Computational Physics Problem Solving with Python

by Landau, Rubin H.; Páez, Manuel J.; Bordeianu, Cristian C.

• ISBN13: 9783527413157
• ISBN10: 3527413154
• Edition: 3rd
• Format: Paperback
• Copyright: 2015-09-08
• Publisher: Wiley-VCH

Summary

The use of computation and simulation has become an essential part of the scientific process. Being able to transform a theory into an algorithm requires significant theoretical insight, detailed physical and mathematical understanding, and a working level of competency in programming.

This upper-division text provides an unusually broad survey of the topics of modern computational physics from a multidisciplinary, computational science point of view. Its philosophy is rooted in learning by doing (assisted by many model programs), with new scientific materials as well as with the Python programming language. Python has become very popular, particularly for physics education and large scientific projects. It is probably the easiest programming language to learn for beginners, yet is also used for mainstream scientific computing, and has packages for excellent graphics and even symbolic manipulations.

The text is designed for an upper-level undergraduate or beginning graduate course and provides the reader with the essential knowledge to understand computational tools and mathematical methods well enough to be successful. As part of the teaching of using computers to solve scientific problems, the reader is encouraged to work through a sample problem stated at the beginning of each chapter or unit, which involves studying the text, writing, debugging and running programs, visualizing the results, and the expressing in words what has been done and what can be concluded. Then there are exercises and problems at the end of each chapter for the reader to work on their own (with model programs given for that purpose).
 

Author Biography

Rubin H. Landau is Professor Emeritus in the Department of Physics at Oregon State University in Corvallis. He has been teaching courses in computational physics for over 25 years, was a founder of the Computational Physics Degree Program and the Northwest Alliance for Computational Science and Engineering, and has been using computers in theoretical physics research ever since graduate school. He is author of more than 90 refereed publications and has also authored books on Quantum Mechanics, Workstations and Supercomputers, the first two editions of Computational Physics, and a First Course in Scientific Computing.

Manuel J. Paez is a professor in the Department of Physics at the University of Antioquia in Medellin, Colombia. He has been teaching courses in Modern Physics, Nuclear Physics, Computational Physics, Mathematical Physics as well as programming in Fortran, Pascal and C languages. He and Professor Landau have conducted pioneering computational investigations in the interactions of mesons and nucleons with nuclei.

Cristian C. Bordeianu teaches Physics and Computer Science at the Military College "?tefan cel Mare" in Campulung Moldovenesc, Romania. He has over twenty years of experience in developing educational software for high school and university curricula. He is winner of the 2008 Undergraduate Computational Engineering and Science Award by the US Department of Energy and the Krell Institute. His current research interests include chaotic dynamics in nuclear multifragmentation and plasma of quarks and gluons.

