Numerical Methods

Jean-Michel Tanguy, Ministry of Sustainable Development, France.

• Introduction xiiiPART 1. GENERAL CONSIDERATIONS CONCERNING NUMERICAL TOOLS 1Chapter 1. Feedback on the Notion of a Model and the Need for Calibration 3Denis DARTUS1.1. “Static” and “dynamic” calibrations of a model 61.2. “Dynamic” calibration of a model or data assimilation 101.3. Bibliography 10Chapter 2. Engineering Model and Real-Time Model 11Jean-Michel TANGUY2.1. Categories of modeling tools 112.2. Weather forecasting at Météo France 122.3. Flood forecasting 182.4. Characteristics of real-time models 232.5. Environment of real-time platforms 252.6. Interpretation of hydrological forecasting by those responsible for civil protection 272.7. Conclusion 292.8. Bibliography 30Chapter 3. From Mathematical Model to Numerical Model 31Jean-Michel TANGUY3.1. Classification of the systems of differential equations 323.3. Discrete systems and continuous systems 403.4. Equilibrium and propagation problems 413.5. Linear and non-linear systems 433.6. Conclusion 573.7. Bibliography 57PART 2. DISCRETIZATION METHODS 59Chapter 4. Problematic Issues Encountered 61Marie-Madeleine MAUBOURGUET4.1. Examples of unstable problems 624.2. Loss of material 634.3. Unsuitable scheme 664.4. Bibliography 69Chapter 5. General Presentation of Numerical Methods 71Serge PIPERNO and Alexandre ERN5.1. Introduction 715.2. Finite difference method 725.3. Finite volume method 775.4. Finite element method 785.5. Comparison of the different methods on a convection/diffusion problem 925.6. Bibliography 93Chapter 6. Finite Differences 95Marie-Madeleine MAUBOURGUET and Jean-Michel TANGUY6.1. General principles of the finite difference method 956.2. Discretization of initial and boundary conditions 1026.3. Resolution on a 2D domain 105Chapter 7. Introduction to the Finite Element Method 109Jean-Michel TANGUY7.1. Elementary FEM concepts and presentation of the section 1097.2. Method of approximation by finite elements 1117.3. Geometric transformation 1147.4. Transformation of derivation and integration operators 1217.5. Geometric definition of the elements 1257.6. Method of weighted residuals 1287.7. Transformation of integral forms 1307.8. Matrix presentation of the finite element method 1337.9. Integral form of We on the reference element 1407.10. Introduction of the Dirichlet-type boundary conditions 1487.11. Summary: implementation of the finite element method 1517.12. Application example: wave propagation 1517.13. Bibliography 158Chapter 8. Presentation of the Finite Volume Method 161Alexandre ERN and Serge PIPERNO, section 8.6 written by Dominique THIÉRY8.1. 1D conservation equations 1628.2. Classical, weak and entropic solutions 1708.3. Numerical solution of a conservation law 1758.4. Numerical solution of hyperbolic systems 1838.5. High-order, finite volume methods 1948.6. Application of the finite volume method to the flow development of groundwater 1958.7. Bibliography 210Chapter 9. Spectral Methods in Meteorology 213Jean COIFFIER9.1. Introduction 2139.2. Using finite series expansion of functions 2149.3. The spectral method on the sphere 2169.4. The spectral method on a biperiodic domain 2279.5. Bibliography 232Chapter 10. Numerical-Scheme Study 235Jean-Michel TANGUY10.1. Reminder of the notion of the numerical scheme 23510.2. Time discretization 23610.3. Space discretization 24010.4. Scheme study: notions of consistency, stability and convergence 24110.5. Bibliography 264Chapter 11. Resolution Methods 267Marie-Madeleine MAUBOURGUET11.1. Temporal integration methods 26811.2. Linearization methods for non-linear systems 27011.3. Methods for solving linear systems AX = B 27111.4. Bibliography 272 PART 3. INTRODUCTION TO DATA ASSIMILATION 273Chapter 12. Data Assimilation 275Jean PAILLEUX, Denis DARTUS, Xijun LAI, Jérôme MONNIER and Marc HONNORAT12.1. Several examples of the application of data assimilation 27712.2. Data assimilation in hydraulics with the Dassflow model 28412.3. Bibliography 290Chapter 13. Data Assimilation Methodology 295Hélène BESSIÈRE, Hélène ROUX, François-Xavier LE DIMET and Denis DARTUS13.1. Representation of the system 29513.2. Taking errors into account 29613.3. Simplified approach to optimum static estimation theory 29713.4. Generalization in the multidimensional case 30013.5. The different data assimilation techniques 30313.6. Sequential assimilation method: the Kalman filter 30413.7. Extension to non-linear models: the extended Kalman filter 30713.8. Assessment of the Kalman filter 30813.9. Variational methods 31213.10. Discreet formulation of the cost function: the 3D-VAR 31313.11. General variational formalism: the 4D-VAR 31413.12. Continuous formulation of the cost function 31413.13. Principle of automatic differentiation 32213.14. Summary of variational methods 32213.15. A complete application example: the Burgers equation 32413.16. Feedback on the notion of a model and the need for calibration 33513.17. Bibliography 343List of Authors 349Index 351General Index of Authors 353Summary of the Other Volumes in the Series . . . 355

"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)