كتاب An Introduction to Scientific Computing - Twelve Computational Projects Solved with MATLAB
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 كتاب An Introduction to Scientific Computing - Twelve Computational Projects Solved with MATLAB

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An Introduction to Scientific Computing - Twelve Computational Projects Solved with MATLAB
Ionut Danaila
Pascal Joly
Sidi Mahmoud Kaber
Marie Postel  

كتاب An Introduction to Scientific Computing - Twelve Computational Projects Solved with MATLAB  M_a_i_14
و المحتوى كما يلي :


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
Chapter References . 30
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
Chapter References . 48
3 Polynomial Approximation 49
3.1 Introduction 49
3.2 Polynomial Interpolation . 50
3.2.1 Lagrange Interpolation . 51
3.2.2 Hermite Interpolation 57XII Contents
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
3.5 Further Reading . 69
3.6 Solutions and Programs 70
Chapter References . 83
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
Chapter References . 108
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
Chapter References . 125
6 Signal Processing: Multiresolution Analysis 127
6.1 Introduction 127
6.2 Approximation of a Function: Theoretical Aspect 127
6.2.1 Piecewise Constant Functions . 127
6.2.2 Decomposition of the Space VJ 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 Introduction to Wavelet Theory . 137
6.5.1 Scaling Functions and Wavelets . 137Contents XIII
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
Chapter References . 150
7 Elasticity: Elastic Deformation of a Thin Plate 151
7.1 Introduction 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
7.8.1 Further Comments 162
Chapter References . 164
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
Chapter References . 190
9 Geometrical Design: B´ezier Curves and Surfaces . 193
9.1 Introduction 193
9.2 B´ezier Curves . 193
9.3 Basic Properties of B´ezier 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 B´ezier Curve . 197XIV Contents
9.3.4 Junction of B´ezier Curves 197
9.3.5 Generation of the Point P(t) 198
9.4 Generation of B´ezier Curves 200
9.5 Splitting B´ezier Curves . 201
9.6 Intersection of B´ezier Curves 203
9.6.1 Implementation . 205
9.7 B´ezier Surfaces 206
9.8 Basic properties of B´ezier Surfaces . 206
9.8.1 Convex Hull 206
9.8.2 Tangent Vector . 207
9.8.3 Junction of B´ezier Patches 207
9.8.4 Construction of the Point P(t) 208
9.9 Construction of B´ezier Surfaces . 209
9.10 Solutions and Programs 210
Chapter References . 212
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
Chapter References . 233
11 Thermal Engineering: Optimization of an Industrial
Furnace 235
11.1 Introduction 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
11.8.1 Further Comments 249
Chapter References . 250Contents XV
12 Fluid Dynamics: Solving the Two-Dimensional
Navier–Stokes Equations 251
12.1 Introduction 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
Chapter References . 284
Bibliography . 285
Index 289
Index
absorption equation, 8
Adams–Bashforth scheme, 7, 252,
253, 273
Adams–Moulton scheme, 7
