Numerical Methods

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 Météo 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 THIÉRY

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, Jérôme 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
Hélène BESSIÈRE, Hélène 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