Fundamentals of Engineering Numerical Analysis

by Parviz Moin

• ISBN13: 9780521711234
• ISBN10: 0521711231
• Edition: 2nd
• Format: Paperback
• Copyright: 2010-08-23
• Publisher: Cambridge University Press

Summary

Since the original publication of this book, available computer power has increased greatly. Today, scientific computing is playing an ever more prominent role as a tool in scientific discovery and engineering analysis. In this second edition, the key addition is an introduction to the finite element method. This is a widely used technique for solving partial differential equations (PDEs) in complex domains. This text introduces numerical methods and shows how to develop, analyze, and use them. Complete MATLAB programs for all the worked examples are now available at www.cambridge.org/Moin, and more than 30 exercises have been added. This thorough and practical book is intended as a first course in numerical analysis, primarily for new graduate students in engineering and physical science. Along with mastering the fundamentals of numerical methods, students will learn to write their own computer programs using standard numerical methods.

Author Biography

Parviz Moin is the Franklin P. and Caroline M. Johnson Professor of Mechanical Engineering at Stanford University. He is the founder of the Center for Turbulence Research and the Stanford Institute for Computational and Mathematical Engineering. He pioneered the use of high-fidelity numerical simulations and massively parallel computers for the study of turbulence physics. Professor Moin is a Fellow of the American Physical Society, American Institute of Aeronautics and Astronautics, and the American Academy of Arts and Sciences. He is a Member of the National Academy of Engineering.

Table of Contents

Preface to the Second Edition	p. ix
Preface to the First Edition	p. xi
Interpolation	p. 1
Lagrange Polynomial Interpolation	p. 1
Cubic Spline Interpolation	p. 4
Exercises	p. 8
Further Reading	p. 12
Numerical Differentiation - Finite Differences	p. 13
Construction of Difference Formulas Using Taylor Series	p. 13
A General Technique for Construction of Finite Difference Schemes	p. 15
An Alternative Measure for the Accuracy of Finite Differences	p. 17
Padé Approximations	p. 20
Non-Uniform Grids	p. 23
Exercises	p. 25
Further Reading	p. 29
Numerical Integration	p. 30
Trapezoidal and Simpson's Rules	p. 30
Error Analysis	p. 31
Trapezoidal Rule with End-Correction	p. 34
Romberg Integration and Richardson Extrapolation	p. 35
Adaptive Quadrature	p. 37
Gauss Quadrature	p. 40
Exercises	p. 44
Further Reading	p. 47
Numerical Solution of Ordinary Differential Equations	p. 48
Initial Value Problems	p. 48
Numerical Stability	p. 50
Stability Analysis for the Euler Method	p. 52
Implicit or Backward Euler	p. 55
Numerical Accuracy Revisited	p. 56
Trapezoidal Method	p. 58
Linearization for Implicit Methods	p. 62
Runge-Kutta Methods	p. 64
Multi-Step Methods	p. 70
System of First-Order Ordinary Differential Equations	p. 74
Boundary Value Problems	p. 78
Shooting Method	p. 79
Direct Methods	p. 82
Exercises	p. 84
Further Reading	p. 100
Numerical Solution of Partial Differential Equations	p. 101
Semi-Discretization	p. 102
von Neumann Stability Analysis	p. 109
Modified Wavenumber Analysis	p. 111
Implicit Time Advancement	p. 116
Accuracy via Modified Equation	p. ll9
Du Fort-Frankel Method: An Inconsistent Scheme	p. 121
Multi-Dimensions	p. 124
Implicit Methods in Higher Dimensions	p. 126
Approximate Factorization	p. 128
Stability of the Factored Scheme	p. 133
Alternating Direction Implicit Methods	p. 134
Mixed and Fractional Step Methods	p. 136
Elliptic Partial Differential Equations	p. 137
Iterative Solution Methods	p. 140
The Point Jacobi Method	p. l41
Gauss-Seidel Method	p. 143
Successive Over Relaxation Scheme	p. 144
Multigrid Acceleration	p. 147
Exercises	p. 154
Further Reading	p. 166
Discrete Transform Methods	p. 167
Fourier Series	p. 167
Discrete Fourier Series	p. 168
Fast Fourier Transform	p. 169
Fourier Transform of a Real Function	p. 170
Discrete Fourier Series in Higher Dimensions	p. 172
Discrete Fourier Transform of a Product of Two Functions	p. 173
Discrete Sine and Cosine Transforms	p. 175
Applications of Discrete Fourier Series	p. 176
Direct Solution of Finite Differenced Elliptic Equations	p. 176
Differentiation of a Periodic Function Using Fourier Spectral Method	p. 180
Numerical Solution of Linear, Constant Coefficient Differential Equations with Periodic Boundary Conditions	p. 182
Matrix Operator for Fourier Spectral Numerical Differentiation	p. 185
Discrete Chebyshev Transform and Applications	p. 188
Numerical Differentiation Using Chebyshev Polynomials	p. 192
Quadrature Using Chebyshev Polynomials	p. 195
Matrix Form of Chebyshev Collocation Derivative	p. 196
Method of Weighted Residuals	p. 200
The Finite Element Method	p. 201
Application of the Finite Element Method to a Boundary Value Problem	p. 202
Comparison with Finite Difference Method	p. 207
Comparison with a Padé Scheme	p. 209
A Time-Dependent Problem	p. 210
Application to Complex Domains	p. 213
Constructing the Basis Functions	p. 215
Exercises	p. 221
Further Reading	p. 226
A Review of Linear Algebra	p. 227
Vectors, Matrices and Elementary Operations	p. 227
System of Linear Algebraic Equations	p. 230
Effects of Round-off Error	p. 230
Operations Counts	p. 231
Eigenvalues and Eigenvectors	p. 232
Index	p. 235
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