CONTENTS COMPUTATIONAL METHODS AND ALGORITHMS

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1 CONTENTS CONTENTS COMPUTATIONAL METHODS AND ALGORITHMS Computational Methods and Algorithms - Volume 1 No. of Pages: 542 ISBN: (ebook) ISBN: (Print Volume) Computational Methods and Algorithms - Volume 2 No. of Pages: 422 ISBN: (ebook) ISBN: (Print Volume) For more information of e-book and Print Volume(s) order, please click here Or contact : eolssunesco@gmail.com

2 CONTENTS VOLUME I Computational Methods and Algorithms 1 V.V. Shaidurov,Institute of Computational Modelling, Russian Academy of Sciences, Krasnoyarsk, Russia 2. Mathematical modeling 3. Discretization process 4. Combination of the discretization and solution process 5. Parallelizm and decomposition 5.1. Schwarz methods and the domain decomposition 6. Solution process 6.1. Efficient algorithms 6.2. Hierarchical structures Fast Fourier transform Wavelets Solving sparse systems by the multigrig method 6.3. Difficulties due to complicated geometries 6.4. Robustness versus efficiency 6.5. The solving time-dependent problems 7. Implementation aspects Basic Methods for Solving Equations of Mathematical Physics 29 V.K. Andreev,Institute of Computational Modeling Russian Academy of Sciences, Russia Introduction 1. Analytical methods for problems of mathematical physics 1.1. Methods of the Potential Theory Green s Formulae Fundamental Solutions. Parametrix Functional Spaces and a priori Estimates of the Operator T x Local Existence Theorems Internal Estimates of the Shauder Type and Smoothness of a Solution Volume Potential. Simple and Double Layer Potentials Heat Potentials Green s Function 1.2. EigenvalueProblems. Methods of Eigenfunctions General Description of the Method of Separation of Variables The Method of Separation of Variables for Boundary Value Problems of Mathematical Physics Curvilinear Coordinate Systems 1.3. Methods of Integral Transforms 1.4. Methods of Transformation Groups 2. Approximate methods 2.1. Discretization Methods in Mathematical Physics The Finite Difference Method The Finite Element Method 2.2. Methods for Non-linear Problems Differentiation in Normed Spaces Newton s Method for Nonlinear Operators Monotone Operators in Partially Ordered Banach Spaces 2.3. Variational Formulations of Problems. Variational Methods Sobolev Spaces Weak Solutions of Problems of Mathematical Physics i

3 Variational Methods Projection Methods Methods of Potential Theory 79 V.I. Agoshkov,Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia P.B. Dubovski,Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia 2. Fundamentals of the Potential Theory 2.1. Some Elements from Calculus Basic Orthogonal Coordinates Basic Differential Operations on a Vector Field Formulae from the Field Theory Basic Properties of Harmonic Functions 2.2. Volume Mass or Charge potential Newton s (Coulomb s) potential Properties of Newton s Potential Potential of a Homogenous Sphere Properties of the Potential of Volume-distributed Masses 2.3. Logarithmic Potentials Definition of the Logarithmic Potential Properties of the Logarithmic Potential Logarithmic Potential of a Disk of Constant Density 2.4. Simple Layer Potential Simple Layer Potential in the 3D space Properties of the Simple Layer Potential Potential of a Homogenous Sphere Simple Layer Potential on the Plane 2.5. Double Layer Potential Dipole Potential Double Layer Potential in the Space and its Properties Double Layer Logarithmic Potential and its Properties 3. Application of the Potential Theory to the Classical Problems of Mathematical Physics 3.1. Solution of the Laplace and Poisson Equations Formulation of Boundary Value Problems for the Laplace Equation The Dirichlet Problem in the 3D Space The Dirichlet Problem on the Plane The Neumann Problem The Third Boundary Value Problem for the Laplace Equation The Boundary Value Problem for the Poisson Equation 3.2. Green s Function of the Laplace Operator The Poisson Equation Green s Function The Dirichlet Problem in a Simple Domain 3.3. The Laplace Equation in a Complex-Shaped Domain The Schwarz Method Sweeping-out Method 4. Other Applications of the Potential Method 4.1. Application of the Potential Method to the Helmholtz Equation Basic facts Boundary Value Problems for the Helmholtz Equation Green s Functions The Equation 4.2. Non-stationary Potentials Potentials for the 1D Heat Conductivity Equation Heat Sources in a Multi-dimensional Case Boundary Value Problem for the Telegraph Equation ii

