolutionofBoundary Value Problems 288
9.5.1 The Shooting Method 289
9.5.2 Tri Diagonal Elimination 291
9.5.3 Newton s Method 291
Exercises 292
9.6 Saturation of Finite Difference Methods by Smoothness 293
Exercises 300
9.7 The Notion of Spectral Methods 301
Exercises 306
10 Finite Difference Scheines for Partial Differential Equations 307
10.1 Key Definitions and Ulustrating Examples 307
10.1.1 Definition of Convergence 307
10.1.2 Definition of Consistency 309
10.1.3 Definition of Stability 312
10.1.4 The Courant, Friedrichs, and Lewy Condition 317
10.1.5 TheMechanismoflnstability 319
10.1.6 The Kantorovich Theorem 320
10.1.7 On the Efficacy of Finite Difference Schemes 322
10.1.8 Bibliography Comments 323
Exercises 324
10.2 Construction of Consistent Difference Schemes 327
10.2.1 Replacement of Derivatives by Difference Quotients .... 327
10.2.2 The Method of Undetermined Coefficients 333
10.2.3 Other Methods. Phase Error 340
10.2.4 Predictor Corrector Schemes 344
Exercises 345
10.3 Spectral Stability Criterion for Finite Difference Cauchy Problems . 349
10.3.1 Stability with Respect to Initial Data 349
10.3.2 A Necessary Spectral Condition for Stability 350
10.3.3 Examples 352
10.3.4 Stability in C 362
10.3.5 Sufficiency of the Spectral Stability Condition in h 362
10.3.6 Scalar Equations vs. Systems 365
Exercises 367
10.4 Stability for Problems with Variable Coefficients 369
10.4.1 The Principle of Frozen Coefficients 369
10.4.2 Dissipation of Finite Difference Schemes 372
Exercises 377
10.5 Stability for Initial Boundary Value Problems 377
10.5.1 The Babenko Gelfand Criterion 377
10.5.2 Spectra of the Families of Operators. The Godunov
Ryaben kii Criterion 385
10.5.3 The Energy Method 402
ix
10.5.4 A Necessary and Sufficient Condition of Stability. The
Kreiss Criterion 409
Exercises 418
10.6 Maximum Principle for the Heat Equation 422
10.6.1 An Explicit Scheme 422
10.6.2 An Implicit Scheme 425
Exercises 426
11 Discontinuous Solutions and Methods of Their Computation 427
11.1 Differential Form of an Integral Conservation Law 428
11.1.1 Differential Equation in the Case of Smooth Solutions . . . 428
11.1.2 The Mechanism of Formation of Discontinuities 429
11.1.3 Condition at the Discontinuity 431
11.1.4 Generalized Solution of a Differential Problem 433
11.1.5 The Riemann Problem 434
Exercises 436
11.2 Construction of Difference Schemes 436
11.2.1 Artificial Viscosity 437
11.2.2 The Method of Characteristics 438
11.2.3 Conservative Schemes. The Godunov Scheme 439
Exercises 444
12 Discrete Methods for Elliptic Problems 445
12.1 A Simple Finite Difference Scheme. The Maximum Principle . . . 446
12.1.1 Consistency 447
12.1.2 Maximum Principle and Stability 448
12.1.3 Variable Coefficients 451
Exercises 452
12.2 The Notion of Finite Elements. Ritz and Galerkin Approximations . 453
12.2.1 Variational Problem 454
12.2.2 The Ritz Method 458
12.2.3 The Galerkin Method 460
12.2.4 An Exampleof Finite Element Discretization 464
12.2.5 Convergence of Finite Element Approximations 466
Exercises 469
IV The Methods of Boundary Equations for the Numerical
Solution of Boundary Value Problems 471
13 Boundary Integral Equations and the Method of Boundary Elements 475
13.1 Reduction of Boundary Value Problems to Integral Equations ... 475
13.2 Discretization of Integral Equations and Boundary Elements .... 479
13.3 The Range ofApplicability for Boundary Elements 480
X
14 Boundary Equations with Projections and the Method of Difference Po¬
tentials 483
14.1 Formulation of Model Problems 484
14.1.1 Interior Boundary Value Problem 485
14.1.2 Exterior Boundary Value Problem 485
14.1.3 Problem of Artificial Boundary Conditions 485
14.1.4 Problem of Two Subdomains 486
