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

309 Pages·2018·5.49 MB·English

by  George F. Pinder

Numerical Methods for Solving
Partial Differential Equations

Numerical Methods for Solving
Partial Differential Equations
A Comprehensive Introduction
for Scientists and Engineers
George F. Pinder
This edition first published 2018 by John Wiley and Sons, Inc.
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Contents
Preface xi
1 Interpolation 1
1.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 De nitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Weirstraus Approximation Theorem . . . . . . . . . . . . . . . . . . . . 3
1.5 Lagrange Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Compare P ((cid:18)) and f^((cid:18)) . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2
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 Di erentiation 33
2.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Two-Point Di erence Formulae . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.1 Forward Di erence Formula . . . . . . . . . . . . . . . . . . . . . 35
2.2.2 Backward Di erence 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 Di erence Formulae . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 First-Order Derivative Di erence 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
vii
viii CONTENTS
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 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.1 Example of Stability . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Lax Equivalence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6 Runge Kutta Type Formulae . . . . . . . . . . . . . . . . . . . . . . . . 75
(cid:0)
4.6.1 General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.6.2 Runge Kutta First-Order Form (Euler’s Method) . . . . . . . . 75
(cid:0)
4.6.3 Runge Kutta Second-Order Form . . . . . . . . . . . . . . . . . 75
(cid:0)
4.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Weighted Residuals Methods 83
5.1 Finite Volume or Subdomain Method . . . . . . . . . . . . . . . . . . . . 84
5.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.2 Finite Di erence Interpretation of the Finite Volume Method . . 93
5.2 Galerkin Method for First Order Equations . . . . . . . . . . . . . . . . 94
5.2.1 Finite-Di erence Interpretation of the Galerkin
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 Galerkin Method for Second-Order Equations . . . . . . . . . . . . . . . 102
5.3.1 Finite Di erence Interpretation ofSecond-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 Di erence Representation 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 Approximations . . . . . . . . . . . . . . . 139
6.2.1 Example of a Two Dimensional Polynomial
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.3 Finite Di erence Approximation . . . . . . . . . . . . . . . . . . . . . . 141
6.3.1 Example of Implicit First-Order Accurate Finite Di erence
Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.3.2 Example of Second Order Accurate Finite Di erence
Approximation in Space . . . . . . . . . . . . . . . . . . . . . . . 146
6.4 Stability of Finite Di erence Approximations . . . . . . . . . . . . . . . 149
6.4.1 Example of Stability . . . . . . . . . . . . . . . . . . . . . . . . . 153
CONTENTS ix
6.4.2 Example Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.5 Galerkin Finite Element Approximations in Time . . . . . . . . . . . . . 158
6.5.1 Strategy One: Forward Di erence Approximation . . . . . . . . . 160
6.5.2 Strategy Two: Backward Di erence Approximation . . . . . . . 161
6.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7 Finite Di erence Methods in Two Space 169
7.1 Example Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.2 Chapter 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 Functions . . . . . . . . . . . . . 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 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . 246
9.2.2 Weighted Residual Formulation . . . . . . . . . . . . . . . . . . . 246
9.2.3 Element Coe cient 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 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . 262
9.3.2 Weighted Residual Formulation . . . . . . . . . . . . . . . . . . . 262
9.3.3 Element Coe cient 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