Table of Contents

Dedication V

Preface XIX

1 Introduction 1

1.1 Computational Physics and Computational Science 1

1.2 This Book’s Subjects 3

1.3 This Book’s Problems 4

1.4 This Book’s Language: The Python Ecosystem 8

1.4.1 Python Packages (Libraries) 9

1.4.2 This Book’s Packages 10

1.4.3 The EasyWay: Python Distributions (Package Collections) 12

1.5 Python’s Visualization Tools 13

1.5.1 Visual (VPython)’s 2D Plots 14

1.5.2 VPython’s Animations 17

1.5.3 Matplotlib’s 2D Plots 17

1.5.4 Matplotlib’s 3D Surface Plots 22

1.5.5 Matplotlib’s Animations 24

1.5.6 Mayavi’s Visualizations Beyond Plotting 26

1.6 Plotting Exercises 30

1.7 Python’s Algebraic Tools 31

2 Computing Software Basics 33

2.1 Making Computers Obey 33

2.2 ProgrammingWarmup 35

2.2.1 Structured and Reproducible Program Design 36

2.2.2 Shells, Editors, and Execution 37

2.3 Python I/O 39

2.4 Computer Number Representations (Theory) 40

2.4.1 IEEE Floating-Point Numbers 41

2.4.2 Python and the IEEE 754 Standard 47

2.4.3 Over and Underflow Exercises 48

2.4.4 Machine Precision (Model) 49

2.4.5 Experiment: Your Machine’s Precision 50

2.5 Problem: Summing Series 51

2.5.1 Numerical Summation (Method) 51

2.5.2 Implementation and Assessment 52

3 Errors and Uncertainties in Computations 53

3.1 Types of Errors (Theory) 53

3.1.1 Model for Disaster: Subtractive Cancelation 55

3.1.2 Subtractive Cancelation Exercises 56

3.1.3 Round-off Errors 57

3.1.4 Round-off Error Accumulation 58

3.2 Error in Bessel Functions (Problem) 58

3.2.1 Numerical Recursion (Method) 59

3.2.2 Implementation and Assessment: Recursion Relations 61

3.3 Experimental Error Investigation 62

3.3.1 Error Assessment 65

4 Monte Carlo: Randomness, Walks, and Decays 69

4.1 Deterministic Randomness 69

4.2 Random Sequences (Theory) 69

4.2.1 Random-Number Generation (Algorithm) 70

4.2.2 Implementation: Random Sequences 72

4.2.3 Assessing Randomness and Uniformity 73

4.3 RandomWalks (Problem) 75

4.3.1 Random-Walk Simulation 76

4.3.2 Implementation: RandomWalk 77

4.4 Extension: Protein Folding and Self-Avoiding RandomWalks 79

4.5 Spontaneous Decay (Problem) 80

4.5.1 Discrete Decay (Model) 81

4.5.2 Continuous Decay (Model) 82

4.5.3 Decay Simulation with Geiger Counter Sound 82

4.6 Decay Implementation and Visualization 84

5 Differentiation and Integration 85

5.1 Differentiation 85

5.2 Forward Difference (Algorithm) 86

5.3 Central Difference (Algorithm) 87

5.4 Extrapolated Difference (Algorithm) 87

5.5 Error Assessment 88

5.6 Second Derivatives (Problem) 90

5.6.1 Second-Derivative Assessment 90

5.7 Integration 91

5.8 Quadrature as Box Counting (Math) 91

5.9 Algorithm: Trapezoid Rule 93

5.10 Algorithm: Simpson’s Rule 94

5.11 Integration Error (Assessment) 96

5.12 Algorithm: Gaussian Quadrature 97

5.12.1 Mapping Integration Points 98

5.12.2 Gaussian Points Derivation 99

5.12.3 Integration Error Assessment 100

5.13 Higher Order Rules (Algorithm) 103

5.14 Monte Carlo Integration by Stone Throwing (Problem) 104

5.14.1 Stone Throwing Implementation 104

5.15 Mean Value Integration (Theory and Math) 105

5.16 Integration Exercises 106

5.17 Multidimensional Monte Carlo Integration (Problem) 108

5.17.1 Multi Dimension Integration Error Assessment 109

5.17.2 Implementation: 10D Monte Carlo Integration 110

5.18 Integrating Rapidly Varying Functions (Problem) 110

5.19 Variance Reduction (Method) 110

5.20 Importance Sampling (Method) 111

5.21 von Neumann Rejection (Method) 111

5.21.1 Simple Random Gaussian Distribution 113

5.22 Nonuniform Assessment ¡Ñ 113

5.22.1 Implementation ¡Ñ 114

6 Matrix Computing 117

6.1 Problem 3: N–D Newton–Raphson; Two Masses on a String 117

6.1.1 Theory: Statics 118

6.1.2 Algorithm: Multidimensional Searching 119

