Numerical Methods

This series of five volumes proposes an integrated description of physical processes modeling used by scientific disciplines from meteorology to coastal morphodynamics. Volume 1 describes the physical processes and identifies the main measurement devices used to measure the main parameters that are indispensable to implement all these simulation tools. Volume 2 presents the different theories in an integrated approach: mathematical models as well as conceptual models, used by all disciplines to represent these processes. Volume 3 identifies the main numerical methods used in all these scientific fields to translate mathematical models into numerical tools. Volume 4 is composed of a series of case studies, dedicated to practical applications of these tools in engineering problems. To complete this presentation, volume 5 identifies and describes the modeling software in each discipline.

Table of Contents:
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