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

"An inventory of ground measurement instruments, which provide necessary input data for the various modeling tools described in the book, is drawn up, and mathematical models describing each field within the overall subject area are detailed by a series of system equations." (Live-PR (EN), 19 April 2011)

<b>Jean-Michel Tanguy</b>, Ministry of Sustainable Development, France.

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