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Numerical Methods for Solving Partial Differential Equations: A Comprehensive Introduction for Scientists and Engineers

Numerical Methods for Solving Partial Differential Equations: A Comprehensive Introduction for Scientists and Engineers

George F. Pinder

ISBN: 978-1-119-31638-1

Feb 2018

320 pages

$100.99

Description

A comprehensive guide to numerical methods for simulating physical-chemical systems 

This book offers a systematic, highly accessible presentation of numerical methods used to simulate the behavior of physical-chemical systems. Unlike most books on the subject, it focuses on methodology rather than specific applications. Written for students and professionals across an array of scientific and engineering disciplines and with varying levels of experience with applied mathematics, it provides comprehensive descriptions of numerical methods without requiring an advanced mathematical background.

Based on its author’s more than forty years of experience teaching numerical methods to engineering students, Numerical Methods for Solving Partial Differential Equations presents the fundamentals of all of the commonly used numerical methods for solving differential equations at a level appropriate for advanced undergraduates and first-year graduate students in science and engineering. Throughout, elementary examples show how numerical methods are used to solve generic versions of equations that arise in many scientific and engineering disciplines. In writing it, the author took pains to ensure that no assumptions were made about the background discipline of the reader.

  • Covers the spectrum of numerical methods that are used to simulate the behavior of physical-chemical systems that occur in science and engineering
  • Written by a professor of engineering with more than forty years of experience teaching numerical methods to engineers
  • Requires only elementary knowledge of differential equations and matrix algebra to master the material
  • Designed to teach students to understand, appreciate and apply the basic mathematics and equations on which Mathcad and similar commercial software packages are based

Comprehensive yet accessible to readers with limited mathematical knowledge, Numerical Methods for Solving Partial Differential Equations is an excellent text for advanced undergraduates and first-year graduate students in the sciences and engineering. It is also a valuable working reference for professionals in engineering, physics, chemistry, computer science, and applied mathematics.

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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