6.2 Why Matrix Computing? 122

6.3 Classes ofMatrix Problems (Math) 122

6.3.1 Practical Matrix Computing 124

6.4 Python Lists as Arrays 126

6.5 Numerical Python (NumPy) Arrays 127

6.5.1 NumPy’s linalg Package 132

6.6 Exercise: TestingMatrix Programs 134

6.6.1 Matrix Solution of the String Problem 137

6.6.2 Explorations 139

7 Trial-and-Error Searching and Data Fitting 141

7.1 Problem 1: A Search for Quantum States in a Box 141

7.2 Algorithm: Trial-and-Error Roots via Bisection 142

7.2.1 Implementation: Bisection Algorithm 144

7.3 Improved Algorithm: Newton–Raphson Searching 145

7.3.1 Newton–Raphson with Backtracking 147

7.3.2 Implementation: Newton–Raphson Algorithm 148

7.4 Problem 2: Temperature Dependence ofMagnetization 148

7.4.1 Searching Exercise 150

7.5 Problem 3: Fitting An Experimental Spectrum 150

7.5.1 Lagrange Implementation, Assessment 152

7.5.2 Cubic Spline Interpolation (Method) 153

7.6 Problem 4: Fitting Exponential Decay 156

7.7 Least-Squares Fitting (Theory) 158

7.7.1 Least-Squares Fitting: Theory and Implementation 160

7.8 Exercises: Fitting Exponential Decay,Heat Flow andHubble’s Law 162

7.8.1 Linear Quadratic Fit 164

7.8.2 Problem 5: Nonlinear Fit to a Breit–Wigner 167

8 Solving Differential Equations: Nonlinear Oscillations 171

8.1 Free Nonlinear Oscillations 171

8.2 Nonlinear Oscillators (Models) 171

8.3 Types of Differential Equations (Math) 173

8.4 Dynamic Form for ODEs (Theory) 175

8.5 ODE Algorithms 177

8.5.1 Euler’s Rule 177

8.6 Runge–Kutta Rule 178

8.7 Adams–Bashforth–Moulton Predictor–Corrector Rule 183

8.7.1 Assessment: rk2 vs. rk4 vs. rk45 185

8.8 Solution for Nonlinear Oscillations (Assessment) 187

8.8.1 Precision Assessment: Energy Conservation 188

8.9 Extensions: Nonlinear Resonances, Beats, Friction 189

8.9.1 Friction (Model) 189

8.9.2 Resonances and Beats: Model, Implementation 190

8.10 Extension: Time-Dependent Forces 190

9 ODE Applications: Eigenvalues, Scattering, and Projectiles 193

9.1 Problem: Quantum Eigenvalues in Arbitrary Potential 193

9.1.1 Model: Nucleon in a Box 194

9.2 Algorithms: Eigenvalues via ODE Solver + Search 195

9.2.1 Numerov Algorithm for Schrödinger ODE ¡Ñ 197

9.2.2 Implementation: Eigenvalues viaODESolver + BisectionAlgorithm 200

9.3 Explorations 203

9.4 Problem: Classical Chaotic Scattering 203

9.4.1 Model and Theory 204

9.4.2 Implementation 206

9.4.3 Assessment 207

9.5 Problem: Balls Falling Out of the Sky 208

9.6 Theory: Projectile Motion with Drag 208

9.6.1 Simultaneous Second-Order ODEs 209

9.6.2 Assessment 210

9.7 Exercises: 2- and 3-Body Planet Orbits and ChaoticWeather 211

10 High-Performance Hardware and Parallel Computers 215

10.1 High-Performance Computers 215

10.2 Memory Hierarchy 216

10.3 The Central Processing Unit 219

10.4 CPU Design: Reduced Instruction Set Processors 220

10.5 CPU Design:Multiple-Core Processors 221

10.6 CPU Design: Vector Processors 222

10.7 Introduction to Parallel Computing 223

10.8 Parallel Semantics (Theory) 224

10.9 Distributed Memory Programming 226

10.10 Parallel Performance 227

10.10.1 Communication Overhead 229

10.11 Parallelization Strategies 230

10.12 Practical Aspects of MIMD Message Passing 231

10.12.1 High-Level View ofMessage Passing 233

10.12.2 Message Passing Example and Exercise 234

10.13 Scalability 236

10.13.1 Scalability Exercises 238

10.14 Data Parallelism and Domain Decomposition 239

10.14.1 Domain Decomposition Exercises 242

10.15 Example: The IBM Blue Gene Supercomputers 243

10.16 Exascale Computing via Multinode-Multicore GPUs 245

11 Applied HPC: Optimization, Tuning, and GPU Programming 247

11.1 General Program Optimization 247

11.1.1 Programming for Virtual Memory (Method) 248

11.1.2 Optimization Exercises 249

