An Introduction to Scientific Computing

Twelve Computational Projects Solved with MATLAB

By: Ionut Danaila, Pascal Joly, Sidi Mahmoud Kaber

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Hardcover | 27 November 2006

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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 imageprocessing, 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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From the reviews:

"In An Introduction to Scientific Computing, the authors present approaches to the numerical solution of problems drawn from a variety of applications. ... This is a graduate-level introduction and the pace is brisk. ... This is a strong text on scientific computing for advanced students in applied mathematics. ... the book is most appropriate for students with some prior experience in scientific computing ... ." (William J. Satzer, MathDL, February, 2007)

"The book is based on material offered by the authors at Universite Pierre et Marie Curie (Paris, France) and different engineering schools. It is intended as a graduate-level text in applied mathematics, but it may also be used by students in engineering or physical sciences. It may also be used as a reference for researchers and practicing engineers. Since different possible levels of each project are suggested, the text can be used to propose assignments at different graduate levels." (I. N. Katz, Zentralblatt MATH, Vol. 1119 (21), 2007)

"An Introduction to Scientific Computing plunges into solving PDEs by numerical approximation. ... the book is an attempt to completely discuss numerical issues for reasonably complex problems at the level of a graduate textbook. A project-based approach is used. ... Overall, this is a pleasing and useful companion to more complete expositions of the topic. ... If you're preparing advanced students for a workshop, or organizing a numerical analysis club for the semester, then the book is perfect." (Sorin Mitran, SIAM Review, Vol. 50 (1), 2008)

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Non-Fiction Science Science in General Mathematics Applied Mathematics Engineering & Technology Electronics & Communications Engineering Electronics Engineering Technology in General Engineering in General Physics Mathematical Physics