4 Eigenvalue Problems: Methods of Eigenfunctions 125 V.I. Agoshkov,Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia V.P. Shutyaev,Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia 2. Eigenvalue problems 2.1. The Formulation of an Eigenvalue Problem and its Physical Meaning 2.2. Eigenvalue problems for differential operators 2.3. Properties of Eigenvalues and Eigenfunctions 2.4. The Fourier Series 2.5. Eigenfunctions of some One-dimensional Problems 3. Special functions 3.1. Spherical Functions 3.2. The Legendre Polynomials 3.3. Cylindrical Functions 3.4. The Chebyshev, Laguerre, and Hermite Polynomials 3.5. The Mathieu Functions and Hypergeometric Functions 4. The method of eigenfunctions 4.1. A General Description of the Method of Eigenfunctions 4.2. The method of Eigenfunctions for Differential Equations of Mathematical Physics 4.3. On the Solution of Problems with Nonhomogeneous Boundary Conditions 5. The method of eigenfunctions for some problems of the theory of electromagnetism 5.1. The Problem on a Bounded Transmission Line 5.2. A Field inside a Sphere with Potential given on its Surface 6. The method of eigenfunctions for the heat conductivity problem 6.1. The Heat Conductivity Problem for a Bounded Rod 7. The method of eigenfunctions for problems of the oscillation theory 7.1. Free Oscillations of a Homogeneous String 7.2. Oscillations of a String with a Moving Endpoint 7.3. The Problem of Free Oscillations of Gas 7.4. Oscillations of a Membrane with Fixed Edge Methods of Integral Transforms 165 V. I. Agoshkov,Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia P. B. Dubovski,Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia 2. Basic integral transforms 2.1. The Fourier Transform Basic Properties of the Fourier Transform The Multiple Fourier Transforms 2.2. The Laplace Transform The Laplace Integral The Inversion Formula for the Laplace Transform Limit Theorems 2.3. The Mellin Transform 2.4. The Hankel Transform 2.5. The Meyer Transform 2.6. The Kontorovich-Lebedev Transform 2.7. The Mehler-Foque Transform 2.8. The Hilbert Transform 2.9. The Laguerre and Legendre Transforms The Bochner Transform, the Convolution Transform, the Wavelet and Chain Transforms 3. The application of integral transforms to problems of the oscillation theory 3.1. Electric Oscillations 3.2. Transverse Oscillations of a String 3.3. Transverse Oscillations of an Infinite Round Membrane iii

5 4. The application of integral transforms to heat conductivity problems 4.1. The Solution of the Heat Conductivity Problem by the use of the Laplace Transform 4.2. The Solution of the Heat Conductivity Problem by the use of the Fourier Transform 4.3. The Problem of Temperature Condition of a Sphere 5. The application of integral transforms in the theory of neutron slow-down and diffusion 5.1. The Solution of the Equation of Neutron Slow-down for Moderator of Infinite Size 5.2. The Problem of Diffusion of Thermal Neutrons 6. The application of integral transforms to problems of hydrodynamics 6.1. Two-dimensional Irrotational Flow of an Ideal Liquid 6.2. Flow of an Ideal Liquid through a Gap 6.3. Outflow of an Ideal Liquid through a Round Aperture 7. The application of integral transforms in the elasticity theory 7.1. Axially Symmetric Stresses in a Cylinder 7.2. The Boussinesq Problem in a Half-space 7.3. Determination of Stresses in a Wedge 8. The application of integral transforms in the coagulation kinetics 8.1. The exact Solution of the Coagulation Equation 8.2. The Violation of the Law of Conservation of Mass 9. Brief instructions for the application of integral transforms Discretization Methods for Problems of Mathematical Physics 207 V.I. Agoshkov,Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia 2. Finite difference methods 2.1. The Grid Method Basic Ideas of the Method General Definitions. Convergence Theorem The Grid Method for Partial Differential Equations 3. Variational methods 3.1. Basic Notions in Variational Methods Variational Formulation of a Problem Basic Ideas behind Direct Methods in Calculus of Variations 3.2. The Ritz Method The classical Ritz Method The Ritz Method in an Energy space 3.3. The Method of Least Squares 4. Projection methods 4.1. The Bubnov-Galerkin Method The Bubnov-Galerkin Method (General Case) The Bubnov-Galerkin Method (The Case of A = A 0 + B ) 4.2. Method of Moments 4.3. Projection Methods in Hilbert and Banach Spaces Projection Method in a Hilbert Space The Galerkin-Petrov Method The Projection Method in a Banach Space The Collocation Method 4.4. Basic Notions in ProjectionGrid Methods Variational Formulation of Problems and Variational Methods 244 Brigitte LUCQUIN-DESREUX,Laboratoire d Analyse Numérique, Université Pierre et Marie Curie, Paris, France 1.1. Motivation iv

