Environmental Hydraulics – Numerical Methods V 3

<p>Introduction xiii</p>
<p>PART 1. GENERAL CONSIDERATIONS CONCERNING NUMERICAL TOOLS 1</p>
<p>Chapter 1. Feedback on the Notion of a Model and the Need for Calibration 3<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>Chapter 2. Engineering Model and Real–Time Model 11<br /> Jean–Michel TANGUY</p>
<p>2.1. Categories of modeling tools 11</p>
<p>2.2. Weather forecasting at M&eacute;t&eacute;o 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>Chapter 3. From Mathematical Model to Numerical Model 31<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>PART 2. DISCRETIZATION METHODS 59</p>
<p>Chapter 4. Problematic Issues Encountered 61<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>Chapter 5. General Presentation of Numerical Methods 71<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>Chapter 6. Finite Differences 95<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>Chapter 7. Introduction to the Finite Element Method 109<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>Chapter 8. Presentation of the Finite Volume Method 161<br /> Alexandre ERN and Serge PIPERNO, section 8.6 written by Dominique THI&Eacute;RY</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>Chapter 9. Spectral Methods in Meteorology 213<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>Chapter 10. Numerical–Scheme Study 235<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>Chapter 11. Resolution Methods 267<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 /> PART 3. INTRODUCTION TO DATA ASSIMILATION 273</p>
<p>Chapter 12. Data Assimilation 275<br /> Jean PAILLEUX, Denis DARTUS, Xijun LAI, J&eacute;r&ocirc;me 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>Chapter 13. Data Assimilation Methodology 295<br /> H&eacute;l&egrave;ne BESSI&Egrave;RE, H&eacute;l&egrave;ne ROUX, Fran&ccedil;ois–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>
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