14.1.5 Problem of Active Shielding 487
14.2 Difference Potentials 488
14.2.1 Auxiliary Difference Problem 488
14.2.2 The Potential u+= P+ vy 489
14.2.3 Difference Potential u~ =P^vr 492
14.2.4 Cauchy Type Difference Potential w±=P±vy 493
14.2.5 Analogy with Classical Cauchy Type Integral 497
14.3 Solutionof Model Problems 498
14.3.1 Interior Boundary Value Problem 498
14.3.2 Exterior Boundary Value Problem 500
14.3.3 Problem of Artificial Boundary Conditions 501
14.3.4 Problem of Two Subdomains 501
14.3.5 Problem of Active Shielding 503
14.4 General Remarks 505
14.5 Bibliography Comments 506
ListofFigures 507
Referenced Books 509
Referenced Journal Articles 517
Index 521
|
| adam_txt |
Contents
Preface xj
Acknowledgments xiii
1 Introduction 1
1.1 Discretization 4
Exercises 5
1.2 Conditioning 6
Exercises 7
1.3 Error 7
1.3.1 Unavoidable Error 8
1.3.2 Error oftheMethod 10
1.3.3 Round off Error 10
Exercises 11
1.4 On Methods of Computation 12
1.4.1 Accuracy 13
1.4.2 Operation Count 14
1.4.3 Stability 14
1.4.4 Loss of Significant Digits 15
1.4.5 Convergence 18
1.4.6 General Comments 18
Exercises 19
1 Interpolation of Functions. Quadratures 21
2 Algebraic Interpolation 25
2.1 Existence and Uniqueness of Interpolating Polynomial 25
2.1.1 The Lagrange Form of Interpolating Polynomial 25
2.1.2 The Newton Form of Interpolating Polynomial. Divided Dif
ferences 26
2.1.3 Comparison of the Lagrange and Newton Forms 31
2.1.4 Conditioning of the Interpolating Polynomial 32
2.1.5 On Poor Convergence of Interpolation with Equidistant
Nodes 33
Exercises 34
2.2 Classical Piecewise Polynomial Interpolation 35
2.2.1 Definition of Piecewise Polynomial Interpolation 35
iii
iv
2.2.2 Formula for the Interpolation Error 35
2.2.3 Approximation of Derivatives for a Grid Function 38
2.2.4 Estimate of the Unavoidable Error and the Choice of Degree
for Piecewise Polynomial Interpolation 40
2.2.5 Saturation of Piecewise Polynomial Interpolation 42
Exercises • 42
2.3 Smooth Piecewise Polynomial Interpolation (Splines) 43
2.3.1 Local Interpolation of Smoothness ä and Its Properties . 43
2.3.2 Nonlocal Smooth Piecewise Polynomial Interpolation . 48
2.3.3 Proofof Theorem 2.11 • 53
Exercises • 56
2.4 Interpolation of Functions of Two Variables 57
2.4.1 Structured Grids • 57
2.4.2 Unstructured Grids 59
Exercises • 60
3 Trigonometrie Interpolation 61
3.1 Interpolation of Periodic Functions 62
3.1.1 An Important Particular Choice of Interpolation Nodes . 62
3.1.2 Sensitivity of the Interpolating Polynomial to Perturbations
of the Function Values 67
3.1.3 Estimate of Interpolation Error 68
3.1.4 An Alternative Choice of Interpolation Nodes . 72
3.2 Interpolation of Functions on an Interval. Relation between Alge
braic and Trigonometrie Interpolation 73
3.2.1 Periodization 73
3.2.2 Trigonometrie Interpolation 75
3.2.3 Chebyshev Polynomials. Relation between Algebraic and
Trigonometrie Interpolation 75
3.2.4 Properties of Algebraic Interpolation with poots of the
Chebyshev Polynomial Tn+\ (x) as Nodes . 77
3.2.5 An Algorithm for Evaluating the Interpolating Polynomial . 78
3.2.6 Algebraic Interpolation with Extrema of the Chebyshev
Polynomial T„(x) as Nodes 79
3.2.7 More on the Lebesgue Constants and Convergence of Inter
polants 80
Exercises 89
4 Computation of Definite Integrals. Quadratures 91
4.1 Trapezoidal Rule, Simpson's Formula, and the Like . 91
4.1.1 General Construction of Quadrature Formulae . ¦ 92
4.1.2 Trapezoidal Rule 93
4.1.3 Simpson's Formula 98
Exercises 102
4.2 Quadrature Formulae with No Saturation. Gaussian Quadratures . 102
V
Exercises 107
4.3 Improper Integrals. Combination of Numerical and Analytical Meth
ods 108
Exercises HO
4.4 Multiple Integrals 110
4.4.1 Repeated Integrals and Quadrature Formulae 111