6 1.2. Principle of the Method 2. The variational method 2.1. The Functional Framework 2.2. The Variational Formulation 2.3. The Lax-Milgram Theorem 2.4. The Symmetric Case 3. Applications of the Lax-Milgram theorem 3.1. The Non-homogeneous Dirichlet Problem 3.2. The Neumann Problem 3.3. Problem with Robin Boundary Conditions 3.4. Problem with Mixed Boundary Conditions 3.5. General Symmetric Second Order Elliptic Problems 3.6. Non-symmetric Problems 3.7. A 4-th Order Problem 4. Extensions of the variational theory 4.1. Linearized elasticity 4.2. The Stokes System 4.3. Elliptic Variational Inequalities 4.4. The Galerkin Method 4.5. A Simple Variation of the Projection Theorem 5. Conclusion Methods of Transformation Groups 281 V.K. Andreev,Department of Nonlinear Problems in Mechanics, Institute of Computational Modeling Russian Academy of Sciences, Krasnoyarsk, Russia 1. Continuous Transformation Groups 1.1. Local Transformation Groups 1.2. Lie Equations 1.3. Invariants 1.4. Invariant Manifolds 2. Invariant Differential Equations 2.1. The Continuation of Point Transformations 2.2. Defining Equation 2.3. Invariant and Partly Invariant Solutions 3. Tangential Transformations 3.1. Contact Transformations 3.2. Tangential Transformations of Finite and Infinite Orders 4. Conservation Laws 4.1. Noether Theorem 4.2. The Action of Connected Algebra 4.3. Examples 5. Bäcklund Transformations 6. Sine-Gordon Equation 7. Korteweg de Vries Equation and Lax Pairs 8. Hirota Transformation and Penleve Property 9. Method of Inverse Scattering Problem 10. Schrodinger Equation Numerical Analysis and Methods for Ordinary Differential Equations 306 N.N. Kalitkin,Institute for Mathematical Modeling, Russian Academy of Sciences, Moscow, Russia S.S. Filippov,Keldysh Institute of Applied Mathematics, Moscow, Russia 1.1. Problems of Numerical Analysis Well-posed Problems v

7 Rigor of Studies 1.2. Sources of Error Types of Data and Unknowns and Their Norms Irreducible Error Error of a Method Round-off Error 2. The solution of systems of linear equations 2.1. Problems in Linear Algebra 2.2. Systems of Linear Equations A Review of Methods Gauss Elimination The Choice of a Pivot Special Matrices The Square Root Method Ill-conditioned Systems Iterative Methods The Fast Fourier Transform 2.3. The Determinant and Inverse of a Matrix Calculation of the Determinant Inverse Matrix 2.4. An Eigenvalue Problem Similarity Transformations The Givens Method The Jacobi Method The QR-algorithm The Inverse Iteration 3. The solution of nonlinear equations and systems 3.1. One-step Methods The Jacobi Iteration Rate of convergence Systems of Equations 3.2. Newton s Method The Case of One Equation The Continuous Analogue Systems of Equations 3.3. Multistep Methods The Bisection Method The Method of Secants The Method of Parabolas 3.4. Elimination of Roots Roots of a Polynomial The Case of an Arbitrary Function 4. Numerical integration 4.1. Interpolation Formulae The Trapezoidal Formula The Euler-Maclaurin Formulae The Mean Value Formula 4.2. Grid Refinement Uniform Grids Recursive Refinement Quasiuniform Grids 4.3. The Gauss-Christoffel Method General Formulae Special Cases 4.4. The Collocation Method General Formulae Improper integrals 4.5. Multiple Integrals vi

