## An Introduction to Scientific Computing: Twelve Computational Projects Solved with MATLABTeaching 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 |

285 | |

289 | |

293 | |

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### Common terms and phrases

approximation B´ezier curve basis boundary conditions coeﬃcients components compute constant control points convergence corresponding curve Bm Daubechies wavelet deﬁned deﬁnition Diﬀerential Equations diﬀusion Dirichlet Dirichlet boundary condition discretization displayed in Fig element method error Exact sol exact solution example exercise is proposed expansion ﬁgure ﬁle ﬁnd finite ﬁnite diﬀerence ﬁrst ﬁxed ﬂow ﬂux formula function f global grid Haar wavelet heat equation Implementation initial condition input integration interpolation interval iterations Kelvin–Helmholtz instability Lagrange Lagrange polynomials Laplacian Legendre linear system MATLAB matrix mesh multiresolution analysis nodes nonlinear numerical scheme numerical solution obtained parameters Partial Diﬀerential PDEs piecewise points xi polynomial polynomial interpolant procedure proposed in Sect quadrature reconstruction algorithm right-hand side right-hand-side function satisﬁed script shock tube Signal Reconstructed Solution of Exercise solver spectral method stability step subdomains temperature ﬁeld tion tridiagonal values variables vector velocity wavelet Write a program

### References to this book

Numerical Analysis and Optimization: An Introduction to Mathematical ... Grégoire Allaire No preview available - 2007 |