ADI method, 259, 260, 273
algorithm
de Casteljau, 200
divided differences, 54
Remez, 60
Thomas, 252, 268
approximation
best
Hilbertian, 62
uniform, 60
piecewise
affine, 67
constant, 66
cubic, 68
basis
canonical, 52
Haar, 131
Hilbertian, 62
Lagrange, 52
Legendre, 112
Bernstein polynomial, 194
best Hilbertian approximation, 62
best uniform approximation, 60
bilaplacian, 155
boundary condition, 120, 241
Dirichlet, 14, 17, 171, 236
Fourier, 179, 183, 236
homogeneous, 14
inhomogeneous, 171
Neumann, 179, 183, 236
periodic, 16, 26, 251, 257
boundary layer, 98
boundary value problem, 85
Brusselator, 33
B´ezier
curve, 193
patch, 206
surface, 206
CAGD, 193
CFL condition, 13, 15, 23, 224, 264,
272
characteristic curve, 12, 15, 22, 217,
219
characteristic equation, 38
Chebyshev
expansion, 79
points, 55
polynomial, 55
compressible fluid, 215
condensation, 95
condition number, 52
consistency, 9
contact discontinuity, 214, 220
convection
equation, 11, 21, 216, 217, 224
phenomenon, 30
convection-diffusion equation, 265
convergence, 9, 57, 187
fast, 116
slow, 116
Crank–Nicolson scheme, 7, 252, 253,
273
critical point, 34, 37
data compression, 133
Daubechies wavelet, 142
de Casteljau algorithm, 200
delayed differential equation, 37
density, 215
differences (divided –), 54
differential equation, 33, 111, 165
diffusion, 17, 18, 226, 254
numerical, 27
phenomenon, 29
diffusivity (thermal –), 29
Dirichlet boundary condition, 14,
17, 153, 171, 236
discontinuity, 116290 Index
contact, 214, 220
dissipation, 25
artificial, 226
divergence, 45, 252
divided differences, 53, 54
domain decomposition, 166
domain of dependence, 15
elasticity, 152
energy (total –), 215
enthalpy, 215
equioscillatory, 59
erf (error function), 18
Euler
explicit modified scheme, 6, 7
explicit scheme, 5, 7, 20, 39,
46, 272
implicit scheme, 5, 7
system of equations, 215
expansion
Chebyshev, 79
fan, 220
Fourier, 16
Legendre, 62
wave, 214
extrapolation, 49
FEM, 87, 237
FFT, 70, 252, 262
finite difference, 125, 155, 167, 171
backward, 3, 224, 226
centered, 3, 4, 224, 257
forward, 3, 224, 226
finite element, 86, 87, 90, 237
P1, 87
P2, 90
fluid
compressible, 215
incompressible, 251
flux, 215, 229
splitting, 228
formulation (variational –), 86
Fourier
boundary condition, 179, 183,
236
expansion, 16
FFT, 70, 252, 262
series, 261
Galerkin approximation, 117
Gauss quadrature, 113
Gibbs phenomenon, 116
Godunov scheme, 228
gradient, 252
Green formula, 237
grid (computational –), 172, 182,
256
Haar
basis, 131
wavelet, 137
heat
coefficient, 215, 217
equation, 17, 29, 226, 271, 275
steady equation, 171
Helmholtz equation, 252, 255, 258
Hermite, 119
interpolation, 57
polynomial, 58
Heun scheme, 7
Hopf bifurcation, 42
hyperbolic system, 216
ill-conditioned, 56
incompressible fluid, 251, 252
interpolation, 49, 51
Hermite, 57, 58
Lagrange, 51, 57
stability, 56
inverse problem, 244
isentropic flow, 217
isocontours, 274
Jacobian matrix, 35, 38, 216
jet flow, 266
junction
of curves, 197
of patches, 207
Kelvin–Helmholtz instability, 252,
265Index 291
Lagrange
basis, 52
interpolation, 51
polynomial, 51, 52
Laplacian, 152, 155, 166, 172, 252
Lax–Wendroff scheme, 223
leapfrog scheme, 5, 7
