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تذكير بمساهمة فاتح الموضوع :أخوانى فى الله أحضرت لكم كتاب Numerical PDEs for Environmental Scientists and Engineers -
A First Practical Course
Daniel R. Lynch Dartmouth College Dartmouth, New Hampshire
ويتناول الموضوعات الأتية :
Preface Synopsis I The Finite Difference Method Introduction 1.1 From Algebra to Calculus and Back 1.2 DistributedLumpedDiscrete Systems 1.3 PDE Solutions 1.4 IC'sBC'sClassification A uniqueness proof: Poisson Equation Classification of BC's Classification of Equations 2 Finite Difference Calculus 2.1 1-D Differences on a Uniform Mesh SummaryU niform Mesh 2.2 Use of the Error Term 2.3 1-D Differences on Nonuniform Meshes 2.4 Polynomial Fit 2.5 Cross-Derivatives 3 Elliptic Equations 3.1 Introduction 3.2 1-D Example 3.3 2-D Example Molecules Matrix Assembly and Direct Solution Iterative Solution 3.4 Operation Counts 3.5 Advective-Diffusive Equation 4 Elliptic Iterations 4.1 Bare Essentials 4.2 Point Methods 4.3 Block Methods Alternating Direction Methods 4.4 Helmholtz Equation CONTENTS 4.5 Gradient Descent Methods 47 5 Parabolic Equations 51 5.1 Introduction 51 5.2 Examples: Discrete Systems 53 Euler 53 Leapfrog 54 Backward Euler 55 2-Level Implicit 55 5.3 Boundary Conditions 57 5.4 Stability, Consistency, Convergence 58 Convergence Lumped System 59 Convergence - Discrete System 60 Consistency 61 Stability 61 5.5 Accuracy: Fourier Analysis 64 Continuous System 64 Lumped System 65 Discrete System 67 Example: Implicit Leapfrog System 71 5.6 Conservation Laws 76 5.7 Two-Dimensional Problems 82 5.8 Nonlinear Problems 85 6 Hyperbolic Equations 89 6.1 Introduction 89 6.2 Lumped Systems 93 6.3 Harmonic Approach 94 6.4 More Lumped Systems 97 6.5 Dispersion Relationship 99 Continuous System 99 Lumped System # 1 100 Lumped System # 2 101 Lumped System # 3 102 Lumped System # 4 103 6.6 Discrete Systems 104 Discrete System 1 (Telegraph Equation) 106 Discrete Systems 3: Coupled lSt Order Equations 109 Discrete System 4: Implicit Four-Point Primitive 115 6.7 Lumped Systems in Higher Dimensions 116 I1 The Finite Element Method 121 7 General Principles 123 7.1 The Method of Weighted Residuals 123 7.2 MWR Examples 125 7.3 Weak Forms 128 7.4 Discrete Form 129 7.5 Boundary Conditions 129 7.6 Variational Principles 130 7.7 Weak Forms and Conservation Properties 133 8 A 1-D Tutorial 139 8.1 Polynomial Basesth e Lagrange Family 1 39 8.2 Global and Local Interpolation 140 8.3 Local Interpolation on Elements 142 8.4 Continuity - Hermite Polynomials 143 8.5 Example 146 8.6 Boundary Conditions 150 8.7 The Element Matrix 152 8.8 Assembly and the Incidence List 157 8.9 Matrix Structure 158 8.10 Variable Coefficients 161 8.1 1 Numerical Integration 162 8.12 Assembly with Quadrature 164 9 Multi-Dimensional Elements 167 9.1 Linear Triangular Elements 167 Local Interpolation 167 Differentiation 169 Integration ; 170 9.2 Example: Helmholtz Equation on Linear Triangles 170 9.3 Higher Order Triangular Elements 172 Local Coordinate System 172 Higher-Order Local Interpolation on Triangles 173 Differentiation 175 Numerical Integration 177 9.4 Isoparametric Transformation 179 9.5 Quadrilateral Elements 181 The Bilinear Element 181 Higher-Order Quadrilateral Elements 183 Isoparametric Quadrilaterals 183 10 Time-Dependent Problems 189 10.1 General Approach 189 10.2 Lumped and Discrete Systems 189 10.3 Example: Diffusion Equation 190 10.4 Example: Advection-Diffusion Equation 192 10.5 Example: Wave Equation 193 10.6 Example: Telegraph Equation 195 11 Vector Problems 197 11.1 Introduction 197 11.2 Gradient of a Scalar 197 Galerkin Form 198 Natural Local Coordinate Systems and Neumann Boundaries 1 99 Dirichlet Boundaries 201 Elasticity 202 Weak Form 202 Constitutive Relations 2 03 Galerkin Approximation 204 Natural Local Coordinate Systems 204 ReferencesS olid Mechanics 2 05 11.4 Electromagnetics 205 Governing Equations 206 Potentials and Gauge 2 06 Helmholtz Equations in the Potentials 2 07 Weak Form 208 Boundary Conditions 209 Reconstructing E and H 2 09 ReferencesE &M 2 09 11.5 Fluid Mechanics with Mixed Interpolation 2 10 Governing equations 210 Bases and Weights 2 11 Mixed Elements 2 11 Weak Form 212 Galerkin Equations 2 12 Numbering Convention 2 13 Coordinate Rotation 2 