Introduction xiii
PART 1. GENERAL CONSIDERATIONS CONCERNING NUMERICAL TOOLS 1
Chapter 1. Feedback on the Notion of a Model and the Need for Calibration 3
Denis DARTUS
1.1. "Static" and "dynamic" calibrations of a model 6
1.2. "Dynamic" calibration of a model or data assimilation 10
1.3. Bibliography 10
Chapter 2. Engineering Model and Real-Time Model 11
Jean-Michel TANGUY
2.1. Categories of modeling tools 11
2.2. Weather forecasting at Meteo France 12
2.3. Flood forecasting 18
2.4. Characteristics of real-time models 23
2.5. Environment of real-time platforms 25
2.6. Interpretation of hydrological forecasting by those responsible for civil protection 27
2.7. Conclusion 29
2.8. Bibliography 30
Chapter 3. From Mathematical Model to Numerical Model 31
Jean-Michel TANGUY
3.1. Classification of the systems of differential equations 32
3.3. Discrete systems and continuous systems 40
3.4. Equilibrium and propagation problems 41
3.5. Linear and non-linear systems 43
3.6. Conclusion 57
3.7. Bibliography 57
PART 2. DISCRETIZATION METHODS 59
Chapter 4. Problematic Issues Encountered 61
Marie-Madeleine MAUBOURGUET
4.1. Examples of unstable problems 62
4.2. Loss of material 63
4.3. Unsuitable scheme 66
4.4. Bibliography 69
Chapter 5. General Presentation of Numerical Methods 71
Serge PIPERNO and Alexandre ERN
5.1. Introduction 71
5.2. Finite difference method 72
5.3. Finite volume method 77
5.4. Finite element method 78
5.5. Comparison of the different methods on a convection/diffusion problem 92
5.6. Bibliography 93
Chapter 6. Finite Differences 95
Marie-Madeleine MAUBOURGUET and Jean-Michel TANGUY
6.1. General principles of the finite difference method 95
6.2. Discretization of initial and boundary conditions 102
6.3. Resolution on a 2D domain 105
Chapter 7. Introduction to the Finite Element Method 109
Jean-Michel TANGUY
7.1. Elementary FEM concepts and presentation of the section 109
7.2. Method of approximation by finite elements 111
7.3. Geometric transformation 114
7.4. Transformation of derivation and integration operators 121
7.5. Geometric definition of the elements 125
7.6. Method of weighted residuals 128
7.7. Transformation of integral forms 130
7.8. Matrix presentation of the finite element method 133
7.9. Integral form of We on the reference element 140
7.10. Introduction of the Dirichlet-type boundary conditions 148
7.11. Summary: implementation of the finite element method 151
7.12. Application example: wave propagation 151
7.13. Bibliography 158
Chapter 8. Presentation of the Finite Volume Method 161
Alexandre ERN and Serge PIPERNO, section 8.6 written by Dominique THIERY
8.1. 1D conservation equations 162
8.2. Classical, weak and entropic solutions 170
8.3. Numerical solution of a conservation law 175
8.4. Numerical solution of hyperbolic systems 183
8.5. High-order, finite volume methods 194
8.6. Application of the finite volume method to the flow development of groundwater 195
8.7. Bibliography 210
Chapter 9. Spectral Methods in Meteorology 213
Jean COIFFIER
9.1. Introduction 213
9.2. Using finite series expansion of functions 214
9.3. The spectral method on the sphere 216
9.4. The spectral method on a biperiodic domain 227
9.5. Bibliography 232
Chapter 10. Numerical-Scheme Study 235
Jean-Michel TANGUY
10.1. Reminder of the notion of the numerical scheme 235
10.2. Time discretization 236
10.3. Space discretization 240
10.4. Scheme study: notions of consistency, stability and convergence 241
10.5. Bibliography 264
Chapter 11. Resolution Methods 267
Marie-Madeleine MAUBOURGUET
11.1. Temporal integration methods 268
11.2. Linearization methods for non-linear systems 270
11.3. Methods for solving linear systems AX = B 271
11.4. Bibliography 272
PART 3. INTRODUCTION TO DATA ASSIMILATION 273
Chapter 12. Data Assimilation 275
Jean PAILLEUX, Denis DARTUS, Xijun LAI, Jerôme MONNIER and Marc HONNORAT
12.1. Several examples of the application of data assimilation 277
12.2. Data assimilation in hydraulics with the Dassflow model 284
12.3. Bibliography 290
Chapter 13. Data Assimilation Methodology 295
Helene BESSIÈRE, Helene ROUX, François-Xavier LE DIMET and Denis DARTUS
13.1. Representation of the system 295
13.2. Taking errors into account 296
13.3. Simplified approach to optimum static estimation theory 297
13.4. Generalization in the multidimensional case 300
13.5. The different data assimilation techniques 303
13.6. Sequential assimilation method: the Kalman filter 304
13.7. Extension to non-linear models: the extended Kalman filter 307
13.8. Assessment of the Kalman filter 308
13.9. Variational methods 312
13.10. Discreet formulation of the cost function: the 3D-VAR 313
13.11. General variational formalism: the 4D-VAR 314
13.12. Continuous formulation of the cost function 314
13.13. Principle of automatic differentiation 322
13.14. Summary of variational methods 322
13.15. A complete application example: the Burgers equation 324
13.16. Feedback on the notion of a model and the need for calibration 335
13.17. Bibliography 343
List of Authors 349
Index 351
General Index of Authors 353
Summary of the Other Volumes in the Series . . . 355