8 The Mean Value Formulae Other Formulae 5. Interpolation and approximation of functions 5.1. Interpolational Polynomials Approximation Problems The Lagrange Interpolational Polynomial The Newton polynomial Convergence 5.2. Mean Square Approximation by Polynomials Generalized Polynomials Power Series The Chebyshev Polynomials 5.3. Trigonometric Series The Fourier Series The Bessel Formula 5.4. Spline Approximation Forms of a Spline B-splines Finite Elements Mean Square Splines Interpolational Splines 6. Numerical differentiation 6.1. The Derivative of an Interpolational Polynomial General Formulae The Simplest Formulae Uniform Grids Round-off Errors 6.2. Other Methods Differentiation of a Spline Differentiation of the Fourier Series Derivatives of the Chebyshev Series 6.3. Applications to Problems of Mathematical Physics Ordinary Differential Equations Partial Differential Equations Higher-order Accuracy 7. Two-sided methods and interval analysis 7.1. Two-sided Methods 7.2. Interval Analysis 8. Numerical methods for ordinary differential equations 8.1. Representation of Ordinary Differential Equations and Formulations of Problems The Standard Form of ODE Dynamical Systems The Cauchy Problem A Boundary Value Problem Differential-algebraic Equations Implicit Differential Equations 8.2. Approximate Methods for the Cauchy Problem Analytic Methods Numerical Methods Stiff Problems 8.3. One-step Methods Runge-Kutta Methods Rosenbrock Type Methods 8.4. Linear Multistep Methods General Properties The Adams Methods Upwind Methods Nordsieck s Representation vii

9 8.5. Differential Equations with a Retarded Argument 9. Conclusion Solution of Systems of Linear Algebraic Equations 382 Pascal Joly, Research Engineer, CNRS, France 1. An Unusable Formula 1.1. The Case of a Triangular Matrix 2. Direct methods Factorization Method 2.3. Numerical Stability 2.4. Gauss Factorization Algorithm 2.5. Computation Cost 2.6. Crout Factorization 2.7. Cholesky Factorization 2.8. Computational Cost 2.9. Skyline of a Matrix 3. Iterative methods Regular Decomposition 3.3. Jacobi Method 3.4. Gauss-Seidel Method 3.5. Relaxation Method 3.6. Double Regular Decomposition 3.7. Symmetric Relaxation Method (S.S.O.R.) 3.8. Comparison of the Methods 3.9. Condensed Storage 4. The conjugate gradient method A Minimization Problem 4.3. A First Formulation 4.4. Computational Cost 4.5. Convergence Rate 4.6. Preconditioning the Iterations 5. Conjugate gradient method: general case Orthomin Algorithm 5.3. Biconjugate Gradient Algorithm 5.4. A Different Approach : GMRES and QMR 5.5. Preconditioning Nonsymmetric Matrices 5.6. An Awkward Question 6. Domain decomposition methods Domain Decomposition 6.3. Generalization 7. Conclusion Numerical Integration 426 M.V. Noskov,Department of Applied Mathematics, Krasnoyarsk State Technical University, Russia 1. Statements of Problems 1.1. Algebraically Accurate Formulae 1.2. Statements of Problems of Numerical Integration in Terms of Functional Analysis 1.3. Multi-dimensional Case 2. Quadrature Formulae 2.1. Interpolatory Quadrature Formulae viii

