An Introduction to Scientific Computing: Twelve Computational Projects Solved with MATLAB

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Springer Science & Business Media, Nov 27, 2006 - Mathematics - 294 pages
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Teaching or learning numerical methods in applied mathematics cannot be conceived nowadays without numerical experimentation on computers. There is a vast literature devoted either to theoretical numerical methods or - merical programming of basic algorithms, but there are few texts o?ering a complete discussion of numerical issues involved in the solution of concrete and relatively complex problems. This book is an attempt to ?ll this need. It is our belief that advantages and drawbacks of a numerical method cannot be accounted for without one’s experiencing all the steps of scienti?c comp- ing, from physical and mathematical description of the problem to numerical formulation and programming and, ?nally, to critical discussion of numerical results. The book provides twelve computational projects aimed at numerically solving problems selected to cover a broad spectrum of applications, from ?uid mechanics, chemistry, elasticity, thermal science, computer-aided design, signal and image processing, etc. Even though the main volume of this text concerns the numerical analysis of computational methods and their imp- mentation, we have tried to start, when possible, from realistic problems of practical interest for researchers and engineers. For each project, an introductory record card summarizes the mathem- ical and numerical topics explained and the ?elds of application of the - proach. A level of di?culty, scaling from 1 to 3, is assigned to each project.
 

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Contents

Numerical Approximation of Model Partial Differential Equations
1
111 Construction of Numerical Integration Schemes
2
112 General Form of Numerical Schemes
6
113 Application to the Absorption Equation
8
114 Stability of a Numerical Scheme
9
12 Model Partial Differential Equations
11
122 The Wave Equation
14
123 The Heat Equation
17
752 Program Validation
158
76 Solving the Linear Problem
159
772 Numerical Solution
160
78 Solutions and Programs
162
Chapter References
164
Domain Decomposition Using a Schwarz Method
165
82 OneDimensional Finite Difference Solution
166
83 Schwarz Method in One Dimension
167

13 Solutions and Programs
19
Chapter References
30
Nonlinear Differential Equations Application to Chemical Kinetics
33
22 Stability of the System
34
23 Model for the Maintained Reaction
36
232 Numerical Solution
37
25 Solutions and Programs
41
Chapter References
48
Polynomial Approximation
49
32 Polynomial Interpolation
50
321 Lagrange Interpolation
51
322 Hermite Interpolation
57
33 Best Polynomial Approximation
59
332 Best Hilbertian Approximation
61
333 Discrete Least Squares Approximation
64
34 Piecewise Polynomial Approximation
65
341 Piecewise Constant Approximation
66
342 Piecewise Affine Approximation
67
343 Piecewise Cubic Approximation
68
35 Further Reading
69
36 Solutions and Programs
70
Chapter References
83
Solving an AdvectionDiffusion Equation by a Finite Element Method
84
42 A P1 Finite Element Method
87
43 A P2 Finite Element Method
90
44 A Stabilization Method
93
442 Analysis of the Stabilized Method
95
45 The Case of a Variable Source Term
97
Chapter References
108
Solving a Differential Equation by a Spectral Method
111
51 Some Properties of the Legendre Polynomials
112
52 GaussLegendre Quadrature
113
53 Legendre Expansions
115
54 A Spectral Discretization
117
55 Possible Extensions
119
56 Solutions and Programs
120
Chapter References
125
Signal Processing Multiresolution Analysis
126
622 Decomposition of the Space VJ
129
623 Decomposition and Reconstruction Algorithms
132
624 Importance of Multiresolution Analysis
133
Practical Aspect
134
Implementation
135
65 Introduction to Wavelet Theory
137
652 The Schauder Wavelet
139
653 Implementation of the Schauder Wavelet
141
654 The Daubechies Wavelet
142
655 Implementation of the Daubechies Wavelet D4
144
Image Processing
146
Implementation
147
67 Solutions and Programs
148
Chapter References
150
Elasticity Elastic Deformation of a Thin Plate
151
72 Modeling Elastic Deformations Linear Problem
152
73 Modeling Electrostatic Forces Nonlinear Problem
153
74 Numerical Discretization of the Problem
154
75 Programming Tips
157
831 Discretization
168
84 Extension to the TwoDimensional Case
171
842 Domain Decomposition in the TwoDimensional Case
175
843 Implementation of Realistic Boundary Conditions
178
844 Possible Extensions
180
85 Solutions and Programs
181
Chapter References
190
Geometrical Design Bézier Curves and Surfaces
193
93 Basic Properties of Bézier Curves
195
932 Multiple Control Points
196
933 Tangent Vector to a Bezier Curve
197
935 Generation of the Point Pt
198
94 Generation of Bézier Curves
200
95 Splitting Bézier Curves
201
96 Intersection of Bézier Curves
203
961 Implementation
205
97 Bézier Surfaces
206
982 Tangent Vector
207
984 Construction of the Point Pt
208
99 Construction of Bezier Surfaces
209
910 Solutions and Programs Solution of Exercise 91
210
Chapter References
212
Gas Dynamics The Riemann Problem and Discontinuous Solutions Application to the Shock Tube Problem
213
102 Euler Equations of Gas Dynamics
215
1021 Dimensionless Equations
218
103 Numerical Solution
222
1032 Upwind Schemes Roes Approximate Solver
227
104 Solutions and Programs
232
Chapter References
233
Thermal Engineering Optimization of an Industrial Furnace
234
112 Formulation of the Problem
236
113 Finite Element Discretization
237
114 Implementation
239
115 Boundary Conditions
241
1151 Modular Implementation
242
116 Inverse Problem Formulation
244
117 Implementation of the Inverse Problem
245
118 Solutions and Programs Solution of Exercise 111
248
1181 Further Comments
249
Chapter References
250
Fluid Dynamics Solving the TwoDimensional NavierStokes Equations
251
122 The Incompressible NavierStokes Equations
252
123 Numerical Algorithm
253
124 Computational Domain Staggered Grids and Boundary Conditions
255
125 Finite Difference Discretization
256
126 Flow Visualization
264
127 Initial Condition
265
128 StepbyStep Implementation
268
1282 Solving the Unsteady Heat Equation
271
1283 Solving the Steady Heat Equation Using FFTs
275
129 Solutions and Programs
277
Chapter References
284
Bibliography
285
Index
289
Index of Programs
293
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