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(s): Parviz Moin
Edition: 2
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 258
Tags: Математика;Вычислительная математика;
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Dedication......Page 10
Preface to the Second Edition......Page 11
Preface to the First Edition......Page 13
1.1 Lagrange Polynomial Interpolation......Page 17
1.2 Cubic Spline Interpolation......Page 20
EXERCISES......Page 24
FURTHER READING......Page 28
2.1 Construction of Difference Formulas Using Taylor Series......Page 29
2.2 A General Technique for Construction of Finite Difference Schemes......Page 31
2.3 An Alternative Measure for the Accuracy of Finite Differences......Page 33
2.4 Padé Approximations......Page 36
2.5 Non-Uniform Grids......Page 39
EXERCISES......Page 41
FURTHER READING......Page 45
3.1 Trapezoidal and Simpsons Rules......Page 46
3.2 Error Analysis......Page 47
3.3 Trapezoidal Rule with End-Correction......Page 50
3.4 Romberg Integration and Richardson Extrapolation......Page 51
3.5 Adaptive Quadrature......Page 53
3.6 Gauss Quadrature......Page 56
EXERCISES......Page 60
FURTHER READING......Page 63
4.1 Initial Value Problems......Page 64
4.2 Numerical Stability......Page 66
4.3 Stability Analysis for the Euler Method......Page 68
4.4 Implicit or Backward Euler......Page 71
4.5 Numerical Accuracy Revisited......Page 72
4.6 Trapezoidal Method......Page 74
4.7 Linearization for Implicit Methods......Page 78
4.8 Runge--Kutta Methods......Page 80
4.9 Multi-Step Methods......Page 86
4.10 System of First-Order Ordinary Differential Equations......Page 90
4.11 Boundary Value Problems......Page 94
4.11.1 Shooting Method......Page 95
4.11.2 Direct Methods......Page 98
EXERCISES......Page 100
FURTHER READING......Page 116
5 Numerical Solution of Partial Differential Equations......Page 117
5.1 Semi-Discretization......Page 118
5.2 von Neumann Stability Analysis......Page 125
5.3 Modified Wavenumber Analysis......Page 127
5.4 Implicit Time Advancement......Page 132
5.5 Accuracy via Modified Equation......Page 135
5.6 Du Fort--Frankel Method: An Inconsistent Scheme......Page 137
5.7 Multi-Dimensions......Page 140
5.8 Implicit Methods in Higher Dimensions......Page 142
5.9 Approximate Factorization......Page 144
5.9.1 Stability of the Factored Scheme......Page 149
5.9.2 Alternating Direction Implicit Methods......Page 150
5.9.3 Mixed and Fractional Step Methods......Page 152
5.10 Elliptic Partial Differential Equations......Page 153
5.10.1 Iterative Solution Methods......Page 156
5.10.2 The Point Jacobi Method......Page 157
5.10.3 Gauss--Seidel Method......Page 159
5.10.4 Successive Over Relaxation Scheme......Page 160
5.10.5 Multigrid Acceleration......Page 163
EXERCISES......Page 170
FURTHER READING......Page 182
6.1 Fourier Series......Page 183
6.1.1 Discrete Fourier Series......Page 184
6.1.2 Fast Fourier Transform......Page 185
6.1.3 Fourier Transform of a Real Function......Page 186
6.1.4 Discrete Fourier Series in Higher Dimensions......Page 188
6.1.5 Discrete Fourier Transform of a Product of Two Functions......Page 189
6.1.6 Discrete Sine and Cosine Transforms......Page 191
6.2.1 Direct Solution of Finite Differenced Elliptic Equations......Page 192
6.2.2 Differentiation of a Periodic Function Using Fourier Spectral Method......Page 196
6.2.3 Numerical Solution of Linear, Constant Coefficient Differential Equations with Periodic Boundary Conditions......Page 198
6.3 Matrix Operator for Fourier Spectral Numerical Differentiation......Page 201
6.4 Discrete Chebyshev Transform and Applications......Page 204
6.4.1 Numerical Differentiation Using Chebyshev Polynomials......Page 208
6.4.2 Quadrature Using Chebyshev Polynomials......Page 211
6.4.3 Matrix Form of Chebyshev Collocation Derivative......Page 212
6.5 Method of Weighted Residuals......Page 216
6.6 The Finite Element Method......Page 217
6.6.1 Application of the Finite Element Method to a Boundary Value Problem......Page 218
6.6.2 Comparison with Finite Difference Method......Page 223
6.6.3 Comparison with a Padé Scheme......Page 225
6.6.4 A Time-Dependent Problem......Page 226
The One-Dimensional Heat Equation......Page 227
6.7 Application to Complex Domains......Page 229
6.7.1 Constructing the Basis Functions......Page 231
EXERCISES......Page 237
FURTHER READING......Page 242
A.1 Vectors, Matrices and Elementary Operations......Page 243
A.2.1 Effects of Round-off Error......Page 246
A.3 Operations Counts......Page 247
A.4 Eigenvalues and Eigenvectors......Page 248
Index......Page 251