11.2 Optimized Matrix Programming with NumPy 251

11.2.1 NumPy Optimization Exercises 254

11.3 Empirical Performance of Hardware 254

11.3.1 Racing Python vs. Fortran/C 255

11.4 Programming for the Data Cache (Method) 262

11.4.1 Exercise 1: Cache Misses 264

11.4.2 Exercise 2: Cache Flow 264

11.4.3 Exercise 3: Large-Matrix Multiplication 265

11.5 Graphical Processing Units for High Performance Computing 266

11.5.1 The GPU Card 267

11.6 Practical Tips forMulticore and GPU Programming ¡Ñ 267

11.6.1 CUDA Memory Usage 270

11.6.2 CUDA Programming ¡Ñ 271

12 Fourier Analysis: Signals and Filters 275

12.1 Fourier Analysis of Nonlinear Oscillations 275

12.2 Fourier Series (Math) 276

12.2.1 Examples: Sawtooth and Half-Wave Functions 278

12.3 Exercise: Summation of Fourier Series 279

12.4 Fourier Transforms (Theory) 279

12.5 The Discrete Fourier Transform 281

12.5.1 Aliasing (Assessment) 285

12.5.2 Fourier Series DFT (Example) 287

12.5.3 Assessments 288

12.5.4 Nonperiodic Function DFT (Exploration) 290

12.6 Filtering Noisy Signals 290

12.7 Noise Reduction via Autocorrelation (Theory) 290

12.7.1 Autocorrelation Function Exercises 293

12.8 Filtering with Transforms (Theory) 294

12.8.1 Digital Filters:Windowed Sinc Filters (Exploration) ¡Ñ 296

12.9 The Fast Fourier Transform Algorithm ¡Ñ 299

12.9.1 Bit Reversal 301

12.10 FFT Implementation 303

12.11 FFT Assessment 304

13 Wavelet and Principal Components Analyses: Nonstationary Signals and Data Compression 307

13.1 Problem: Spectral Analysis of Nonstationary Signals 307

13.2 Wavelet Basics 307

13.3 Wave Packets and Uncertainty Principle (Theory) 309

13.3.1 Wave Packet Assessment 311

13.4 Short-Time Fourier Transforms (Math) 311

13.5 TheWavelet Transform 313

13.5.1 GeneratingWavelet Basis Functions 313

13.5.2 ContinuousWavelet Transform Implementation 316

13.6 DiscreteWavelet Transforms,Multiresolution Analysis ¡Ñ 317

13.6.1 Pyramid Scheme Implementation ¡Ñ 323

13.6.2 DaubechiesWavelets via Filtering 327

13.6.3 DWT Implementation and Exercise 330

13.7 Principal Components Analysis 332

13.7.1 Demonstration of Principal Component Analysis 334

13.7.2 PCA Exercises 337

14 Nonlinear Population Dynamics 339

14.1 Bug Population Dynamics 339

14.2 The Logistic Map (Model) 339

14.3 Properties of NonlinearMaps (Theory and Exercise) 341

14.3.1 Fixed Points 342

14.3.2 Period Doubling, Attractors 343

14.4 Mapping Implementation 344

14.5 Bifurcation Diagram (Assessment) 345

14.5.1 Bifurcation Diagram Implementation 346

14.5.2 Visualization Algorithm: Binning 347

14.5.3 Feigenbaum Constants (Exploration) 348

14.6 Logistic Map Random Numbers (Exploration) ¡Ñ 348

14.7 Other Maps (Exploration) 348

14.8 Signals of Chaos: Lyapunov Coefficient and Shannon Entropy ¡Ñ 349

14.9 Coupled Predator–PreyModels 353

14.10 Lotka–Volterra Model 354

14.10.1 Lotka–Volterra Assessment 356

14.11 Predator–Prey Chaos 356

14.11.1 Exercises 359

14.11.2 LVM with Prey Limit 359

14.11.3 LVM with Predation Efficiency 360

14.11.4 LVM Implementation and Assessment 361

14.11.5 Two Predators, One Prey (Exploration) 362

15 Continuous Nonlinear Dynamics 363

15.1 Chaotic Pendulum 363

15.1.1 Free Pendulum Oscillations 364

15.1.2 Solution as Elliptic Integrals 365

15.1.3 Implementation and Test: Free Pendulum 366

15.2 Visualization: Phase-Space Orbits 367

15.2.1 Chaos in Phase Space 368

15.2.2 Assessment in Phase Space 372

15.3 Exploration: Bifurcations of Chaotic Pendulums 374

15.4 Alternate Problem: The Double Pendulum 375

15.5 Assessment: Fourier/Wavelet Analysis of Chaos 377

15.6 Exploration: Alternate Phase-Space Plots 378

15.7 Further Explorations 379

16 Fractals and Statistical Growth Models 383

16.1 Fractional Dimension (Math) 383

16.2 The Sierpi¨½ski Gasket (Problem 1) 384