10 2.2. Newton-Cotes Formulae 2.3. Error Analysis for Quadrature Formulae 2.4. Compound Quadrature Formulae 2.5. Quadrature Formulae of Gaussian type 2.6. Generalization of Quadrature Formulae of the Highest Algebraic Accuracy 2.7. Quadrature Formulae of the Highest Trigonometric Accuracy 2.8. Chebyshev Quadrature Formulae 3. Cubature Formulae 4. Conclusion Numerical Methods for Ordinary Differential Equations and Dynamic Systems 447 E.A. Novikov,Institute of Computational Modeling, Russian Academy of Sciences, Krasnoyarsk, Russia 2. Dynamic Systems 3. Analytic Methods 4. One-step Methods 5. Stiff Systems 6. Linear Multistep Methods 7. Error Estimation 8. Delay Differential Equations Index 471 About EOLSS 481 VOLUME II Finite Element Method 1 Jacques-Hervé SAIAC,Départment de Mathématiques, Conservatoire National des Arts et Métiers, Paris, France 1.1. A Simple One-dimensional Problem 1.2. Approximation Process with Linear Elements 1.3. Computation of Matrix Coefficients 2. Other one-dimensional boundary problems 2.1. The Non-homogeneous Dirichlet Problem 2.2. The Neumann Problem 2.3. The Problem with Robin Boundary Conditions 3. Higher order elements in one dimension 3.1. P 2 Elements 3.2. Hermite Cubics 4. Two or Three-dimensional Elliptic Problems 4.1. A Model Problem 4.2. Variational Formulation 4.3. Domain Discretization: The Mesh 4.4. P 1 -Lagrange Triangular Element P 1 Basis Functions P 1 Shape Functions 4.5. The Approximate Problem with P 1 Elements Element Matrix Computation Left-hand Side Computations ix

11 5. Two-dimensional P k Lagrange Elements 5.1. P 1 Triangle 5.2. P 2 Triangle 5.3. P 3 Triangle 6. Three-dimensional P k Elements 6.1. P 1 Tetrahedron 6.2. P 2 Tetrahedron 6.3. P 3 Tetrahedron 7. Isoparametric elements 7.1. Quadrilateral Elements 7.2. Q 1 Elements 7.3. Q 1 Shape Functions 7.4. Q 2 Elements 7.5. Curved P 2 Triangle 7.6. Three-dimensional Isoparametric Elements 8. Error analysis with exact integration 8.1. Internal Approximation 8.2. A General Error Bound for Elliptic Problems 8.3. General Error Bound with P k or Q k Elements and Exact Integration 9. Numerical quadrature formulas 9.1. One Dimensional Formulas The Midpoint Formula The Trapezoidal Rule The Simpson s Rule The Gauss-Legendre Formula 9.2. Two-dimensional Formulas 9.3. Three-dimensional Formulas 10. Error analysis with numerical integration Ellipticity Conditions Error Bound with Numerical Quadrature 11. Some Practical Conclusions One-dimensional Elliptic Problems Elements P P 2 Elements P k Elements Two-dimensional Elliptic Problems P 1 Elements P 2 Elements P k Elements Q 1 Elements Q 2 Elements Counter-examples: Non-elliptic Approximate Bilinear Forms An Introduction to Finite Volume Methods 36 François Dubois,Conservatoire national des Arts et Metiers, France 1. Advection equation and method of characteristics Advection Equation 1.2. Initial-Boundary Value Problems for the Advection Equation 1.3. Inflow and Outflow for the Advection Equation 2. Finite volumes for linear hyperbolic systems Linear Advection x

