Numerical Solution of Differential Equations
- 1st Edition - May 10, 2014
- Latest edition
- Author: Isaac Fried
- Editor: Werner Rheinboldt
- Language: English
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… Read more
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.
Preface
1 Finite Differences
1. Calculus to Algebra to Arithmetic
2. Differentials and Finite Difference Approximations
3. Finite Difference Schemes
4. Accuracy Analysis
5. Higher-Order Schemes
Exercises
Suggested Further Reading
2 Two-Point Boundary Value Problems
1. Finite Difference Approximation of the Loaded String Equation
2. Incorporation of Boundary Conditions
3. Consistency and Stability: Convergence
4. Higher-Order Consistency
5. Finite Difference Approximation of the Beam Equation
6. Splitting of the Beam Equation into Two String Equations
7. Nonlinear Two-Point Boundary Value Problems
Exercises
Suggested Further Reading
3 Variational Formulations
1. Energy Error
2. Principle of Minimum Potential Energy
3. More General Boundary Conditions
4. Complementary Variational Principles
5. Euler-Lagrange Equations
6. Total Potential Energy of the Thin Elastic Beam
7. Indefinite Variational Principles
8. A Bound Theorem
Exercises
Suggested Further Reading
4 Finite Elements
1. The Idea of Ritz
2. Finite Element Basis Functions
3. Finite Element Matrices
4. Assembly of Global Matrices
5. Essential and Natural Boundary Conditions
6. Higher-Order Finite Elements
7. Beam Element
8. Complex Structures
Exercises
Suggested Further Reading
5 Discretization Accuracy
1. Energy Theorems
2. Energy Rates of Convergence
3. Sharpness of the Energy Error Estimate
4. L2 Error Estimate
5. L∞ Error Estimate
6. Richardson's Extrapolation to the Limit
7. Numerical Integration
Exercises
Suggested Further Reading
6 Eigenproblems
1. Stability of Columns
2. Vibration of Elastic Systems
3. Finite Difference Approximation
4. Rayleigh's Quotient
5. Finite Element Approximation
6. The Minmax Principle
7. Discretization Accuracy of Eigenvalues
8. Discretization Accuracy of Eigenfunctions
9. Change of Basis: Condensation
10. Numerical Integration: Lumping
11. Nonlinear Eigenproblems
Exercises
Suggested Further Reading
7 Algebraic Properties of the Global Matrices
1. Eigenvalue Range in Ky = λ My
2. Spectral Norms of K and M
3. Spectral Condition Numbers
4. Irregular Meshes
5. The Influence (Green's) Function
6. Maximum Norms and Condition Numbers
7. Scaling
8. Positive Flexibility Matrices
9. Computational Errors
10. Detection of Computational Errors
Exercises
Suggested Further Reading
8 Equation of Heat Transfer
1. Nonstationary Heat Transfer in a Rod
2. Finite Difference Approximation
3. Modal Analysis
4. Finite Elements
5. Essential Boundary Conditions
6. Euler's Stepwise Integration in Time
7. Explicit Finite Element Schemes
8. Convergence of Euler's Method
9. Stability
10. Stable Time Step Size Estimate
11. Numerical Example
12. Implicit Unconditionally Stable Schemes
13. Higher-Order Single Step Implicit Schemes
14. Superstable Schemes
15. Multistep Schemes
16. Predictor-Corrector Methods
17. Nonlinear Heat Condition and the Runge-Kutta Method
Exercises
Suggested Further Reading
9 Equation of Motion
1. Spring-Mass System
2. Single Step Explicit Scheme
3. Conditionally Stable Schemes
5. Modal Decomposition
6. Stability Conditions for Ky + Mÿ = 0
7. Nonlinear Equation of Motion
8. Single Step Unconditionally Stable Implicit Scheme
9. Unconditionally Stable Semiexplicit Schemes
10. Multistep Methods
11. Runge-Kutta-Nyström Method
12. Shooting in Boundary Value Problems
Exercises
Suggested Further Reading
10 Wave Propagation
1. Standing and Traveling Waves in a String
2. Discretization in Space
3. Spurious Dispersion
4. Effects of (Numerical) Viscosity
5. Higher-Order Elements
6. Spurious Reflection
7. Flexural Waves in a Beam
8. Stiff String
Exercises
Suggested Further Reading
Index
- Edition: 1
- Latest edition
- Published: May 10, 2014
- Language: English
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