4.4.2 The Use of Coordinate Transformations 112
4.4.3 The Notion of Monte Carlo Methods 113
II Systems of Scalar Equations 115
5 Systems of Linear Algebraic Equations: Direct Methods 119
5.1 Different Forms of Consistent Linear Systems 120
5.1.1 Canonical Form of a Linear System 120
5.1.2 Operator Form 121
5.1.3 Finite Difference Dirichlet Problem for the Poisson Equa
tion 121
Exercises 124
5.2 Linear Spaces, Norms, and Operators 124
5.2.1 Normed Spaces 126
5.2.2 Norm of a Linear Operator 129
Exercises 131
5.3 Conditioning of Linear Systems 133
5.3.1 Condition Number 134
5.3.2 Characterization of a Linear System by Means of Its Condi¬
tion Number 136
Exercises 139
5.4 Gaussian Elimination and Its Tri Diagonal Version 140
5.4.1 Standard Gaussian Elimination 141
5.4.2 Tri Diagonal Elimination 145
5.4.3 Cyclic Tri Diagonal Elimination 148
5.4.4 Matrix Interpretation of the Gaussian Elimination. LU Fac
torization 149
5.4.5 Cholesky Factorization 153
5.4.6 Gaussian Elimination with Pivoting 154
5.4.7 An Algorithm with a Guaranteed Error Estimate 155
Exercises 156
5.5 Minimization of Quadratic Functions and Its Relation to Linear Sys¬
tems I57
Exercises 159
5.6 TheMethodofConjugateGradients 159
5.6.1 Construction of the Method 159
5.6.2 Flexibility in Specifying the Operator A 163
5.6.3 Computational Complexity 163
Exercises 163
vi
5.7 Finite Fourier Series 164
5.7.1 Fourier Series for Grid Functions 165
5.7.2 Representation of Solution as a Finite Fourier Series . 168
5.7.3 Fast Fourier Transform 169
Exercises 171
6 Iterative Methods for Solving Linear Systems 173
6.1 Richardson Iterations and the Like 174
6.1.1 Generallteration Scheme 174
6.1.2 A Necessary and Sufficient Condition for Convergence . . . 178
6.1.3 The Richardson Method for A =A* 0 181
6.1.4 Preconditioning 188
6.1.5 Scaling 192
Exercises 193
6.2 Chebyshev Iterations and Conjugate Gradients 194
6.2.1 Chebyshev Iterations 194
6.2.2 Conjugate Gradients 196
Exercises 197
6.3 Krylov Subspace Iterations 198
6.3.1 Definition of Krylov Subspaces 199
6.3.2 GMRES 201
Exercises 204
6.4 Multigrid Iterations 204
6.4.1 Ideaof the Method 205
6.4.2 Description ofthe Algorithm 208
6.4.3 Bibliography Comments 210
Exercises 210
¦ 7 Overdetermined Linear Systems. The Method of Least Squares 211
I 7.1 Examples of Problems that Result in Overdetermined Systems . 211
1 7.1.1 Processing of Experimental Data. Empirical Formulae . 211
7.1.2 Improving the Accuracy of Experimental Results by Increas
ing the Number of Measurements 213
7.2 Weak Solutions of Füll Rank Systems. 0Ä Factorization 214
7.2.1 Existence and Uniquenessof Weak Solutions 214
7.2.2 Computation of Weak Solutions. QR Factorization 217
7.2.3 Geometrie Interpretation of the Method of Least Squares . . 220
7.2.4 Overdetermined Systems in the Operator Form 221
Exercises 222
7.3 Rank Deficient Systems. Singular Value Decomposition 225
7.3.1 Singular Value Decomposition and Moore Penrose Pseu
doinverse 225
7.3.2 Minimum Norm Weak Solution 227
Exercises 229
vii
8 Numerical Solution of Nonlinear Equations and Systems 231
8.1 Commonly Used Methods of Rootfinding 233
8.1.1 The Bisection Method 233
8.1.2 The Chord Method 234
8.1.3 The Secant Method 235
8.1.4 Newton's Method 236
8.2 Fixed Point Iterations 237
8.2.1 TheCaseofOneScalarEquation 237
8.2.2 The Case ofa System of Equations 240
Exercises 242
8.3 Newton's Method 242
8.3.1 Newton's Linearization for One Scalar Equation 242
8.3.2 Newton's Linearization for Systems 244
8.3.3 Modified Newton's Methods 246
Exercises 247
III The Method of Finite Differences for the Numerical Solu¬
tion of Differential Equations 249
9 Numerical Solution of Ordinary Differential Equations 253
