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An introduction to scientific computing : twelve computational projects solved with MATLAB

Author: Ionut Danaila
Publisher: New York : Springer, ©2007.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

This book demonstrates scientific computing by presenting twelve computational projects spanning a broad spectrum of disciplines including Fluid Mechanics, Chemistry, Elasticity, Thermal Science,  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Ionut Danaila
ISBN: 038730889X 9780387308890
OCLC Number: 71336121
Description: xv, 294 pages : illustrations ; 24 cm
Contents: 1 Numerical Approximation of Model Partial Differential Equations 1 --
1.1 Discrete Integration Methods for Ordinary Differential Equations 1 --
1.1.1 Construction of Numerical Integration Schemes 2 --
1.1.2 General Form of Numerical Schemes 6 --
1.1.3 Application to the Absorption Equation 8 --
1.1.4 Stability of a Numerical Scheme 9 --
1.2 Model Partial Differential Equations 11 --
1.2.1 The Convection Equation 11 --
1.2.2 The Wave Equation 14 --
1.2.3 The Heat Equation 17 --
1.3 Solutions and Programs 19 --
2 Nonlinear Differential Equations: Application to Chemical Kinetics 33 --
2.1 Physical Problem and Mathematical Modeling 33 --
2.2 Stability of the System 34 --
2.3 Model for the Maintained Reaction 36 --
2.3.1 Existence of a Critical Point and Stability 36 --
2.3.2 Numerical Solution 37 --
2.4 Model of Reaction with a Delay Term 37 --
2.5 Solutions and Programs 41 --
3 Polynomial Approximation 49 --
3.2 Polynomial Interpolation 50 --
3.2.1 Lagrange Interpolation 51 --
3.2.2 Hermite Interpolation 57 --
3.3 Best Polynomial Approximation 59 --
3.3.1 Best Uniform Approximation 59 --
3.3.2 Best Hilbertian Approximation 61 --
3.3.3 Discrete Least Squares Approximation 64 --
3.4 Piecewise Polynomial Approximation 65 --
3.4.1 Piecewise Constant Approximation 66 --
3.4.2 Piecewise Affine Approximation 67 --
3.4.3 Piecewise Cubic Approximation 68 --
3.6 Solutions and Programs 70 --
4 Solving an Advection-Diffusion Equation by a Finite Element Method 85 --
4.1 Variational Formulation of the Problem 85 --
4.2 A P1 Finite Element Method 87 --
4.3 A P2 Finite Element Method 90 --
4.4 A Stabilization Method 93 --
4.4.1 Computation of the Solution at the Endpoints of the Intervals 93 --
4.4.2 Analysis of the Stabilized Method 95 --
4.5 The Case of a Variable Source Term 97 --
4.6 Solutions and Programs 97 --
5 Solving a Differential Equation by a Spectral Method 111 --
5.1 Some Properties of the Legendre Polynomials 112 --
5.2 Gauss-Legendre Quadrature 113 --
5.3 Legendre Expansions 115 --
5.4 A Spectral Discretization 117 --
5.5 Possible Extensions 119 --
5.6 Solutions and Programs 120 --
6 Signal Processing: Multiresolution Analysis 127 --
6.2 Approximation of a Function: Theoretical Aspect 127 --
6.2.1 Piecewise Constant Functions 127 --
6.2.2 Decomposition of the Space V[subscript J] 129 --
6.2.3 Decomposition and Reconstruction Algorithms 132 --
6.2.4 Importance of Multiresolution Analysis 133 --
6.3 Multiresolution Analysis: Practical Aspect 134 --
6.4 Multiresolution Analysis: Implementation 135 --
6.5.1 Scaling Functions and Wavelets 137 --
6.5.2 The Schauder Wavelet 139 --
6.5.3 Implementation of the Schauder Wavelet 141 --
6.5.4 The Daubechies Wavelet 142 --
6.5.5 Implementation of the Daubechies Wavelet D4 144 --
6.6 Generalization: Image Processing 146 --
6.6.1 Image Processing: Implementation 147 --
