# Details

## Numerical Methods for Solving Partial Differential Equations

A Comprehensive Introduction for Scientists and Engineers
1. Aufl.
 von: George F. Pinder 96,99 € Verlag: Wiley Format: EPUB Veröffentl.: 05.02.2018 ISBN/EAN: 9781119316381 Sprache: englisch Anzahl Seiten: 320

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## Beschreibungen

#### Inhaltsverzeichnis

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

#### Autorenportrait

George F. Pinder, PhD, is a Distinguished Professor of Civil and Environmental Engineering with a secondary appointments in Mathematics and Statistics and Computer Science at the University of Vermont, Burlington, Vermont. He is the author or co-author of ten books in numerical mathematics and engineering. Dr. Pinder is the recipient of numerous national and international honors and is a member of the National Academy of Engineering.