Numerical methods for science and engineering. (2018)

• Record Type: Book
• Further Information: Note: George F. Pinder.
• Authors: Pinder, George Francis, 1942-
• Contents: Preface xi 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 4 1.5.1 Example 6 1.6 Compare P2 (_) and f (_) 8 1.7 Error of Approximation 9 1.8 Multiple Elements 14 1.8.1 Example 17 1.9 Hermite Polynomials 20 1.10 Error in Approximation by Hermites 23 1.11 Chapter Summary 24 1.12 Problems 24 2 Numerical Dierentiation 33 2.1 General Theory 33 2.2 Two-Point Dierence Formulae 34 2.2.1 Forward Dierence Formula 35 2.2.2 Backward Dierence Formula 36 2.2.3 Example 36 2.2.4 Error of the Approximation 36 2.3 Two-Point Formulae from Taylor Series 37 2.4 Three-point Dierence Formulae 40 2.4.1 First-Order Derivative Dierence Formulae 41 2.4.2 Second-Order Derivatives 43 2.5 Chapter Summary 46 2.6 Problems 46 3 Numerical Integration 55 3.1 Newton-Cotes Quadrature Formula 55 3.1.1 Lagrange Interpolation 55 3.1.2 Trapezoidal Rule 56 3.1.3 Simpson’s Rule 57 3.1.4 General Form 58 3.1.5 Example using Simpson’s Rule 59 3.1.6 Gauss Legendre Quadrature 59 3.2 Chapter Summary 62 3.3 Problems 63 4 Initial Value Problems 67 4.1 Euler Forward Integration Method Example 68 4.2 Convergence 69 4.3 Consistency 72 4.4 St ability 73 4.4.1 Example of Stability 74 4.5 Lax Equivalence Theorem 74 4.6 Runge Kut t a Type Formulae 75 4.6.1 General Form 75 4.6.2 Runge Kut ta First-Order Form (Euler’s Method) 75 4.6.3 Runge Kut ta Second-Order Form 75 4.7 Chapter Summary 78 4.8 Problems 78 5 Weight ed Residuals Methods 83 5.1 Finite Volume or Subdomain Method 84 5.1.1 Example 86 5.1.2 Finite Dierence Interpretation of the Finite Volume Method 93 5.2 Galerkin Method for First Order Equations 94 5.2.1 Finite-Dierence Interpretation of the Galerkin Approximation 102 5.3 Galerkin Method for Second-Order Equations 102 5.3.1 Finite Dierence Interpretation of Second-Order Galerkin Method111 5.4 Finite Volume Method for Second-Order Equations 112 5.4.1 Example of Finite Volume Solution of a Second-Order Equation 116 5.4.2 Finite Dierence Representat ion of the Finite-Volume Method for Second-Order Equations 122 5.5 Collocation Method 123 5.5.1 Collocation Method for First-Order Equations 123 5.5.2 Collocation Method for Second-Order Equations 126 5.6 Chapter Summary 133 5.7 Problems 133 6 Initial Boundary-Value Problems 139 6.1 Introduction 139 6.2 Two Dimensional Polynomial Approximat ions 139 6.2.1 Example of a Two Dimensional Polynomial Approximation 140 6.3 Finite Dierence Approximation 141 6.3.1 Example of Implicit First-Order Accurate Finite Dierence Calculation 144 6.3.2 Example of Second Order Accurate Finite Dierence Approximation in Space 146 6.4 St ability of Finite Dierence Approximations 149 6.4.1 Example of Stability 153 6.4.2 Example Simulation 156 6.5 Galerkin Finite Element Approximations in Time 158 6.5.1 Strategy One: Forward Dierence Approximation 160 6.5.2 Strategy Two: Backward Dierence Approximation 161 6.6 Chapter Summary 162 6.7 Problems 162 7 Finite Dierence Methods in Two Space 169 7.1 Example Problem 174 7.2 Chapt er Summary 175 7.3 Problems 176 8 Finite Element Methods in Two Space 181 8.1 Finite Element Approximations over Rectangles 181 8.2 Finite Element Approximations over Triangles 195 8.2.1 Formulation of Triangular Basis Funct ions 196 8.2.2 Example Problem of Finite Element Approximation over Triangles 200 8.2.3 Second Type or Neumann Boundary-Value Problem 206 8.3 Isoparametric Finite Element Approximation 211 8.3.1 Natural Coordinate Systems 211 8.3.2 Basis Functions 217 8.3.3 Calculation of the Jacobian 219 8.3.4 Example of Isoparametric Formulation 223 8.4 Chapter Summary 230 8.5 Problems 230 9 Finite Volume Approximation in Two Space 239 9.1 Finite Volume Formulation 239 9.2 Finite Volume Example Problem 1 246 9.2.1 Problem Definition 246 9.2.2 Weighted Residual Formulation 246 9.2.3 Element Coecient Matrices 248 9.2.4 Evaluation of the Line Integral 249 9.2.5 Evaluation of the Area Integral 256 9.2.6 Global Matrix Assembly 260 9.3 Finite Volume Example Problem Two 262 9.3.1 Problem Denition 262 9.3.2 Weighted Residual Formulation 262 9.3.3 Element Coecient Matrices 263 9.3.4 Evaluation of the Source Term 265 9.4 Chapter Summary 266 9.5 Problems 266 10 Initial Boundary-Value Problems 273 10.1 Mass Lumping 276 10.2 Chapter Summary 276 10.3 Problems 276 11 Boundary-Value Problems in Three Space 279 11.1 Finite Dierence Approximations 279 11.2 Finite Element Approximations 280 11.3 Chapter Summary 285 12 Nomenclature 289 Index 293
• Edition: 1st
• Publisher Details: Hoboken, New Jersey : John Wiley & Sons, Inc
• Publication Date: 2018
• Extent: 1 online resource
• Subjects: 511.0245; Engineering mathematics; Science -- Mathematics; Numerical analysis
• Languages: English
• ISBNs: 9781119316381; 9781119316367
• Related ISBNs: 9781119316114
• Notes: Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.