[eBook Code] Numerical Methods for Solving Partial Differential Equations (eBook Code, 1st)

Preface vii

1 Interpolation 1

1.1 Purpose 1

1.2 Definitions 1

1.3 Example 2

1.4 Weirstraus Approximation Theorem 3

1.5 Lagrange Interpolation 3

1.5.1 Example 6

1.6 Compare P₂ (θ) and f (θ) 8

1.7 Error of Approximation 9

1.8 Multiple Elements 14

1.8.1 Example 17

1.9 Hermite Polynomials 19

1.10 Error in Approximation by Hermites 22

1.11 ChapterSummary 23

1.12 Problems 24

2 Numerical Differentiation 31

2.1 General Theory 31

2.2 Two-Point Difference Formulae 32

2.2.1 Forward Difference Formula 33

2.2.2 Backward Difference Formula 33

2.2.3 Example 34

2.2.4 Error of the Approximation 34

2.3 Two-Point Formulae from Taylor Series 36

2.4 Three-point Difference Formulae 38

2.4.1 First-Order Derivative Difference Formulae 39

2.4.2 Second-Order Derivatives 40

2.5 Chapter Summary 44

2.6 Problems 44

3 Numerical Integration 53

3.1 Newton-Cotes Quadrature Formula 53

3.1.1 Lagrange Interpolation 53

3.1.2 Trapezoidal Rule 54

3.1.3 Simpson’s Rule 55

3.1.4 General Form 56

3.1.5 Example using Simpson’s Rule 56

3.1.6 Gauss Legendre Quadrature 57

3.2 Chapter Summary 60

3.3 Problems 61

4 Initial Value Problems 65

4.1 Euler Forward Integration Method Example 66

4.2 Convergence 67

4.3 Consistency 70

4.4 Stability 71

4.4.1 Example of Stability 72

4.5 Lax Equivalence Theorem 72

4.6 Runge−Kutta Type Formulae 72

4.6.1 GeneralForm 72

4.6.2 Runge−Kutta First-Order Form (Euler’s Method) 73

4.6.3 Runge−Kutta Second-Order Form 73

4.7 ChapterSummary 76

4.8 Problems 76

5 Weighted Residuals Methods 81

5.1 Finite Volume or Subdomain Method 82

5.1.1 Example 84

5.1.2 Finite Difference Interpretation of the Finite Volume Method 91

5.2 Galerkin Method for First Order Equations 92

5.2.1 Finite-Difference Interpretation of the Galerkin Approximation 99

5.3 Galerkin Method for Second-Order Equations 99

5.3.1 Finite Difference Interpretation of Second-Order Galerkin Method 107

5.4 Finite Volume Method for Second-Order Equations 108

5.4.1 Example of Finite Volume Solution of a Second-Order Equation 112

5.4.2 Finite Difference Representation of the Finite-Volume Method for Second-Order Equations 118

5.5 CollocationMethod 119

5.5.1 CollocationMethod forFirst-OrderEquations 119

5.5.2 Collocation Method for Second-Order Equations 122

5.6 ChapterSummary 128

5.7 Problems 128

6 Initial Boundary-Value Problems 133

6.1 Introduction 133

6.2 Two Dimensional Polynomial Approximations 133

6.2.1 Example of a Two Dimensional Polynomial Approximation 134

6.3 Finite Difference Approximation 135

6.3.1 First-Order Accurate Finite Difference Calculation 137

6.3.2 Example of Second Order Accurate Finite Difference Approximation in Space 140

6.4 Stability of Finite Difference Approximations 143

6.4.1 Example of Stability 146

6.4.2 Example Simulation 149

6.5 Galerkin Finite Element Approximations in Time 151

6.5.1 Strategy One: Forward Difference Approximation 153

6.5.2 Strategy Two: Backward Difference Approximation 154

6.6 Chapter Summary 155

6.7 Problems 155

7 Finite Difference Methods in Two Space 161

7.1 Example Problem 166

7.2 Chapter Summary 168

7.3 Problems 168

8 Finite Element Methods in Two Space 173

8.1 Finite Element Approximations over Rectangles 173

8.2 Finite Element Approximations over Triangles 186

8.2.1 Formulation of Triangular Basis Functions 188

8.2.2 Example Problem of Finite Element Approximation over Triangles 191

8.2.3 Second Type or Neumann Boundary-Value Problem 198

8.3 Isoparametric Finite Element Approximation 202

8.3.1 Natural Coordinate Systems 202

8.3.2 Basis Functions 208

8.3.3 Calculation of the Jacobian 209

8.3.4 Example of Isoparametric Formulation 213

8.4 Chapter Summary 220

8.5 Problems 220

9 Finite Volume Approximation in Two Space 229

9.1 Finite Volume Formulation 229

9.2 Finite Volume Example Problem 1 235

9.2.1 Problem Definition 235

9.2.2 Weighted Residual Formulation 236

9.2.3 Element Coefficient Matrices 237

9.2.4 Evaluation of the Line Integral 238

9.2.5 Evaluation of the Area Integral 245

9.2.6 Global Matrix Assembly 249

9.3 Finite Volume Example Problem Two 250

9.3.1 Problem Definition 250

9.3.2 Weighted Residual Formulation 251

9.3.3 Element Coefficient Matrices 252

9.3.4 Evaluation of the Source Term 253

9.4 Chapter Summary 254

9.5 Problems 254

10 Initial Boundary-Value Problems 261

10.1 Mass Lumping 263

10.2 Chapter Summary 264

10.3 Problems 264

11 Boundary-Value Problems in Three Space 267

11.1 Finite Difference Approximations 267

11.2 Finite Element Approximations 268

11.3 Chapter Summary 273

12 Nomenclature 277

Index 281