Fried / Rheinboldt Numerical Solution of Differential Equations

1. Auflage 2014
ISBN: 978-1-4832-6252-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 278 Seiten, Web PDF

Numerical Solution of Differential Equations is a 10-chapter text that provides the numerical solution and practical aspects of differential equations. After a brief overview of the fundamentals of differential equations, this book goes on presenting the principal useful discretization techniques and their theoretical aspects, along with geometrical and physical examples, mainly from continuum mechanics. Considerable chapters are devoted to the development of the techniques of the numerical solution of differential equations and their analysis. The remaining chapters explore the influential invention in computational mechanics-finite elements. Each chapter emphasizes the relationship among the analytic formulation of the physical event, the discretization techniques applied to it, the algebraic properties of the discrete systems created, and the properties of the digital computer. This book will be of great value to undergraduate and graduate mathematics and physics students.

Autoren/Hrsg.

Fried, Isaac

Rheinboldt, Werner

Inhaltsverzeichnis

1;Front Cover;1
2;Numerical Solution of Differential Equations;4
3;Copyright Page;5
4;Table of Contents;8
5;Preface;12
6;CHAPTER 1. FINITE DIFFERENCES;18
6.1;1. Calculus to Algebra to Arithmetic;18
6.2;2. Differentials and Finite Difference Approximations;20
6.3;3. Finite Difference Schemes;22
6.4;4. Accuracy Analysis;24
6.5;5. Higher-Order Schemes;26
6.6;Exercises;28
6.7;Suggested Further Reading;28
7;CHAPTER 2. TWO-POINT BOUNDARY VALUE PROBLEMS;29
7.1;1. Finite Difference Approximation of the Loaded String Equation;29
7.2;2. Incorporation of Boundary Conditions;31
7.3;3. Consistency and Stability: Convergence;33
7.4;4. Higher-Order Consistency;35
7.5;5. Finite Difference Approximation of the Beam Equation;36
7.6;6. Splitting of the Beam Equation into Two String Equations;38
7.7;7. Nonlinear Two-Point Boundary Value Problems;40
7.8;Exercises;41
7.9;Suggested Further Reading;45
8;CHAPTER 3. VARIATIONAL FORMULATIONS;46
8.1;1. Energy Error;46
8.2;2. Principle of Minimum Potential Energy;50
8.3;3. More General Boundary Conditions;53
8.4;4. Complementary Variational Principles;55
8.5;5. Euler-Lagrange Equations;57
8.6;6. Total Potential Energy of the Thin Elastic Beam;59
8.7;7. Indefinite Variational Principles;60
8.8;8. A Bound Theorem;62
8.9;Exercises;62
8.10;Suggested Further Reading;64
9;CHAPTER 4. FINITE ELEMENTS;65
9.1;1. The Idea of Ritz;65
9.2;2. Finite Element Basis Functions;68
9.3;3. Finite Element Matrices;70
9.4;4. Assembly of Global Matrices;72
9.5;5. Essential and Natural Boundary Conditions;76
9.6;6. Higher-Order Finite Elements;78
9.7;7. Beam Element;84
9.8;8. Complex Structures;84
9.9;Exercises;86
9.10;Suggested Further Reading;87
10;CHAPTER 5. DISCRETIZATION ACCURACY;88
10.1;1. Energy Theorems;88
10.2;2. Energy Rates of Convergence;91
10.3;3. Sharpness of the Energy Error Estimate;96
10.4;4. L 2 Error Estimate;98
10.5;5. L8 Error
Estimate;100
10.6;6. Richardson's Extrapolation to the Limit;101
10.7;7. Numerical Integration;102
10.8;Exercises;103
10.9;Suggested Further Reading;105
11;CHAPTER 6. EIGENPROBLEMS;106
11.1;1. Stability of Columns;106
11.2;2. Vibration of Elastic Systems;110
11.3;3. Finite Difference Approximation;112
11.4;4. Rayleigh's Quotient;115
11.5;5. Finite Element Approximation;118
11.6;6. The Minmax Principle;120
11.7;7. Discretization Accuracy of Eigenvalues;122
11.8;8. Discretization Accuracy of Eigenfunctions;128
11.9;9. Change of Basis: Condensation;129
11.10;10. Numerical Integration: Lumping;132
11.11;11. Nonlinear Eigenproblems;135
11.12;Exercises;138
11.13;Suggested Further Reading;141
12;CHAPTER 7. ALGEBRAIC PROPERTIES OF THE GLOBAL MATRICES;142
12.1;1. Eigenvalue Range in Ky = ..
My;142
12.2;2. Spectral Norms of K
and M;145
12.3;3. Spectral Condition Numbers;148
12.4;4. Irregular Meshes;152
12.5;5. The Influence (Green's) Function;155
12.6;6. Maximum Norms and Condition Numbers;158
12.7;7. Scaling;160
12.8;8. Positive Flexibility Matrices;167
12.9;9. Computational Errors;169
12.10;10. Detection of Computational Errors;171
12.11;Exercises;172
12.12;Suggested Further Reading;174
13;CHAPTER 8. EQUATION OF HEAT TRANSFER;175
13.1;1. Nonstationary Heat Transfer in a Rod;175
13.2;2. Finite Difference Approximation;177
13.3;3. Modal Analysis;178
13.4;4. Finite Elements;180
13.5;5. Essential Boundary Conditions;180
13.6;6. Euler's Stepwise Integration in Time;182
13.7;7. Explicit Finite Element Schemes;184
13.8;8. Convergence of Euler's Method;185
13.9;9. Stability;186
13.10;10. Stable Time Step Size Estimate;188
13.11;11. Numerical Example;190
13.12;12. Implicit Unconditionally Stable Schemes;192
13.13;13. Higher-Order Single Step Implicit Schemes;194
13.14;14. Superstable Schemes;196
13.15;15. Multistep Schemes;199
13.16;16. Predictor-Corrector Methods;204
13.17;17. Nonlinear Heat Condition and the Runge-Kutta Method;206
13.18;Exercises;207
13.19;Suggested Further Reading;209
14;CHAPTER 9. EQUATION OF MOTION;210
14.1;1. Spring–Mass
System;210
14.2;2. Single Step Explicit Scheme;211
14.3;3. Conditionally Stable Schemes;213
14.4;4. Lattice of Springs and Masses;217
14.5;5. Modal Decomposition;218
14.6;6. Stability Conditions for Ky + My = 0;220
14.7;7. Nonlinear Equation of Motion;223
14.8;8. Single Step Unconditionally Stable Implicit Scheme;225
14.9;9. Unconditionally Stable Semiexplicit Schemes;228
14.10;10. Multistep Methods;231
14.11;11. Runge-Kutta-Nyström Method;235
14.12;12. Shooting in Boundary Value Problems;235
14.13;Exercises;237
14.14;Suggested Further Reading;245
15;CHAPTER 10. WAVE PROPAGATION;246
15.1;1. Standing and Traveling Waves in a String;246
15.2;2. Discretization in Space;249
15.3;3. Spurious Dispersion;251
15.4;4. Effects of (Numerical) Viscosity;254
15.5;5. Higher-Order Elements;255
15.6;6. Spurious Reflection;260
15.7;7. Flexural Waves in a Beam;262
15.8;8. Stiff String;266
15.9;Exercises;266
15.10;Suggested Further Reading;268
16;Index;270