E-Book, Englisch, 521 Seiten
Based on a Summer School Held in Oxford, August-September 1961
E-Book, Englisch, 521 Seiten
ISBN: 978-1-4831-4947-9
Verlag: Elsevier Reference Monographs
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
The book is organized into four parts. The first three cover the numerical solution of ordinary differential equations, integral equations, and partial differential equations of quasi-linear form. Most of the techniques are evaluated from the standpoints of accuracy, convergence, and stability (in the various senses of these terms) as well as ease of coding and convenience of machine computation. The last part, on practical problems, uses and develops the techniques for the treatment of problems of the greatest difficulty and complexity, which tax not only the best machines but also the best brains.
This book was written for scientists who have problems to solve, and who want to know what methods exist, why and in what circumstances some are better than others, and how to adapt and develop techniques for new problems. The budding numerical analyst should also benefit from this book, and should find some topics for valuable research. The first three parts, in fact, could be used not only by practical men but also by students, though a preliminary elementary course would assist the reading.
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Weitere Infos & Material
1;Front Cover;1
2;Numerical Solution of Ordinary and Partial Differential Equations;4
3;Copyright Page ;5
4;Table of Contents;6
5;PREFACE;8
6;PART I: ORDINARY DIFFERENTIAL EQUATIONS;12
6.1;CHAPTER 1. ORDINARY DIFFERENTIAL EQUATIONS AND FINITE DIFFERENCES;14
6.1.1;Introduction;14
6.1.2;Finite differences;15
6.1.3;Finite-difference formulae;17
6.1.4;Lagrangian formulae and rounding errors;21
6.1.5;Truncation error;22
6.1.6;Conclusion;26
6.2;CHAPTER 2. METHODS OF RUNGE–KUTTA TYPE;27
6.2.1;Introduction;27
6.2.2;Method of Taylor's series;27
6.2.3;Runge–Kutta methods;31
6.2.4;The fourth-order process;33
6.2.5;Evaluation of Runge–Kutta methods;35
6.2.6;The Kutta–Merson process;35
6.2.7;Simultaneous equations;36
6.2.8;Practical details;37
6.3;CHAPTER 3. PREDICTION AND CORRECTION: DEFERRED CORRECTION;39
6.3.1;Introduction;39
6.3.2;Adams–Bashforth method;40
6.3.3;Methods of Milne;40
6.3.4;Convergence of the iteration;41
6.3.5;Simultaneous first-order equations;42
6.3.6;Comparison with Runge–Kutta method;43
6.3.7;Other methods of prediction;43
6.3.8;Mixed methods;44
6.3.9;Change of interval;45
6.3.10;Starting the integrations;47
6.3.11;Comparison of predictor–corrector methods;48
6.3.12;Deferred correction;48
6.3.13;Non-linear equations. Evaluation;51
6.3.14;Convergence of the difference-correction iteration;52
6.3.15;Other methods;54
6.4;CHAPTER 4. STABILITY OF STEP-BY-STEP METHODS;57
6.4.1;Introduction;57
6.4.2;Inherent instability;59
6.4.3;Partial instability;60
6.4.4;Strong instability;62
6.4.5;General theory;63
6.4.6;Non-linear equations. Simultaneous equations;64
6.4.7;Practical considerations;65
6.4.8;Effect of the predictor formula;66
6.4.9;Comparison of predictor–corrector methods;67
6.4.10;Explicit use of differences;68
6.5;CHAPTER 5. BOUNDARY-VALUE PROBLEMS AND METHODS;69
6.5.1;Introduction;69
6.5.2;Initial-value methods. Linear systems;69
6.5.3;Non-linear equations;74
6.5.4;Boundary-value techniques;77
6.5.5;Comparison of methods for time and convenience;81
6.6;CHAPTER 6. EIGENVALUE PROBLEMS;84
6.6.1;Introduction;84
6.6.2;Boundary-value methods;84
6.6.3;Latent roots and vectors of matrices;85