Lebesgue constant, 56
Legendre
basis, 112
coefficients, 63
expansion, 62, 115
polynomials, 62, 111
series, 62
MacCormack scheme, 223
Mach number, 215
Mallat transform, 141
matrix
exponential, 35
inverse, 233
Jacobian, 216
tridiagonal, 268
tridiagonal periodic, 260, 268,
269, 277
Vandermonde, 64, 70
mesh, 238
mother wavelet, 138
multiresolution analysis, 133
multiscale analysis, 133
Navier–Stokes
equations, 252
fractional-step method, 253
projection method, 253
Neumann boundary condition, 153,
179, 183, 236
normal equations, 65
numerical integration, 113
ODE, 1, 33, 111
orthogonal projection, 63
overlap, 166
P1 finite element, 87
P2 finite element, 90
parametric curve, 36, 43
PDE, 1, 111, 165
Peclet number, 90, 265
periodic
boundary condition, 16, 26, 251,
257
trajectory, 42, 46
phenomenon
Gibbs, 116
Runge, 57
Poisson equation, 252, 255, 261
polygon (control –), 196
polynomial
Bernstein, 194
Hermite, 119
Lagrange, 51, 52
Legendre, 62, 111
of best Hilbertian approximation, 62
of best uniform approximation,
60
projection (orthogonal), 63
quadrature, 50, 86
Gauss, 113
rule, 113
Simpson, 87
trapezoidal, 86
Rankine–Hugoniot, 219
rarefaction wave, 214
regression line, 64
Remez algorithm, 60
Reynolds number, 253
Riemann problem, 217
Roe
approximate solver, 229
average, 230
Runge–Kutta scheme, 7, 20, 37, 40
Runge phenomenon, 57
scaling function, 137
Schauder wavelet, 139, 141
scheme
13-point, 156292 Index
5-point, 155, 172
Adams–Bashforth, 7, 253, 273
Adams–Moulton, 7
centered, 19, 222
conservative, 230
Crank–Nicolson, 7, 253, 273
Euler explicit , 7
Euler implicit , 7
explicit, 5, 6, 224, 272
Godunov, 228
Heun, 7
implicit, 5, 6, 273
Lax–Wendroff, 223
leapfrog, 7
MacCormack, 223
Roe, 229
Runge–Kutta, 7
upwind, 13, 23, 25, 224, 227
Schwarz method, 166
series
Chebyshev, 79
Legendre, 62
Taylor, 3
shock
tube, 213
wave, 214, 219
Simpson quadrature, 87
smooth function, 116
Sod shock tube, 221
spectral method, 117
spline, 66
stability, 9, 34, 56
amplification function, 10
CFL condition, 13, 15, 23, 224,
264, 272
region, 10, 20, 26
steady solution, 34
stopping criterion, 169
stream-function, 267
string (vibrating –), 16
Taylor
expansion, 3, 35, 172, 224
formula, 35
thermal
diffusivity, 179
shock, 171
Thomas algorithm, 252, 268, 269
tracer (passive –), 265
trajectory, 42
periodic, 42, 46
trapezoidal quadrature, 86
triangulation, 238
tridiagonal matrix, 167, 173
two-scale relation, 128, 139, 142
upwind scheme, 13, 23, 25, 224, 227
Vandermonde matrix, 64, 70
variational formulation, 86, 237
viscosity
artificial, 226
kinematic, 253
vortex, 264, 267
dipole, 252, 267
vorticity, 264
wave
characteristic, 217, 224
elementary, 16
equation, 14
expansion, 214
number, 16, 262
rarefaction, 214
shock, 214
wavelength, 279
wavelet, 137
Daubechies, 142
Haar, 137
Schauder, 139Index of Programs
APP ApproxScript1.m, 70
APP ApproxScript2.m, 71
APP ApproxScript3.m, 72
APP ApproxScript4.m, 73
APP ApproxScript5.m, 73
APP ApproxScript8.m, 77
APP condVanderMonde.m, 71
APP condVanderMondeBis.m, 71
APP dd.m, 72
APP ddHermite.m, 76
APP equiosc.m, 79
APP interpol.m, 73
APP Interpolation.m, 76
APP Lebesgue.m, 73
APP ls.m, 80
APP Remez.m, 79
APP Runge.m, 76
APP scriptHermite.m, 77
