Fundamentals of Engineering Numerical Analysis

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  • Edition: 2nd
  • Format: Paperback
  • Copyright: 2010-08-23
  • Publisher: Cambridge University Press
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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 Editionp. ix
Preface to the First Editionp. xi
Interpolationp. 1
Lagrange Polynomial Interpolationp. 1
Cubic Spline Interpolationp. 4
Exercisesp. 8
Further Readingp. 12
Numerical Differentiation - Finite Differencesp. 13
Construction of Difference Formulas Using Taylor Seriesp. 13
A General Technique for Construction of Finite Difference Schemesp. 15
An Alternative Measure for the Accuracy of Finite Differencesp. 17
Padé Approximationsp. 20
Non-Uniform Gridsp. 23
Exercisesp. 25
Further Readingp. 29
Numerical Integrationp. 30
Trapezoidal and Simpson's Rulesp. 30
Error Analysisp. 31
Trapezoidal Rule with End-Correctionp. 34
Romberg Integration and Richardson Extrapolationp. 35
Adaptive Quadraturep. 37
Gauss Quadraturep. 40
Exercisesp. 44
Further Readingp. 47
Numerical Solution of Ordinary Differential Equationsp. 48
Initial Value Problemsp. 48
Numerical Stabilityp. 50
Stability Analysis for the Euler Methodp. 52
Implicit or Backward Eulerp. 55
Numerical Accuracy Revisitedp. 56
Trapezoidal Methodp. 58
Linearization for Implicit Methodsp. 62
Runge-Kutta Methodsp. 64
Multi-Step Methodsp. 70
System of First-Order Ordinary Differential Equationsp. 74
Boundary Value Problemsp. 78
Shooting Methodp. 79
Direct Methodsp. 82
Exercisesp. 84
Further Readingp. 100
Numerical Solution of Partial Differential Equationsp. 101
Semi-Discretizationp. 102
von Neumann Stability Analysisp. 109
Modified Wavenumber Analysisp. 111
Implicit Time Advancementp. 116
Accuracy via Modified Equationp. ll9
Du Fort-Frankel Method: An Inconsistent Schemep. 121
Multi-Dimensionsp. 124
Implicit Methods in Higher Dimensionsp. 126
Approximate Factorizationp. 128
Stability of the Factored Schemep. 133
Alternating Direction Implicit Methodsp. 134
Mixed and Fractional Step Methodsp. 136
Elliptic Partial Differential Equationsp. 137
Iterative Solution Methodsp. 140
The Point Jacobi Methodp. l41
Gauss-Seidel Methodp. 143
Successive Over Relaxation Schemep. 144
Multigrid Accelerationp. 147
Exercisesp. 154
Further Readingp. 166
Discrete Transform Methodsp. 167
Fourier Seriesp. 167
Discrete Fourier Seriesp. 168
Fast Fourier Transformp. 169
Fourier Transform of a Real Functionp. 170
Discrete Fourier Series in Higher Dimensionsp. 172
Discrete Fourier Transform of a Product of Two Functionsp. 173
Discrete Sine and Cosine Transformsp. 175
Applications of Discrete Fourier Seriesp. 176
Direct Solution of Finite Differenced Elliptic Equationsp. 176
Differentiation of a Periodic Function Using Fourier Spectral Methodp. 180
Numerical Solution of Linear, Constant Coefficient Differential Equations with Periodic Boundary Conditionsp. 182
Matrix Operator for Fourier Spectral Numerical Differentiationp. 185
Discrete Chebyshev Transform and Applicationsp. 188
Numerical Differentiation Using Chebyshev Polynomialsp. 192
Quadrature Using Chebyshev Polynomialsp. 195
Matrix Form of Chebyshev Collocation Derivativep. 196
Method of Weighted Residualsp. 200
The Finite Element Methodp. 201
Application of the Finite Element Method to a Boundary Value Problemp. 202
Comparison with Finite Difference Methodp. 207
Comparison with a Padé Schemep. 209
A Time-Dependent Problemp. 210
Application to Complex Domainsp. 213
Constructing the Basis Functionsp. 215
Exercisesp. 221
Further Readingp. 226
A Review of Linear Algebrap. 227
Vectors, Matrices and Elementary Operationsp. 227
System of Linear Algebraic Equationsp. 230
Effects of Round-off Errorp. 230
Operations Countsp. 231
Eigenvalues and Eigenvectorsp. 232
Indexp. 235
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