9.1 Examples of Finite Difference Schemes. Convergence 253
9.1.1 Examples of Difference Schemes 254
9.1.2 Convergent Difference Schemes 256
9.1.3 Verification of Convergence for a Difference Scheine . 259
9.2 Approximation of Continuous Problem by a Difference Scheme.
Consistency 260
9.2.1 Truncation Error öfw 261
9.2.2 Evaluation of the Truncation Error 8fw 262
9.2.3 Accuracyof Order/** 264
9.2.4 Examples 265
9.2.5 Replacement of Derivatives by Difference Quotients . 269
9.2.6 Other Approaches to Constructing Difference Schemes . . . 269
Exercises 271
9.3 Stability of Finite Difference Schemes 271
9.3.1 Definition of Stability 272
9.3.2 The Relation between Consistency, Stability, and Conver¬
gence 273
9.3.3 Convergent Scheme for an Integral Equation 277
9.3.4 The Effect of Rounding 278
9.3.5 General Comments. A stability 280
Exercises 283
9.4 The Runge Kutta Methods 284
9.4.1 The Runge Kutta Schemes 284
9.4.2 Extension to Systems 286
Exercises 288
viii
9.5 SolutionofBoundary Value Problems 288
9.5.1 The Shooting Method 289
9.5.2 Tri Diagonal Elimination 291
9.5.3 Newton's Method 291
Exercises 292
9.6 Saturation of Finite Difference Methods by Smoothness 293
Exercises 300
9.7 The Notion of Spectral Methods 301
Exercises 306
10 Finite Difference Scheines for Partial Differential Equations 307
10.1 Key Definitions and Ulustrating Examples 307
10.1.1 Definition of Convergence 307
10.1.2 Definition of Consistency 309
10.1.3 Definition of Stability 312
10.1.4 The Courant, Friedrichs, and Lewy Condition 317
10.1.5 TheMechanismoflnstability 319
10.1.6 The Kantorovich Theorem 320
10.1.7 On the Efficacy of Finite Difference Schemes 322
10.1.8 Bibliography Comments 323
Exercises 324
10.2 Construction of Consistent Difference Schemes 327
10.2.1 Replacement of Derivatives by Difference Quotients . 327
10.2.2 The Method of Undetermined Coefficients 333
10.2.3 Other Methods. Phase Error 340
10.2.4 Predictor Corrector Schemes 344
Exercises 345
10.3 Spectral Stability Criterion for Finite Difference Cauchy Problems . 349
10.3.1 Stability with Respect to Initial Data 349
10.3.2 A Necessary Spectral Condition for Stability 350
10.3.3 Examples 352
10.3.4 Stability in C 362
10.3.5 Sufficiency of the Spectral Stability Condition in h 362
10.3.6 Scalar Equations vs. Systems 365
Exercises 367
10.4 Stability for Problems with Variable Coefficients 369
10.4.1 The Principle of Frozen Coefficients 369
10.4.2 Dissipation of Finite Difference Schemes 372
Exercises 377
10.5 Stability for Initial Boundary Value Problems 377
10.5.1 The Babenko Gelfand Criterion 377
10.5.2 Spectra of the Families of Operators. The Godunov
Ryaben'kii Criterion 385
10.5.3 The Energy Method 402
ix
10.5.4 A Necessary and Sufficient Condition of Stability. The
Kreiss Criterion 409
Exercises 418
10.6 Maximum Principle for the Heat Equation 422
10.6.1 An Explicit Scheme 422
10.6.2 An Implicit Scheme 425
Exercises 426
11 Discontinuous Solutions and Methods of Their Computation 427
11.1 Differential Form of an Integral Conservation Law 428
11.1.1 Differential Equation in the Case of Smooth Solutions . . . 428
11.1.2 The Mechanism of Formation of Discontinuities 429
11.1.3 Condition at the Discontinuity 431
11.1.4 Generalized Solution of a Differential Problem 433
11.1.5 The Riemann Problem 434
Exercises 436
11.2 Construction of Difference Schemes 436
11.2.1 Artificial Viscosity 437
11.2.2 The Method of Characteristics 438
11.2.3 Conservative Schemes. The Godunov Scheme 439
Exercises 444
12 Discrete Methods for Elliptic Problems 445
12.1 A Simple Finite Difference Scheme. The Maximum Principle . . . 446
12.1.1 Consistency 447
12.1.2 Maximum Principle and Stability 448
12.1.3 Variable Coefficients 451
Exercises 452