Calculus & Mathematical Analysis Numerical Analysis

Numerical Approximation of Model Partial Differential Equations	p. 1
Discrete Integration Methods for Ordinary Differential Equations	p. 1
Construction of Numerical Integration Schemes	p. 2
General Form of Numerical Schemes	p. 6
Application to the Absorption Equation	p. 8
Stability of a Numerical Scheme	p. 9
Model Partial Differential Equations	p. 11
The Convection Equation	p. 11
The Wave Equation	p. 14
The Heat Equation	p. 17
Solutions and Programs	p. 19
Chapter References	p. 30
Nonlinear Differential Equations: Application to Chemical Kinetics	p. 33
Physical Problem and Mathematical Modeling	p. 33
Stability of the System	p. 34
Model for the Maintained Reaction	p. 36
Existence of a Critical Point and Stability	p. 36
Numerical Solution	p. 37
Model of Reaction with a Delay Term	p. 37
Solutions and Programs	p. 41
Chapter References	p. 48
Polynomial Approximation	p. 49
Introduction	p. 49
Polynomial Interpolation	p. 50
Lagrange Interpolation	p. 51
Hermite Interpolation	p. 57
Best Polynomial Approximation	p. 59
Best Uniform Approximation	p. 59
Best Hilbertian Approximation	p. 61
Discrete Least Squares Approximation	p. 64
Piecewise Polynomial Approximation	p. 65
Piecewise Constant Approximation	p. 66
Piecewise Affine Approximation	p. 67
Piecewise Cubic Approximation	p. 68
Further Reading	p. 69
Solutions and Programs	p. 70
Chapter References	p. 83
Solving an Advection-Diffusion Equation by a Finite Element Method	p. 85
Variational Formulation of the Problem	p. 85
A P1 Finite Element Method	p. 87
A P2 Finite Element Method	p. 90
A Stabilization Method	p. 93
Computation of the Solution at the Endpoints of the Intervals	p. 93
Analysis of the Stabilized Method	p. 95
The Case of a Variable Source Term	p. 97
Solutions and Programs	p. 97
Chapter References	p. 108
Solving a Differential Equation by a Spectral Method	p. 111
Some Properties of the Legendre Polynomials	p. 112
Gauss-Legendre Quadrature	p. 113
Legendre Expansions	p. 115
A Spectral Discretization	p. 117
Possible Extensions	p. 119
Solutions and Programs	p. 120
Chapter References	p. 125
Signal Processing: Multiresolution Analysis	p. 127
Introduction	p. 127
Approximation of a Function: Theoretical Aspect	p. 127
Piecewise Constant Functions	p. 127
Decomposition of the Space V[subscript J]	p. 129
Decomposition and Reconstruction Algorithms	p. 132
Importance of Multiresolution Analysis	p. 133
Multiresolution Analysis: Practical Aspect	p. 134
Multiresolution Analysis: Implementation	p. 135
Introduction to Wavelet Theory	p. 137
Scaling Functions and Wavelets	p. 137
The Schauder Wavelet	p. 139
Implementation of the Schauder Wavelet	p. 141
The Daubechies Wavelet	p. 142
Implementation of the Daubechies Wavelet D4	p. 144
Generalization: Image Processing	p. 146
Image Processing: Implementation	p. 147
Solutions and Programs	p. 148
Chapter References	p. 150
Elasticity: Elastic Deformation of a Thin Plate	p. 151
Introduction	p. 151
Modeling Elastic Deformations (Linear Problem)	p. 152
Modeling Electrostatic Forces (Nonlinear Problem)	p. 153
Numerical Discretization of the Problem	p. 154
Programming Tips	p. 157
Modular Programming	p. 157
Program Validation	p. 158
Solving the Linear Problem	p. 159
Solving the Nonlinear Problem	p. 159
A Fixed-Point Algorithm	p. 159
Numerical Solution	p. 160
Solutions and Programs	p. 162
Further Comments	p. 162
Chapter References	p. 164
Domain Decomposition Using a Schwarz Method	p. 165
Principle and Application Field of Domain Decomposition	p. 165
One-Dimensional Finite Difference Solution	p. 166
Schwarz Method in One Dimension	p. 167
Discretization	p. 168
Extension to the Two-Dimensional Case	p. 171
Finite Difference Solution	p. 171
Domain Decomposition in the Two-Dimensional Case	p. 175
Implementation of Realistic Boundary Conditions	p. 178
Possible Extensions	p. 180
Solutions and Programs	p. 181
Chapter References	p. 190
Geometrical Design: Bezier Curves and Surfaces	p. 193
Introduction	p. 193
Bezier Curves	p. 193
Basic Properties of Bezier Curves	p. 195
Convex Hull of the Control Points	p. 195
Multiple Control Points	p. 196
Tangent Vector to a Bezier Curve	p. 197
Junction of Bezier Curves	p. 197
Generation of the Point P(t)	p. 198
Generation of Bezier Curves	p. 200
Splitting Bezier Curves	p. 201
Intersection of Bezier Curves	p. 203
Implementation	p. 205
Bezier Surfaces	p. 206
Basic properties of Bezier Surfaces	p. 206
Convex Hull	p. 206
Tangent Vector	p. 207
Junction of Bezier Patches	p. 207
Construction of the Point P(t)	p. 208
Construction of Bezier Surfaces	p. 209
Solutions and Programs	p. 210
Chapter References	p. 212
Gas Dynamics: The Riemann Problem and Discontinuous Solutions: Application to the Shock Tube Problem	p. 213
Physical Description of the Shock Tube Problem	p. 213
Euler Equations of Gas Dynamics	p. 215
Dimensionless Equations	p. 218
Exact Solution	p. 218
Numerical Solution	p. 222
Lax-Wendroff and MacCormack Centered Schemes	p. 222
Upwind Schemes (Roe's Approximate Solver)	p. 227
Solutions and Programs	p. 232
Chapter References	p. 233
Thermal Engineering: Optimization of an Industrial Furnace	p. 235
Introduction	p. 235
Formulation of the Problem	p. 236
Finite Element Discretization	p. 237
Implementation	p. 239
Boundary Conditions	p. 241
Modular Implementation	p. 242
Numerical Solution of the Problem	p. 242
Inverse Problem Formulation	p. 244
Implementation of the Inverse Problem	p. 245
Solutions and Programs	p. 248
Further Comments	p. 249
Chapter References	p. 250
Fluid Dynamics: Solving the Two-Dimensional Navier-Stokes Equations	p. 251
Introduction	p. 251
The Incompressible Navier-Stokes Equations	p. 252
Numerical Algorithm	p. 253
Computational Domain, Staggered Grids, and Boundary Conditions	p. 255
Finite Difference Discretization	p. 256
Flow Visualization	p. 264
Initial Condition	p. 265
Step-by-Step Implementation	p. 268
Solving a Linear System with Tridiagonal, Periodic Matrix	p. 268
Solving the Unsteady Heat Equation	p. 271
Solving the Steady Heat Equation Using FFTs	p. 275
Solving the 2D Navier-Stokes Equations	p. 275
Solutions and Programs	p. 277
Chapter References	p. 284
Bibliography	p. 285
Index	p. 289
Index of Programs	p. 293
Table of Contents provided by Ingram. All Rights Reserved.

An Introduction to Scientific Computing

Twelve Computational Projects Solved with MATLAB

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