Computational Physics Problem Solving with Python
, by Landau, Rubin H.; Páez, Manuel J.; Bordeianu, Cristian C.
- ISBN: 9783527413157 | 3527413154
- Cover: Paperback
- Copyright: 9/8/2015
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).
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).
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.
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.
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
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