12 2.2. Numerical Flux Boundary Conditions 2.3. A Model System with Two Equations 2.4. Unidimensional Linear Acoustics 2.5. Characteristic Variables 2.6. A Family of Model Systems with Three Equations 2.7. First Order Upwind-centered Finite Volumes 3. Gas dynamics with the Roe method Nonlinear Acoustics in One Spatial Dimension 3.2. Linearization of the Gas Dynamics Equations 3.3. Roe Matrix 3.4. Roe Flux 3.5. Entropy Correction 3.6. Nonlinear Flux Boundary Conditions 4. Second order and two space dimensions Towards Second Order Accuracy 4.2. The Method of Lines 4.3. The MUSCL Method of Van Leer 4.4. Second Order Accurate Finite Volume Method for Fluid Problems Explicit Runge-Kutta Integration with respect to Time Numerical Methods for Integral Equations 106 A.M. Denisov, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Russia I.K. Lifanov, Institute of Numerical Mathematics of Russian Academy of Science, Moscow, Russia E.V. Zakharov, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Russia 2. Quadrature methods 2.1. Quadrature Formulae for an Integral over a Segment 2.2. Reducing an Integral Equation to a System of Linear Algebraic Equations 3. Degenerate Kernels. Projection and Collocation Methods 3.1. Method of Degenerate Kernel 3.2. Projection Method 3.3. Collocation Method 4. Iterative methods for linear and nonlinear integral equations 4.1. Successive Approximation Method 4.2. Newton s Method 5. Singular integral equations 5.1. Singular Integral 5.2. Numerical Method for Singular Integral Equations Numerical Algorithms for Inverse and Ill-Posed Problems 133 A.М. Denisov,Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia 2. Inverse Problems 2.1. Reconstruction of Input Signal 2.2. Inverse Problems for Ordinary Differential Equations 2.3. Inverse Problems for Partial Differential Equations 2.4. Computerized Tomography Problem 2.5. Abstract Setting of Inverse Problems 3. Ill-Posed Problems 3.1. Well-posed and Ill-posed Problems 3.2. Integral and Operator Equations of the First Kind 3.3. Inverse Problems with Initial Information given approximately xi

13 3.4. Regularizing Operator 4. Numerical Algorithms for Solving Inverse and Ill-Posed Problems 4.1. Solving of Ill-posed Problems on Compact Sets 4.2. Tikhonov Regularization Method 4.3. The Projective Algorithm for Solving Ill-posed Problems 4.4. The Iterative Algorithms for Solving Linear Ill-posed Problems 4.5. The Iterative Algorithms for Solving Nonlinear Ill-posed Problems 4.6. The Quasi-inversion Method Solution of Electromagnetism Theory Problems 153 V.V. Denisenko, Institute of Computational Modelling SB RAS,Krasnoyarsk,Russia 2. Two-dimensional electrostatics problems 3. Three-dimensional electrostatics problems 4. Two-dimensional magnetostatics problems 5. Three-dimensional magnetostatics problems 6. Electrical conductivity problems 7. Solutions harmonic with respect to time 8. Nonstationary solutions 9. Conclusions Computational Methods in Elasticity 184 Michel SALAUN,Départment de Mathématiques et Informatique, ENSICA, Toulouse, France 2. Basic aspects of continuum mechanics 2.1. Strain Tensor 2.2. Linearized Strain Tensor and Small Displacements Hypothesis 2.3. Stress Tensor 2.4. Generalized Hooke s Law 3. The three-dimensional linearized elasticity 3.1. Variational Formulation 3.2. Equilibrium Equation 4. The three-dimensional elastodynamics problem 4.1. Theoretical Framework 4.2. Characterization of the Eigenvalues 4.3. Modal Analysis 4.4. Step-by-step Methods of Integration General Principles Convergence of Numerical Schemes Examples of Numerical Schemes 5. A particular case of structures: plates Notation 5.3. The Kirchhoff-Love Model for Plates Kirchhoff-Love Hypotheses Kirchhoff-Love Displacement Fields Kirchhoff-Love Hypotheses and Hooke s Law Kirchhoff-Love Plate Model Equilibrated Plates 5.4. The Mindlin- Naghdi-Reissner Model for Plates Mindlin-Naghdi-Reissner Hypotheses Mindlin-Naghdi-Reissner Plate Model Equilibrated Plates 5.5. Finite Elements for Bending Plates xii