6.6.4;Numerical methods for the algebraic problem;86
6.6.5;Comments on the mathematical problem;88
6.6.6;Connexions between differential and algebraic problems;90
6.6.7;Initial-value methods;93
6.6.8;Infinite ranges and stability;95
6.6.9;Comparison of finite-difference methods;96
6.6.10;Solution in series;96
6.7;CHAPTER 7. APPLICATION TO THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION;98
6.7.1;Introduction;98
6.7.2;Boundary-value method;99
6.7.3;Initial-value method;99
6.7.4;Special techniques;101
6.7.5;The matching point;103
6.7.6;An inhomogeneous eigenvalue problem;104
6.8;CHAPTER 8. ACCURACY AND PRECISION OF METHODS;106
6.8.1;Introduction;106
6.8.2;Inherent instability;106
6.8.3;Induced instability;108
6.8.4;Truncation errors;114
6.8.5;The deferred approach to the limit;117
6.8.6;Failure of deferred approach;121
6.8.7;Convergence of the deferred correction iteration;122
6.9;CHAPTER 9. CHEBYSHEV APPROXIMATION;124
6.9.1;Introduction;124
6.9.2;Fourier series;125
6.9.3;The Chebyshev polynomials;126
6.9.4;Polynomials of best fit;127
6.9.5;Convergence of certain series;129
6.9.6;Properties of Chebyshev polynomials;130
6.9.7;Evaluation, differentiation and integration;132
6.9.8;Economized power series;134
6.9.9;Other representations;135
6.9.10;The Lanczos t-method for finite series;136
6.9.11;The infinite series;138
6.10;CHAPTER 10. CHEBYSHEV SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS;140
6.10.1;Introduction;140
6.10.2;Method of Lanczos;140
6.10.3;Overdetermination;142
6.10.4;Equations of higher order;143
6.10.5;Analysis of error;143
6.10.6;Improving the accuracy;145
6.10.7;Method of Clenshaw;146
6.10.8;Reduction in order;148
6.10.9;Eigenvalue problems;150
6.10.10;Evaluation;151
7;PART II: INTEGRAL EQUATIONS;154
7.1;CHAPTER 11. FREDHOLM EQUATIONS OF SECOND KIND;156
7.1.1;Introduction;156
7.1.2;Analogy with algebraic equations;157
7.1.3;Finite-difference methods;157
7.1.4;Gauss quadrature formulae;159
7.1.5;Some simplifications;160
7.1.6;Degenerate kernels;161
7.1.7;The classical iterative method;164
7.2;CHAPTER 12. FREDHOLM EQUATIONS OF FIRST AND THIRD KINDS;165
7.2.1;Introduction;165
7.2.2;The eigenvalue problem;165
7.2.3;The Fredholm equation of the first kind;170
7.2.4;Practical difficulties;171
7.3;CHAPTER 13. EQUATIONS OF VOLTERRA TYPE;176
7.3.1;Introduction;176
7.3.2;Finite-difference methods;177
7.3.3;Solution by Laplace transforms;180
7.3.4;Stability and convergence;180
7.3.5;Mitigation of instability;182
7.3.6;Equations of the first kind;183
7.4;CHAPTER 14. SINGULAR AND NON-LINEAR INTEGRAL EQUATIONS;185
7.4.1;Introduction;185
7.4.2;Infinite interval in Fredholm equation;185
7.4.3;Singular kernel in Fredholm equation;186
7.4.4;Singular Volterra equations;189
7.4.5;Stability;190
7.4.6;Non-linear equations;192
7.4.7;Introduction;195
7.4.8;The integro-differential equations;198
7.4.9;Methods of solution;200
7.4.10;The matrix method of solution;203
7.5;CHAPTER 15. INTEGRO-DIFFERENTIAL EQUATIONS IN NUCLEAR COLLISION PROBLEMS;195
7.5.1;Introduction;195
7.5.2;The integro-differential equations;198
7.5.3;Methods of solution;200
7.5.4;The matrix method of solution;203
7.6;CHAPTER 16. ROOTHAAN'S PROCEDURE FOR SOLVING THE HARTREE–FOCK EQUATION;208
7.6.1;Introduction;208
7.6.2;The approximate wave function;208
7.6.3;The variation principle;209
7.6.4;The general form of the Hartree-Fock equations;210
7.6.5;The numerical method of solution;210
7.6.6;The Roothaan procedure;211
7.6.7;The functions .i4 (l;r);212