16.2.1 Sierpi¨½ski Implementation 384

16.2.2 Assessing Fractal Dimension 385

16.3 Growing Plants (Problem 2) 386

16.3.1 Self-Affine Connection (Theory) 386

16.3.2 Barnsley’s Fern Implementation 387

16.3.3 Self-Affinity in Trees Implementation 389

16.4 Ballistic Deposition (Problem 3) 390

16.4.1 Random Deposition Algorithm 390

16.5 Length of British Coastline (Problem 4) 391

16.5.1 Coastlines as Fractals (Model) 392

16.5.2 Box Counting Algorithm 392

16.5.3 Coastline Implementation and Exercise 393

16.6 Correlated Growth, Forests, Films (Problem 5) 395

16.6.1 Correlated Ballistic Deposition Algorithm 395

16.7 Globular Cluster (Problem 6) 396

16.7.1 Diffusion-Limited Aggregation Algorithm 396

16.7.2 Fractal Analysis of DLA or a Pollock 399

16.8 Fractals in Bifurcation Plot (Problem 7) 400

16.9 Fractals from Cellular Automata 400

16.10 Perlin Noise Adds Realism ¡Ñ 402

16.10.1 Ray Tracing Algorithms 404

16.11 Exercises 407

17 Thermodynamic Simulations and Feynman Path Integrals 409

17.1 Magnets via Metropolis Algorithm 409

17.2 An IsingChain (Model) 410

17.3 Statistical Mechanics (Theory) 412

17.3.1 Analytic Solution 413

17.4 Metropolis Algorithm 413

17.4.1 Metropolis Algorithm Implementation 416

17.4.2 Equilibration, Thermodynamic Properties (Assessment) 417

17.4.3 Beyond Nearest Neighbors, 1D (Exploration) 419

17.5 Magnets viaWang–Landau Sampling ¡Ñ 420

17.6 Wang–Landau Algorithm 423

17.6.1 WLS IsingModel Implementation 425

17.6.2 WLS IsingModel Assessment 428

17.7 Feynman Path Integral Quantum Mechanics ¡Ñ 429

17.8 Feynman’s Space–Time Propagation (Theory) 429

17.8.1 Bound-StateWave Function (Theory) 431

17.8.2 Lattice Path Integration (Algorithm) 432

17.8.3 Lattice Implementation 437

17.8.4 Assessment and Exploration 440

17.9 Exploration: Quantum Bouncer’s Paths ¡Ñ 440

18 Molecular Dynamics Simulations 445

18.1 Molecular Dynamics (Theory) 445

18.1.1 Connection to Thermodynamic Variables 449

18.1.2 Setting Initial Velocities 449

18.1.3 Periodic Boundary Conditions and Potential Cutoff 450

18.2 Verlet and Velocity–Verlet Algorithms 451

18.3 1D Implementation and Exercise 453

18.4 Analysis 456

19 PDE Reviewand Electrostatics via Finite Differences and Electrostatics via Finite Differences 461

19.1 PDE Generalities 461

19.2 Electrostatic Potentials 463

19.2.1 Laplace’s Elliptic PDE (Theory) 463

19.3 Fourier Series Solution of a PDE 464

19.3.1 Polynomial Expansion as an Algorithm 466

19.4 Finite-Difference Algorithm 467

19.4.1 Relaxation and Over-relaxation 469

19.4.2 Lattice PDE Implementation 470

19.5 Assessment via Surface Plot 471

19.6 Alternate Capacitor Problems 471

19.7 Implementation and Assessment 474

19.8 Electric Field Visualization (Exploration) 475

19.9 Review Exercise 476

20 Heat Flow via Time Stepping 477

20.1 Heat Flow via Time-Stepping (Leapfrog) 477

20.2 The Parabolic Heat Equation (Theory) 478

20.2.1 Solution: Analytic Expansion 478

20.2.2 Solution: Time Stepping 479

20.2.3 von Neumann Stability Assessment 481

20.2.4 Heat Equation Implementation 483

20.3 Assessment and Visualization 483

20.4 Improved Heat Flow: Crank–Nicolson Method 484

20.4.1 Solution of Tridiagonal Matrix Equations ¡Ñ 487

20.4.2 Crank–Nicolson Implementation, Assessment 490

21 Wave Equations I: Strings and Membranes 491

21.1 A Vibrating String 491

21.2 The HyperbolicWave Equation (Theory) 491

21.2.1 Solution via Normal-Mode Expansion 493

21.2.2 Algorithm: Time Stepping 494

21.2.3 Wave Equation Implementation 496

21.2.4 Assessment, Exploration 497

21.3 Strings with Friction (Extension) 499

21.4 Strings with Variable Tension and Density 500

21.4.1 Waves on Catenary 501

21.4.2 Derivation of Catenary Shape 501

21.4.3 Catenary and FrictionalWave Exercises 503

