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 |  موضوع: كتاب Numerical PDEs for Environmental Scientists and Engineers - A First Practical Course السبت 17 أغسطس 2013, 11:56 am | |
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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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