APP spline0.m, 81
APP spline1.m, 82
APP spline3.m, 82
CAGD casteljau.m, 210
CAGD cbezier.m, 210
CAGD coox.m, 211
CAGD ex1.m, 210
CAGD ex1b.m, 210
CAGD ex1c.m, 210
CAGD ex2.m, 211
CAGD pbzier.m, 210
CAGD tbezier.m, 210
DDM f1BB.m, 183
DDM f1CT, 183
DDM f1Exact.m, 183
DDM f2CT.m, 183
DDM f2Exact.m, 183
DDM FinDif2dDirichlet.m, 183
DDM FinDif2dFourier.m, 183
DDM FunSchwarz1d.m, 181
DDM g1BB.m, 183
DDM g1CT.m, 183
DDM g1Exact.m, 183
DDM g2Exact.m, 183
DDM LaplaceDirichlet.m, 183
DDM LaplaceFourier.m, 183
DDM Perf.m, 187
DDM rhs1d.m, 181
DDM rhs2dBB.m, 183
DDM rhs2dCT.m, 183
DDM rhs2dExact.m, 183
DDM RightHandSide2dDirichlet.m,
183
DDM RightHandSideFourier.m, 183
DDM Schwarz2dDirichlet.m, 187
DDM Schwarz2dFourier.m, 190
DDM TestFinDif2d.m, 183
DDM TestSchwarz2d.m, 184, 190
ELAS bilap matrix.m, 162
ELAS bilap rhs.m, 162
ELAS lap matrix.m, 162
ELAS lap rhs.m, 162
ELAS plate ex.m, 162
ELAS solution.m, 162
FEM ConvecDiffAP1.m, 101
FEM ConvecDiffAP2.m, 104
FEM ConvecDiffbP1.m, 101
FEM ConvecDiffbP2.m, 104
FEM ConvecDiffscript1.m, 101
FEM ConvecDiffscript2.m, 102
FEM ConvecDiffscript3.m, 102
FEM ConvecDiffscript4.m, 105
FEM ConvecDiffscript5.m, 106
FEM ConvecDiffSolExa.m, 98
HYP calc dt.m, 232
HYP flux roe.m, 233
HYP mach compat.m, 232
HYP plot graph.m, 232
HYP shock tube.m, 232
HYP shock tube exact.m, 232
HYP trans usol w.m, 232294 Index of Programs
HYP trans w f.m, 232
HYP trans w usol.m, 232
MRA daube4.m, 149
MRA daube4 ex1.m, 149
MRA daube4 ex2.m, 149
MRA daube4 ex3.m, 149
MRA haar.m, 148
MRA haar ex1.m, 148
MRA haar ex2.m, 148
MRA haar ex3.m, 149
MRA schauder.m, 148
MRA schauder ex1.m, 149
MRA schauder ex2.m, 149
MRA schauder ex3.m, 149
NSE ADI init.m, 278
NSE ADI step.m, 278
NSE affiche div.m, 279
NSE calc hc.m, 278
NSE calc lap.m, 277
NSE fexact.m, 277
NSE fsource.m, 277
NSE fsource nonl.m, 278
NSE init KH.m, 278
NSE init vortex.m, 278
NSE norm L2.m, 277
NSE Phi init.m, 278
NSE Phi step.m, 278
NSE Q2fft lap.m, 278
NSE Qexp lap.m, 277
NSE Qfft lap.m, 278
NSE Qimp lap.m, 277
NSE Qimp lap nonl.m, 278
NSE QNS.m, 278
NSE test trid.m, 271, 277
NSE trid per c2D.m, 277
NSE visu isos.m, 277
NSE visu sca.m, 278
NSE visu vort.m, 278
ODE Chemistry2.m, 42
ODE Chemistry3.m, 44
ODE DelayEnzyme.m, 46
ODE Enzyme.m, 46
ODE EnzymeCondIni.m, 47
ODE ErrorEnzyme.m, 47
ODE EulerDelay.m, 46
ODE fun2.m, 42
ODE fun3.m, 44
ODE RungeKuttaDelay.m, 46
ODE stab2comp.m, 41
ODE stab3comp.m, 43
ODE StabDelay.m, 45
PDE absorption.m, 20
PDE absorption source.m, 20
PDE conv bound cond.m, 24
PDE conv exact sol.m, 23
PDE conv init cond.m, 24
PDE convection.m, 24
PDE EulerExp.m, 20
PDE heat.m, 29
PDE heat u0.m, 29
PDE heat uex.m, 29
PDE RKutta4.m, 20
PDE wave fstring.m, 28
PDE wave fstring exact.m, 28
PDE wave fstring in.m, 28
PDE wave infstring.m, 26
PDE wave infstring u0.m, 26
PDE wave infstring u1.m, 26
SPE AppLegExp.m, 122
SPE CalcLegExp, 122
SPE fbe.m, 122
SPE LegExpLoop.m, 122
SPE LegLinComb.m, 120
SPE PlotLegPol.m, 120
SPE special.m, 122
SPE SpecMeth.m, 123
SPE specsec.m, 122
SPE TestIntGauss.m, 121
SPE xwGauss.m, 121
THER matrix inv.m, 249
THER oven.m, 249
THER oven ex1.m, 248
THER oven ex2.m, 249

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