12.2 The Notion of Finite Elements. Ritz and Galerkin Approximations . 453
12.2.1 Variational Problem 454
12.2.2 The Ritz Method 458
12.2.3 The Galerkin Method 460
12.2.4 An Exampleof Finite Element Discretization 464
12.2.5 Convergence of Finite Element Approximations 466
Exercises 469
IV The Methods of Boundary Equations for the Numerical
Solution of Boundary Value Problems 471
13 Boundary Integral Equations and the Method of Boundary Elements 475
13.1 Reduction of Boundary Value Problems to Integral Equations . 475
13.2 Discretization of Integral Equations and Boundary Elements . 479
13.3 The Range ofApplicability for Boundary Elements 480
X
14 Boundary Equations with Projections and the Method of Difference Po¬
tentials 483
14.1 Formulation of Model Problems 484
14.1.1 Interior Boundary Value Problem 485
14.1.2 Exterior Boundary Value Problem 485
14.1.3 Problem of Artificial Boundary Conditions 485
14.1.4 Problem of Two Subdomains 486
14.1.5 Problem of Active Shielding 487
14.2 Difference Potentials 488
14.2.1 Auxiliary Difference Problem 488
14.2.2 The Potential u+= P+ vy 489
14.2.3 Difference Potential u~ =P^vr 492
14.2.4 Cauchy Type Difference Potential w±=P±vy 493
14.2.5 Analogy with Classical Cauchy Type Integral 497
14.3 Solutionof Model Problems 498
14.3.1 Interior Boundary Value Problem 498
14.3.2 Exterior Boundary Value Problem 500
14.3.3 Problem of Artificial Boundary Conditions 501
14.3.4 Problem of Two Subdomains 501
14.3.5 Problem of Active Shielding 503
14.4 General Remarks 505
14.5 Bibliography Comments 506
ListofFigures 507
Referenced Books 509
Referenced Journal Articles 517
Index 521 |
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| author | Rjabenʹkij, Viktor Solomonovič 1923- Tsynkov, Semyon V. |
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| illustrated | Illustrated |
| index_date | 2024-07-02T18:52:01Z |
| indexdate | 2024-07-09T21:07:45Z |
| institution | BVB |
| isbn | 9781584886075 1584886072 |
| language | English |
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| spelling | Rjabenʹkij, Viktor Solomonovič 1923- Verfasser (DE-588)1028057067 aut A theoretical introduction to numerical analysis Victor S. Ryaben'kii, Semyon V. Tsynkov Boca Raton, FL [u.a.] Chapman & Hall/CRC c2007 xiii, 537 p. ill. 25 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (p. 509-519) and index Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Numerische Mathematik (DE-588)4042805-9 s DE-604 Tsynkov, Semyon V. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091137&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rjabenʹkij, Viktor Solomonovič 1923- Tsynkov, Semyon V. A theoretical introduction to numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4123623-3 |
| title | A theoretical introduction to numerical analysis |
| title_auth | A theoretical introduction to numerical analysis |
| title_exact_search | A theoretical introduction to numerical analysis |
| title_exact_search_txtP | A theoretical introduction to numerical analysis |
| title_full | A theoretical introduction to numerical analysis Victor S. Ryaben'kii, Semyon V. Tsynkov |
| title_fullStr | A theoretical introduction to numerical analysis Victor S. Ryaben'kii, Semyon V. Tsynkov |
| title_full_unstemmed | A theoretical introduction to numerical analysis Victor S. Ryaben'kii, Semyon V. Tsynkov |
| title_short | A theoretical introduction to numerical analysis |
| title_sort | a theoretical introduction to numerical analysis |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerische Mathematik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091137&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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