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Preface to the Second Edition

Preface to the First Edition

1 Interpolation 1

1.1 Lagrange Polynomial Interpolation 1

1.2 Cubic Spline Interpolation 4

Exercises 8

Further Reading 12

2 Numerical Differentiation - Finite Differences 13

2.1 Construction of Difference Formulas Using Taylor Series 13

2.2 A General Technique for Construction of Finite Difference Schemes 15

2.3 An Alternative Measure for the Accuracy of Finite Differences 17

2.4 Pade Approximations 20

2.5 Non-Uniform Grids 23

Exercises 25

Further Reading 29

3 Numerical Integration 30

3.1 Trapezoidal and Simpson's Rules 30

3.2 Error Analysis 31

3.3 Trapezoidal Rule with End-Correction 34

3.4 Romberg Integration and Richardson Extrapolation 35

3.5 Adaptive Quadrature 37

3.6 Gauss Quadrature 40

Exercises 44

Further Reading 47

4 Numerical Solution of Ordinary Differential Equations 48

4.1 Initial Value Problems 48

4.2 Numerical Stability 50

4.3 Stability Analysis for the Euler Method 52

4.4 Implicit or Backward Euler 55

4.5 Numerical Accuracy Revisited 56

4.6 Trapezoidal Method 58

4.7 Linearization for Implicit Methods 62

4.8 Runge-Kutta Methods 64

4.9 Multi-Step Methods 70

4.10 System of First-Order Ordinary Differential Equations 74

4.11 Boundary Value Problems 78

4.11.1 Shooting Method 79

4.11.2 Direct Methods 82

Exercises 84

Further Reading 100

5 Numerical Solution of Partial Differential Equations 101

5.1 Semi-Discretization 102

5.2 von Neumann Stability Analysis 109

5.3 Modified Wavenumber Analysis 111

5.4 Implicit Time Advancement 116

5.5 Accuracy via Modified Equation 119

5.6 Du Fort-Frankel Method: An Inconsistent Scheme 121

5.7 Multi-Dimensions 124

5.8 Implicit Methods in Higher Dimensions 126

5.9 Approximate Factorization 128

5.9.1 Stability of the Factored Scheme 133

5.9.2 Alternating Direction Implicit Methods 134

5.9.3 Mixed and Fractional Step Methods 136

5.10 Elliptic Partial Differential Equations 137

5.10.1 Iterative Solution Methods 140

5.10.2 The Point Jacobi Method 141

5.10.3 Gauss-Seidel Method 143

5.10.4 Successive Over Relaxation Scheme 144

5.10.5 Multigrid Acceleration 147

Exercises 154

Further Reading 166

6 Discrete Transform Methods 167

6.1 Fourier Series 167

6.1.1 Discrete Fourier Series 168

6.1.2 Fast Fourier Transform 169

6.1.3 Fourier Transform of a Real Function 170

6.1.4 Discrete Fourier Series in Higher Dimensions 172

6.1.5 Discrete Fourier Transform of a Product of Two Functions 173

6.1.6 Discrete Sine and Cosine Transforms 175

6.2 Applications of Discrete Fourier Series 176

6.2.1 Direct Solution of Finite Differenced Elliptic Equations 176

6.2.2 Differentiation of a Periodic Function Using Fourier Spectral Method 180

6.2.3 Numerical Solution of Linear, Constant Coefficient Differential Equations with Periodic Boundary Conditions 182

6.3 Matrix Operator for Fourier Spectral Numerical Differentiation 185

6.4 Discrete Chebyshev Transform and Applications 188

6.4.1 Numerical Differentiation Using Chebyshev Polynomials 192

6.4.2 Quadrature Using Chebyshev Polynomials 195

6.4.3 Matrix Form of Chebyshev Collocation Derivative 196

6.5 Method of Weighted Residuals 200

6.6 The Finite Element Method 201

6.6.1 Application of the Finite Element Method to a Boundary Value Problem 202

6.6.2 Comparison with Finite Difference Method 207

6.6.3 Comparison with a Pade Scheme 209

6.6.4 A Time-Dependent Problem 210

6.7 Application to Complex Domains 213

6.7.1 Constructing the Basis Functions 215

Exercises 221

Further Reading 226

A A Review of Linear Algebra 227

A.1 Vectors, Matrices and Elementary Operations 227

A.2 System of Linear Algebraic Equations 230

A.2.1 Effects of Round-off Error 230

A.3 Operations Counts 231

A.4 Eigenvalues and Eigenvectors 232

Index 235

Engineers need hands-on experience in solving complex engineering problems with computers. This text introduces numerical methods and shows how to develop, analyze, and use them. A thorough and practical book, it is intended for use in a first course in numerical analysis. Along with mastering the fundamentals of numerical methods, students will learn to write their own computer programs using standard numerical methods. They will learn what factors affect accuracy, stability, and convergence, and how to evaluate critically the numerical output from a computer. Special features are the numerous examples and exercises that give students first-hand experience.