Environmental Hydraulics: Numerical Methods / Edition 1

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This series of five volumes proposes an integrated description ofphysical processes modeling used by scientific disciplines frommeteorology to coastal morphodynamics. Volume 1 describes thephysical processes and identifies the main measurement devices usedto measure the main parameters that are indispensable to implementall these simulation tools. Volume 2 presents the differenttheories in an integrated approach: mathematical models as well asconceptual models, used by all disciplines to represent theseprocesses. Volume 3 identifies the main numerical methods used inall these scientific fields to translate mathematical models intonumerical tools. Volume 4 is composed of a series of case studies,dedicated to practical applications of these tools in engineeringproblems. To complete this presentation, volume 5 identifies anddescribes the modeling software in each discipline.

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Editorial Reviews

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"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)
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Product Details

  • ISBN-13: 9781848211551
  • Publisher: Wiley
  • Publication date: 11/30/2010
  • Series: ISTE Series , #435
  • Edition description: Adopted
  • Edition number: 1
  • Pages: 352
  • Product dimensions: 6.30 (w) x 9.30 (h) x 1.20 (d)

Table of Contents

Introduction xiii


Chapter 1. Feedback on the Notion of a Model and the Need forCalibration 3

1.1. “Static” and “dynamic” calibrationsof a model 6

1.2. “Dynamic” calibration of a model or dataassimilation 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 thoseresponsible for civil protection 27

2.7. Conclusion 29

2.8. Bibliography 30

Chapter 3. From Mathematical Model to Numerical Model31
Jean-Michel TANGUY

3.1. Classification of the systems of differential equations32

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


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 Methods71
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 aconvection/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 Method109
Jean-Michel TANGUY

7.1. Elementary FEM concepts and presentation of the section109

7.2. Method of approximation by finite elements 111

7.3. Geometric transformation 114

7.4. Transformation of derivation and integration operators121

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 conditions148

7.11. Summary: implementation of the finite element method151

7.12. Application example: wave propagation 151

7.13. Bibliography 158

Chapter 8. Presentation of the Finite Volume Method161
Alexandre ERN and Serge PIPERNO, section 8.6 written by DominiqueTHIÉRY

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 flowdevelopment of groundwater 195

8.7. Bibliography 210

Chapter 9. Spectral Methods in Meteorology 213

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 andconvergence 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


Chapter 12. Data Assimilation 275

12.1. Several examples of the application of data assimilation277

12.2. Data assimilation in hydraulics with the Dassflow model284

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 theory297

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 filter307

13.8. Assessment of the Kalman filter 308

13.9. Variational methods 312

13.10. Discreet formulation of the cost function: the 3D-VAR313

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 equation324

13.16. Feedback on the notion of a model and the need forcalibration 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

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