14 References: Fluid Mechanics 2 14 11.6 Oceanic Tides 214 Weak Form and Galerkin Helmholtz Equation 215 Velocity Solution 2 16 ReferencesO ceanic Tides 217 12 Numerical Analysis 219 12.1 1-D Elliptic Equations 219 Laplace Equation on 1-D Linear Elements 219 Advective-Diffusive Equation on 1-D Linear Elements 2 19 Helmholtz Equation on 1-D Linear Elements 221 Poisson Equation on 1-D Linear Elements 2 23 Inhomogeneous Helmholtz Equation on 1-D Linear Elements 2 26 12.2 Fourier Transforms for Difference Expressions 230 12.3 2-D Elliptic Equations 2 36 Laplace Equation on Bilinear Rectangles 2 36 Helmholtz Equation on Bilinear Rectangles 2 38 12.4 Diffusion Equation 2 40 Stability 2 41 Monotonicity2 42 Accuracy 2 43 Leapfrog Time-Stepping 2 43 3-level Implicit Time-Stepping 2 45 12.5 Explicit Wave Equation 2 47 Stability 2 48 Accuracy 2 48 CONTENTS ix 12.6 Implicit Wave Equation 250 Stability 250 Accuracy 251 12.7 Advection Equation 251 Euler Advection 252 Two-Level Implicit Advection 253 Leapfrog Advection 253 12.8 Advective-Diffusive Equation 255 Euler 256 2-Level Implicit 257 Leapfrog 258 I11 Inverse MeBhods 263 13 Inverse NoiseSVDand LLS 265 13.1 Matrix Inversion and Inverse Noise 266 Mean and Variability 266 Covariance 266 Variance 268 Noise Models 268 EigenTheory 270 13.2 The Singular Value Decomposition 272 SVDBasics 273 The SquareNonsingular Case 274 The SquareSingular Case 275 The SquareNearly-Singular Case 277 The Over-Determined Case 277 The Under-Determined Case 278 SVD Covariance 278 SVD References 279 13.3 Linear Least Squares and the Normal Equations 279 Quadratic Forms and Gradient 279 Ordinary Least Squares 280 Weighted Least Squares 281 General Least Squares 282 14 Fitting Models to Data 285 14.1 Inverting Data 285 Model-Data Misfit 285 Direct Solution Strategies and Inverse Noise 287 More on the Model-Data Misfit 288 14.2 Constrained Minimization and Gradient Descent 289 Generalized Least Squares as Constrained Minimization 289 The Adjoint Method 290 Gradient Descent 291 Summary - Adjoint Method with Gradient Descent 293 Monte Carlo Variance Estimation - Inverse Noise 293 14.3 Inverting Data With Representers 294 CONTENTS The Procedure 295 Inverse Noise 296 14.4 Inverting Data with Unit Responses 296 Procedure 296 14.5 Summary: GLS Data Inversion 297 14.6 Parameter Estimation 298 GLS Objective 299 First-Order Conditions for GLS Extremum 299 The Gradient in Parameter Space 300 An Adjoint Method for Parameter Estimation 302 14.7 Summary - Terminology 302 15 Dynamic Inversion 305 15.1 Parabolic Model: Advective-Diffusive Transport 305 Forward Model in Discrete Form 306 Objective and First-Order Conditions 307 Adjoint Model 308 Direct Solution An Elliptic Problem in Time 309 Iterative Solution by Gradient Descent 310 Special Case #1: "Shooting" 312 Special Case #2: Agnostic p 313 Parameter Estimation 313 15.2 Hyperbolic Model: Telegraph Equation 315 Problem Statement 315 Optimal Fit: GLS Objective and First-Order Conditions 316 Gradient Descent Algorithms 318 Conjugate Gradient Descent 319 Solution by Representers 319 15.3 Regularization 321 Reduction of the DoF's 321 The Weight Matrix 322 Heuristic Specification of [W] using FEM 322 15.4 Example: Nonlinear Inversion 323 16 Time Conventions for Real-Time Assimilation 329 16.1Time 329 16.2 Observational Data 329 16.3 Simulation Data Products 330 16.4 Sequential Simulation 331 16.5 What Time Is It? 332 16.6 Example: R-T Operations, Cruise EL 9904 332 17 Skill Assessment for Data Assimilative Models 335 17.1 Vocabulary 335 Forward and Inverse Models 335 TruthDataPrediction 335 Skill 336 Accuracy/BiasPrecision/Noise 336 17.2 Observational System Simulation Experiments: Example 337 18 Statistical Interpolation 341 18.1 Introduction: Point Estimation 3 41 18.2 Interpolation and the Gauss-Markov Theorem 3 43 18.3 Interpolating and Sampling Finite Fields 3 45 18.4 Analytic Covariance Functions 3 48 18.5 Stochastically-Forced Differential Equation (SDE) 3 50 Example1 351 Example2 356 18.6 OA-GLS Equivalence 3 56 18.7 Kriging 358 18.8 Concluding Remarks 359 Appendices A1 Vector Identities A2 Coordinate Systems A3 Stability of Quadratic Roots A4 Inversion Notes A5 Time Conventions Bibliography 377 Index 385
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