6.7 Solutions and Programs 148 --
7 Elasticity: Elastic Deformation of a Thin Plate 151 --
7.2 Modeling Elastic Deformations (Linear Problem) 152 --
7.3 Modeling Electrostatic Forces (Nonlinear Problem) 153 --
7.4 Numerical Discretization of the Problem 154 --
7.5 Programming Tips 157 --
7.5.1 Modular Programming 157 --
7.5.2 Program Validation 158 --
7.6 Solving the Linear Problem 159 --
7.7 Solving the Nonlinear Problem 159 --
7.7.1 A Fixed-Point Algorithm 159 --
7.7.2 Numerical Solution 160 --
7.8 Solutions and Programs 162 --
8 Domain Decomposition Using a Schwarz Method 165 --
8.1 Principle and Application Field of Domain Decomposition 165 --
8.2 One-Dimensional Finite Difference Solution 166 --
8.3 Schwarz Method in One Dimension 167 --
8.3.1 Discretization 168 --
8.4 Extension to the Two-Dimensional Case 171 --
8.4.1 Finite Difference Solution 171 --
8.4.2 Domain Decomposition in the Two-Dimensional Case 175 --
8.4.3 Implementation of Realistic Boundary Conditions 178 --
8.4.4 Possible Extensions 180 --
8.5 Solutions and Programs 181 --
9 Geometrical Design: Bezier Curves and Surfaces 193 --
9.2 Bezier Curves 193 --
9.3 Basic Properties of Bezier Curves 195 --
9.3.1 Convex Hull of the Control Points 195 --
9.3.2 Multiple Control Points 196 --
9.3.3 Tangent Vector to a Bezier Curve 197 --
9.3.4 Junction of Bezier Curves 197 --
9.3.5 Generation of the Point P(t) 198 --
9.4 Generation of Bezier Curves 200 --
9.5 Splitting Bezier Curves 201 --
9.6 Intersection of Bezier Curves 203 --
9.6.1 Implementation 205 --
9.7 Bezier Surfaces 206 --
9.8 Basic properties of Bezier Surfaces 206 --
9.8.1 Convex Hull 206 --
9.8.2 Tangent Vector 207 --
9.8.3 Junction of Bezier Patches 207 --
9.8.4 Construction of the Point P(t) 208 --
9.9 Construction of Bezier Surfaces 209 --
9.10 Solutions and Programs 210 --
10 Gas Dynamics: The Riemann Problem and Discontinuous Solutions: Application to the Shock Tube Problem 213 --
10.1 Physical Description of the Shock Tube Problem 213 --
10.2 Euler Equations of Gas Dynamics 215 --
10.2.1 Dimensionless Equations 218 --
10.2.2 Exact Solution 218 --
10.3 Numerical Solution 222 --
10.3.1 Lax-Wendroff and MacCormack Centered Schemes 222 --
10.3.2 Upwind Schemes (Roe's Approximate Solver) 227 --
10.4 Solutions and Programs 232 --
11 Thermal Engineering: Optimization of an Industrial Furnace 235 --
11.2 Formulation of the Problem 236 --
11.3 Finite Element Discretization 237 --
11.4 Implementation 239 --
11.5 Boundary Conditions 241 --
11.5.1 Modular Implementation 242 --
11.5.2 Numerical Solution of the Problem 242 --
11.6 Inverse Problem Formulation 244 --
11.7 Implementation of the Inverse Problem 245 --
11.8 Solutions and Programs 248 --
12 Fluid Dynamics: Solving the Two-Dimensional Navier-Stokes Equations 251 --
12.2 The Incompressible Navier-Stokes Equations 252 --
12.3 Numerical Algorithm 253 --
12.4 Computational Domain, Staggered Grids, and Boundary Conditions 255 --
12.5 Finite Difference Discretization 256 --
12.6 Flow Visualization 264 --
12.7 Initial Condition 265 --
12.8 Step-by-Step Implementation 268 --
12.8.1 Solving a Linear System with Tridiagonal, Periodic Matrix 268 --
12.8.2 Solving the Unsteady Heat Equation 271 --
12.8.3 Solving the Steady Heat Equation Using FFTs 275 --
12.8.4 Solving the 2D Navier-Stokes Equations 275 --
12.9 Solutions and Programs 277.
Responsibility: Ionut Danaila [and others].
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