14 General Aspects Examples 6. Conclusion Computational Methods for Compressible Flow Problems 223 Rémi Abgrall,Université Bordeaux I, Talence Cedex, France 2. A Brief Description of the Solutions 2.1. On the Notion of Solutions 2.2. Hyperbolicity: Characteristic Form of the Equations 3. Numerical Schemes for 1-D Problems 3.1. The Principle of Conservative Schemes 3.2. The Godunov Scheme 3.3. The Roe Scheme 3.4. Kinetic Schemes 3.5. High Order Accuracy Scheme 4. Schemes for Multidimensional Problems 5. Numerical Examples 5.1. A One-dimensional Example: The Sod Problem 5.2. A Two-dimensional Example 6. Conclusions Methods of Nonlinear Kinetics 259 A.N. Gorban,Institute of Computational Modeling, Siberian Branch of Russian Academy of Sciences, Krasnoyarsk, Russia I.V. Karlin,Institute of Energy Technology, ETH-Zentrum, Zűrich, Switzerland 1. The Boltzmann equation 1.1. The Equation 1.2. The Basic Properties of the Boltzmann Equation 1.3. Linearized Collision Integral 2. Phenomenology and Quasi-chemical representation of the Boltzmann equation 3. Kinetic models 4. Methods of reduced description 4.1. The Hilbert Method 4.2. The Chapman-Enskog Method 4.3. The Grad Moment Method 4.4. Special Approximations 4.5. The Method of Invariant Manifold Thermodynamic Projector Iterations for the Invariance Condition 4.6. Quasi-equilibrium Approximations 5. Discrete velocity models 6. Direct simulation 7. Lattice Gas and Lattice Boltzmann models 8. Other kinetic equations 8.1. The Enskog Equation for Hard Spheres 8.2. The Vlasov Equation 8.3. The Smoluchowski Equation Methods for Magnetosphere and Near-Space Problems 280 N. V. Erkaev,Institute of Computational Modeling, Russian Academy of Sciences, Krasnoyarsk, Russia H. K. Biernat,Space Research Institute, Austrian Academy of Sciences, Graz, Austria xiii

15 C. J. Farrugia, Institute for the Study of Earth, Oceans and Space, University of New Hampshire, Durham, NH, USA 2. MHD model of solar wind flow around the magnetosphere 3. Mathematical statement of the flow problem: Basic equations 4. Thermal anisotropy of the magnetosheath plasma 5. Reconnection problem 6. Conclusions Numerical Simulation of Climate Problems 298 V.N. Krupchatnikoff, Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of Russian Academy of Sciences, Russia V.I. Kuzin,Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of Russian Academy of Sciences, Russia 2. Climate, Climatic Variability and Climate Changes 3. Atmosphere & Ocean Circulation Models 3.1. Atmospheric General Circulation Models (AGCM) 3.2. Ocean General Circulation Models (OGCM) 4. Numerical Modeling of Climatic Variability and Climate Changes 4.1. Inter-Annual Climate Variability Modeling and Predictability 4.2. Climate Decadal-to-Centennial Variability Modeling 4.3. Anthropogenic Climate Change 4.4. Regional Climate Modeling 5. Conclusion Numerical Simulation of Biosphere Dynamics 321 S.I.Bartsev, Institute of biophysics SB RAS, Krasnoyarsk,Russia R.G.Khlebopros,Institute of biophysics SB RAS, Krasnoyarsk,Russia 2. Models of Global Dynamics by Club of Rome 3. The Problem of the Earth's Biosphere Stability 4. Canadian Climate Change Model 5. Global Models of Biosphere Dynamics 6. Problems of Biosphere Dynamics Prediction 7. Numerical Simulation and Experimental Models of the Biosphere 8. Is Uncertainty of Global Models Principal? 9. Resume Numerical Methods for Weather Forecasting Problems 335 A.A. Fomenko,Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of Russian Academy of Sciences, Russia. 2. Data assimilation system. 3. Numerical data analysis and initialization. 4. Mathematical Models for Numerical Weather Prediction 4.1. Filtered Models 4.2. Global Models 4.3. Regional and Mesoscale Models 5. Numerical Methods in Weather Forecast 5.1. Spectral and Finite-element Methods xiv

16 5.2. Finite-difference Method (Grid Method) 6. Parameterization schemes Atmospheric Turbulence Parameterization 6.2. Cloudiness and Precipitation Parameterization 6.3. Radiation Transfer Parameterization 6.4. Surface Processes Parameterization 6.5. Orographic Effects Parameterization 7. The use of numerical weather forecasting products. 8. Resume. Index 353 About EOLSS 361 xv