Numerical Solution of Differential Equations

Numerical Solution of Differential Equations

1st Edition - January 1, 1979

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  • Author: Isaac Fried
  • eBook ISBN: 9781483262529

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Description

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.

Table of Contents


  • 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

Product details

  • No. of pages: 278
  • Language: English
  • Copyright: © Academic Press 1979
  • Published: January 1, 1979
  • Imprint: Academic Press
  • eBook ISBN: 9781483262529

About the Author

Isaac Fried

About the Editor

Werner Rheinboldt

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