7.6.8;Comparison with other procedures;213
8;PART III: INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS;214
8.1;CHAPTER 17. GENERAL CLASSIFICATION: HYPERBOLIC EQUATIONS AND CHARACTERISTICS;216
8.1.1;Introduction;216
8.1.2;First-order equations;216
8.1.3;Second-order equations;221
8.1.4;Hyperbolic equations: Solution by characteristics;223
8.1.5;Solution by characteristics on a rectangular grid;226
8.1.6;Simultaneous first-order equations;227
8.2;CHAPTER 18. FINITE-DIFFERENCE METHODS FOR HYPERBOLIC EQUATIONS;229
8.2.1;Introduction;229
8.2.2;Explicit finite-difference methods;229
8.2.3;Stability;231
8.2.4;Implicit methods;232
8.2.5;Stability by matrix methods;233
8.2.6;Non-linear equations;234
8.2.7;Simultaneous first-order equations;234
8.2.8;Discontinuities;237
8.3;CHAPTER 19. PARABOLIC EQUATIONS IN TWO DIMENSIONS: I;241
8.3.1;Introduction;241
8.3.2;An explicit finite-difference scheme;242
8.3.3;Finite x-region;244
8.3.4;Stability by Fourier method;245
8.3.5;Stability by matrix method;246
8.3.6;Compatibility, stability and convergence;249
8.3.7;The deferred approach to the limit;251
8.3.8;A Runge–Kutta method;251
8.4;CHAPTER 20. PARABOLIC EQUATIONS IN TWO DIMENSIONS: II;253
8.4.1;Reducing the truncation error;253
8.4.2;More difficult linear problems;256
8.4.3;Non-linear problems;260
8.4.4;Singularities;263
8.5;CHAPTER 21. FINITE-DIFFERENCE FORMULAE FOR ELLIPTIC EQUATIONS IN TWO DIMENSIONS;266
8.5.1;Introduction;266
8.5.2;Finite-difference equations, specified boundary values;269
8.5.3;Normal derivatives specified;275
8.5.4;Eigenvalue problems;277
8.5.5;Biharmonic problems;278
8.5.6;Convergence and truncation error;279
8.6;CHAPTER 22. DIRECT SOLUTION OF ELLIPTIC FINITE-DIFFERENCE EQUATIONS;283
8.6.1;Introduction;283
8.6.2;Form of the equations;283
8.6.3;Solution in band form;286
8.6.4;Organization of the calculation;287
8.6.5;Use of partitioned matrices;289
8.6.6;Accuracy and stability;292
8.7;CHAPTER 23. ITERATIVE SOLUTION OF ELLIPTIC FINITE-DIFFERENCE EQUATIONS;295
8.7.1;Introduction;295
8.7.2;General theory of iterative methods;296
8.7.3;Methods of Jacobi and Gauss–Seidel;298
8.7.4;Successive over-relaxation;300
8.7.5;Property (A);302
8.7.6;Block methods: Line iteration;305
8.7.7;Alternating direction methods;307
8.7.8;Accelerating devices;309
8.7.9;Chebyshev acceleration;309
8.8;CHAPTER 24. SINGULARITIES IN ELLIPTIC EQUATIONS;312
8.8.1;Introduction;312
8.8.2;Discontinuous boundary values;313
8.8.3;Change of direction of boundary;314
8.8.4;Methods of Motz and Woods;315
8.8.5;Difficult eigenvalue problems;317
8.8.6;The L-shaped membrane;319
8.8.7;Other comments;322
9;PART IV: PRACTICAL PROBLEMS IN PARTIAL DIFFERENTIAL EQUATIONS;324
9.1;CHAPTER 25. ELLIPTIC EQUATIONS IN NUCLEAR REACTOR PROBLEMS;326
9.1.1;Introduction: The diffusion equation;326
9.1.2;Finite-difference equations;327
9.1.3;Solution of the flux equations;329
9.1.4;Alternating-direction methods;330
9.1.5;Extrapolation of the source term;332
9.1.6;Organization of the calculation;334
9.2;CHAPTER 26. SOLUTION BY CHARACTERISTICS OF THE EQUATIONS OF ONE-DIMENSIONAL UNSTEADY FLOW;336
9.2.1;Introduction;336
9.2.2;The equations of compressible flow;336
9.2.3;Conditions at shock discontinuities;338
9.2.4;Conditions at interfaces, free surfaces, etc.;339
9.2.5;Configurations in the net of characteristics;339
9.2.6;Interaction of a shock and an interface;341
9.2.7;The solution of the shock relations;342
9.2.8;The intersection of characteristics;344
9.2.9;Calculation of the shock wave: Case 1;345
9.2.10;Calculation at a free surface;345