21.5 Vibrating Membrane (2DWaves) 504

21.6 Analytical Solution 505

21.7 Numerical Solution for 2DWaves 508

22 Wave Equations II: QuantumPackets and Electromagnetic 511

22.1 Quantum Wave Packets 511

22.2 Time-Dependent Schrödinger Equation (Theory) 511

22.2.1 Finite-Difference Algorithm 513

22.2.2 Wave Packet Implementation, Animation 514

22.2.3 Wave Packets in OtherWells (Exploration) 516

22.3 Algorithm for the 2D Schrödinger Equation 517

22.3.1 Exploration: Bound and Diffracted 2D Packet 518

22.4 Wave Packet–Wave Packet Scattering 518

22.4.1 Algorithm 520

22.4.2 Implementation 520

22.4.3 Results and Visualization 522

22.5 E&MWaves via Finite-Difference Time Domain 525

22.6 Maxwell’s Equations 525

22.7 FDTD Algorithm 526

22.7.1 Implementation 530

22.7.2 Assessment 530

22.7.3 Extension: Circularly PolarizedWaves 531

22.8 Application:Wave Plates 533

22.9 Algorithm 534

22.10 FDTD Exercise and Assessment 535

23 Electrostatics via Finite Elements 537

23.1 Finite-Element Method ¡Ñ 537

23.2 Electric Field from Charge Density (Problem) 538

23.3 Analytic Solution 538

23.4 Finite-Element (Not Difference) Methods, 1D 539

23.4.1 Weak Form of PDE 539

23.4.2 Galerkin Spectral Decomposition 540

23.5 1D FEMImplementation and Exercises 544

23.5.1 1D Exploration 547

23.6 Extension to 2D Finite Elements 547

23.6.1 Weak Form of PDE 548

23.6.2 Galerkin’s Spectral Decomposition 548

23.6.3 Triangular Elements 549

23.6.4 Solution as Linear Equations 551

23.6.5 Imposing Boundary Conditions 552

23.6.6 FEM2D Implementation and Exercise 554

23.6.7 FEM2D Exercises 554

24 Shocks Waves and Solitons 555

24.1 Shocks and Solitons in ShallowWater 555

24.2 Theory: Continuity and Advection Equations 556

24.2.1 Advection Implementation 558

24.3 Theory: ShockWaves via Burgers’ Equation 559

24.3.1 Lax–Wendroff Algorithm for Burgers’ Equation 560

24.3.2 Implementation and Assessment of Burgers’ Shock Equation 561

24.4 Including Dispersion 562

24.5 Shallow-Water Solitons: The KdeV Equation 563

24.5.1 Analytic Soliton Solution 563

24.5.2 Algorithm for KdeV Solitons 564

24.5.3 Implementation: KdeV Solitons 565

24.5.4 Exploration: Solitons in Phase Space, Crossing 567

24.6 Solitons on Pendulum Chain 567

24.6.1 Including Dispersion 568

24.6.2 Continuum Limit, the Sine-Gordon Equation 570

24.6.3 Analytic SGE Solution 571

24.6.4 Numeric Solution: 2D SGE Solitons 571

24.6.5 2D Soliton Implementation 573

24.6.6 SGE Soliton Visualization 574

25 Fluid Dynamics 575

25.1 River Hydrodynamics 575

25.2 Navier–Stokes Equation (Theory) 576

25.2.1 Boundary Conditions for Parallel Plates 578

25.2.2 Finite-Difference Algorithm and Overrelaxation 580

25.2.3 Successive Overrelaxation Implementation 581

25.3 2D Flow over a Beam 581

25.4 Theory: Vorticity Form of Navier–Stokes Equation 582

25.4.1 Finite Differences and the SOR Algorithm 584

25.4.2 Boundary Conditions for a Beam 585

25.4.3 SOR on a Grid 587

25.4.4 Flow Assessment 589

25.4.5 Exploration 590

26 Integral Equations of QuantumMechanics 591

26.1 Bound States of Nonlocal Potentials 591

26.2 Momentum–Space Schrödinger Equation (Theory) 592

26.2.1 Integral toMatrix Equations 593

26.2.2 Delta-Shell Potential (Model) 595

26.2.3 Binding Energies Solution 595

26.2.4 Wave Function (Exploration) 597

26.3 Scattering States of Nonlocal Potentials ¡Ñ 597

26.4 Lippmann–Schwinger Equation (Theory) 598

26.4.1 Singular Integrals (Math) 599

26.4.2 Numerical Principal Values 600

26.4.3 Reducing Integral Equations to Matrix Equations (Method) 600

26.4.4 Solution via Inversion, Elimination 602

26.4.5 Scattering Implementation 603

26.4.6 ScatteringWave Function (Exploration) 604

Appendix A Codes, Applets, and Animations 607

Bibliography 609

Index 615