Numerical Methods for Engineers and Scientists

Transcription

1 Numerical Methods for Engineers and Scientists Second Edition Revised and Expanded Joe D. Hoffman Department of Mechanical Engineering Purdue University West Lafayette, Indiana m MARCEL D E К К E R MARCEL DEKKER, INC. NEW YORK BASEL

2 Preface er Objectives and Approach Organization of the Book Examples Problems Significant Digits, Precision, Accuracy, Errors, and Number Representation Software Packages and Libraries The Taylor Series and the Taylor Polynomial 7 Parti. Basic Tools of Numerical Analysis Systems of Linear Algebraic Equations Eigenproblems Roots of Nonlinear Equations Polynomial Approximation and Interpolation Numerical Differentiation and Difference Formulas Numerical Integration er 1. Systems of Linear Algebraic Equations Properties of Matrices and Determinants Direct Elimination Methods LU Factorization Tridiagonal Systems of Equations Pitfalls of Elimination Methods Iterative Methods er 2. Eigenproblems Mathematical Characteristics of Eigenproblems The Power Method 89 v VII

3 VIII The Direct Method The QR Method Eigenvectors Other Methods гг 3. Nonlinear Equations General Features of Root Finding Closed Domain (Bracketing) Methods Open Domain Methods Polynomials Pitfalls of Root Finding Methods and Other Methods of Root Finding Systems of Nonlinear Equations er 4. Polynomial Approximation and Interpolation Properties of Polynomials Direct Fit Polynomials Lagrange Polynomials Divided Difference Tables and Divided Difference Polynomials Difference Tables and Difference Polynomials Inverse Interpolation Multivariate Approximation Cubic Splines Least Squares Approximation er 5. Numerical Differentiation and Difference Formulas Unequally Spaced Data Equally Spaced Data Taylor Series Approach Difference Formulas Error Estimation and Extrapolation er 6. Numerical Integration Direct Fit Polynomials Newton-Cotes Formulas

4 IX 6.4. Extrapolation and Romberg Integration Adaptive Integration Gaussian Quadrature Multiple Integrals Part II. Ordinary Differential Equations 323 II General Features of Ordinary Differential Equations Classification of Ordinary Differential Equations Classification of Physical Problems Initial-Value Ordinary Differential Equations 327 П.6. Boundary-Value Ordinary Differential Equations 330 II er 7. One-Dimensional Initial-Value Ordinary Differential Equations General Features of Initial-Value ODEs The Taylor Series Method The Finite Difference Method The First-Order Euler Methods Consistency, Order, Stability, and Convergence Single-Point Methods Extrapolation Methods Multipoint Methods of Methods and Results Nonlinear Implicit Finite Difference Equations Higher-Order Ordinary Differential Equations Systems of First-Order Ordinary Differential Equations Stiff Ordinary Differential Equations er 8. One-Dimensional Boundary-Value Ordinary Differential Equations General Features of Boundary-Value ODEs The Shooting (Initial-Value) Method The Equilibrium (Boundary-Value) Method Derivative (and Other) Boundary Conditions Higher-Order Equilibrium Methods The Equilibrium Method for Nonlinear Boundary-Value Problems The Equilibrium Method on Nonuniform Grids Eigenproblems

5 X Part III. Partial Differential Equations 501 III General Features of Partial Differential Equations Classification of Partial Differential Equations Classification of Physical Problems Elliptic Partial Differential Equations Parabolic Partial Differential Equations Hyperbolic Partial Differential Equations The Convection-Diffusion Equation Initial Values and Boundary Conditions 524 ШЛО. Well-Posed Problems 525 III.ll. 526 er 9. Elliptic Partial Differential Equations General Features of Elliptic PDEs The Finite Difference Method Finite Difference Solution of the Laplace Equation Consistency, Order, and Convergence Iterative Methods of Solution Derivative Boundary Conditions Finite Difference Solution of the Poisson Equation Higher-Order Methods Nonrectangular Domains Nonlinear Equations and Three-Dimensional Problems The Control Volume Method er 10. Parabolic Partial Differential Equations General Features of Parabolic PDEs The Finite Difference Method The Forward-Time Centered-Space (FTCS) Method Consistency, Order, Stability, and Convergence The Richardson and DuFort-Frankel Methods Implicit Methods Derivative Boundary Conditions Nonlinear Equations and Multidimensional Problems The Convection-Diffusion Equation Asymptotic Steady State Solution to Propagation Problems er 11. Hyperbolic Partial Differential Equations General Features of Hyperbolic PDEs 655

6 XI The Finite Difference Method The Forward-Time Centered-Space (FTCS) Method and the Lax Method Lax-Wendroff Type Methods Upwind Methods The Backward-Time Centered-Space (BTCS) Method Nonlinear Equations and Multidimensional Problems The Wave Equation er 12. The Finite Element Method The Rayleigh-Ritz, Collocation, and Galerkin Methods The Finite Element Method for Boundary Value Problems The Finite Element Method for the Laplace (Poisson) Equation The Finite Element Method for the Diffusion Equation References 775 Answers to Selected Problems 779 Index 795