9.2.11;Calculation of the shock wave: Case 2;346
9.2.12;Calculation of interface points;347
9.2.13;Initial solutions at singularities;347
9.2.14;On programming;348
9.2.15;Conclusion;349
9.3;CHAPTER 27. FINITE-DIFFERENCE METHODS FOR ONE-DIMENSIONAL UNSTEADY FLOW;350
9.3.1;Introduction;350
9.3.2;Eulerian and Lagrangian equations;353
9.3.3;Conservative form;355
9.3.4;Finite-difference schemes;358
9.3.5;Discontinuities in the flow;360
9.3.6;Interfaces;360
9.3.7;Shock waves;362
9.3.8;Methods of treating shocks;363
9.3.9;The method of von Neumann and Richtmyer;364
9.3.10;The method of Lax;366
9.3.11;Other artificial dissipative methods;369
9.3.12;Errors in the artificial dissipative method;372
9.3.13;Exact treatment of the shock;374
9.3.14;A method using characteristics;374
9.4;CHAPTER 28. CHARACTERISTICS IN THREE INDEPENDENT VARIABLES;377
9.4.1;Introduction;377
9.4.2;Characteristics and bicharacteristics;378
9.4.3;Methods of computation;382
9.4.4;Comparison with other methods;387
9.5;CHAPTER 29. QUASI-LINEAR PARABOLIC EQUATIONS IN MORE THAN TWO DIMENSIONS: I;389
9.5.1;Introduction;389
9.5.2;The spatial finite-difference scheme;389
9.5.3;An explicit method of solution;392
9.5.4;Implicit methods;394
9.6;CHAPTER 30. QUASI-LINEAR PARABOLIC EQUATIONS IN MORE THAN TWO DIMENSIONS: II;399
9.6.1;Introduction;399
9.6.2;Time-dependent coefficients;399
9.6.3;Relaxation methods for the finite-difference equations;400
9.6.4;Alternating-direction methods;403
9.7;CHAPTER 31. THE LINEAR TRANSPORT EQUATION IN ONE AND TWO DIMENSIONS;409
9.7.1;Introduction;409
9.7.2;Eigenvalue problems;410
9.7.3;Multi-group eigenvalue problems;415
9.7.4;Other types of calculations;419
9.7.5;Integration of the transport operator;420
9.7.6;The spherical harmonics method;420
9.7.7;The Sn method;422
9.7.8;Conservation relations;425
9.7.9;The discrete Sn method;426
9.7.10;The finite cylinder;429
9.8;CHAPTER 32. MONTE CARLO METHODS FOR NEUTRONICS PROBLEMS;434
9.8.1;Introduction;434
9.8.2;The use of random numbers;436
9.8.3;External source problems;439
9.8.4;Eigenvalue problems;440
9.8.5;The utilization of electronic computers;444
9.8.6;Statistical considerations;447
9.8.7;Specimen calculations;447
9.8.8;Special techniques in Monte Carlo;450
9.9;CHAPTER 33. SPECIAL TECHNIQUES OF THE MONTE CARLO METHOD;453
9.9.1;Introduction;453
9.9.2;Variance reduction;455
9.9.3;Penetration through a thick shield;458
9.9.4;The matrix method;462
9.10;CHAPTER 34. SOME PROBLEMS IN PLASMA PHYSICS;469
9.10.1;Introduction;469
9.10.2;Mathematical model;469
9.10.3;Magneto-hydrodynamic equations;470
9.10.4;Finite-difference equations;471
9.10.5;Cylindrically symmetrical hydromagnetic disturbance in a plasma;472
9.10.6;Numerical Analysis;475
9.10.7;Particular solutions;476
9.11;CHAPTER 35. SELF-CONSISTENT SOLUTION OF A NON-LINEAR PROBLEM IN PLASMA PHYSICS;480
9.11.1;Introduction;480
9.11.2;Basic theory;480
9.11.3;Choice of the cavity and field-mode;481
9.11.4;Confinement equations for the cylindrical cavity;483
9.11.5;Method of computation;483
9.11.6;Results of a plasma confinement computation;485
9.11.7;Limitations;487
9.12;CHAPTER 36. NUMERICAL WEATHER PREDICTION;489
9.12.1;Introduction;489
9.12.2;The basic equations;490
9.12.3;The model equations;491
9.12.4;The practical solution of the equations;493
9.12.5;The effect of the boundary values;498
9.12.6;An example of the computation;499
9.12.7;A non-linear differential equation;501
10;REFERENCES;505
10.1;Part I;505
10.2;Part II;506
10.3;Part III;507
10.4;Part IV;509
11;INDEX;513
