Computational Physics

Transcription

1 Rubin H. Landau, Manuel J. Paez, and Cristian C. Bordeianu Computational Physics Problem Solving with Computers 2nd, Revised and Enlarged Edition BICENTENNIAL 1 8 O 7 WILEY 2 OO 7 ICINTINNIAL WILEY-VCH Verlag GmbH & Co. KGaA

2 I VI Contents 1 Introduction Computational Physics and Computational Science How to Use this Book 3 2 Computing Software Basics Making Computers Obey Computer Languages Programming Warmup Java-Scanner Implementation C Implementation Fortran Implementation Shells, Editors, and Programs Limited Range and Precision of Numbers Number Representation IEEE Floating Point Numbers Over/Underflows/Exercise Machine Precision Determine Your Machine Precision Structured Program Design Summing Series Numeric Summation Good and Bad Pseudocode Assessment 27 3 Errors and Uncertainties in Computations Living with Errors Types of Errors Model for Disaster: Subtractive Cancellation Subtractive Cancellation Exercises Model for Roundoff Error Accumulation 34 Computationyal Physics. Problem Solving with Computers (2nd edn). Rubin H. Landau, Manuel Jose Paez, Cristian C. Bordeianu Copyright 2007 WILEY-VCH Verlae GmbH & Co. KGaA. Weinheim

3 VIM I Contents 3.6 Errors in Spherical Bessel Functions (Problem) Numeric Recursion Relations (Method) Implementation and Assessment: Recursion Relations Experimental Error Determination Errors in Algorithms Minimizing the Error Error Assessment 42 4 Object-Oriented Programming: Kinematics Problem: Superposition of Motions Theory: Object-Oriented Programming OOP Fundamentals Theory: Newton's Laws, Equation of Motion OOP Method: Class Structure Implementation: Uniform ID Motion, unimld.cpp Uniform Motion in ID, Class UmlD Implementation: Uniform Motion in 2D/Child Um2D, unimot2d.cpp Class Um2D: Uniform Motion in 2D Implementation: Projectile Motion, Child Accm2D, accm2d.cpp Accelerated Motion in Two Directions Assessment: Exploration, shms.cpp 56 5 Integration Problem: Integrating a Spectrum Quadrature as Box Counting (Math) Algorithm: Trapezoid Rule Algorithm: Simpson's Rule Integration Error Algorithm: Gaussian Quadrature Mapping Integration Points Gauss Implementation Empirical Error Estimate (Assessment) Experimentation Higher Order Rules 72 6 Differentiation Problem 1: Numerical Limits Method: Numeric Forward Difference Central Difference Extrapolated Difference 77

4 Contents IX 6.6 Error Analysis Error Analysis (Implementation and Assessment) Second Derivatives Second Derivative Assessment 80 7 Trial and Error Searching Quantum States in Square Well Trial-and-Error Root Finding via Bisection Algorithm Bisection Algorithm Implementation Newton-Raphson Algorithm Newton-Raphson with Backtracking Newton-Raphson Implementation 87 8 Matrix Computing and N-D Newton Raphson Two Masses on a String Statics Multidimensional Newton-Raphson Searching Classes of Matrix Problems Practical Aspects of Matrix Computing Implementation: Scientific Libraries, WWW Exercises for Testing Matrix Calls Matrix Solution of Problem Explorations Data Fitting Fitting Experimental Spectrum Lagrange Interpolation Lagrange Implementation and Assessment Explore Extrapolation Cubic Splines Spline Fit of Cross Section Fitting Exponential Decay Theory to Fit Theory: Probability and Statistics Least-Squares Fitting Goodness of Fit Least-Squares Fits Implementation Exponential Decay Fit Assessment Exercise: Fitting Heat Flow Nonlinear Fit of Breit-Wigner to Cross Section Appendix: Calling LAPACK from C Calling LAPACK Fortran from C Compiling C Programs with Fortran Calls 234

5 X Contents 10 Deterministic Randomness Random Sequences Random-Number Generation Implementation: Random Sequence Assessing Randomness and Uniformity Monte Carlo Applications A Random Walk Simulation Implementation: Random Walk Radioactive Decay Discrete Decay Continuous Decay Simulation Implementation and Visualization Integration by Stone Throwing Integration by Rejection 253 ' Implementation Integration by Mean Value High-Dimensional Integration Multidimensional Monte Carlo Error in N-D Integration Implementation: 10D Monte Carlo Integration Integrating Rapidly Varying Functions Variance Reduction (Method) Importance Sampling Implementation: Nonuniform Randomness von Neumann Rejection Nonuniform Assessment Thermodynamic Simulations: Ising Model Statistical Mechanics An Ising Chain (Model) Analytic Solutions The Metropolis Algorithm Implementation Equilibration Thermodynamic Properties Beyond Nearest Neighbors and ID 277

6 Contents I XI 13 Computer Hardware Basics: Memory and CPU High-Performance Computers Memory Hierarchy The Central Processing Unit CPU Design: RISC Vector Processor High-Performance Computing: Profiling and Tuning Rules for Optimization Programming for Virtual Memory Optimizing Programs; Java vs. Fortran/C Good, Bad Virtual Memory Use Experimental Effects of Hardware on Performance Java versus Fortran/C Programming for Data Cache Exercise 1: Cache Misses Exercise 2: Cache Flow Exercise 3: Large Matrix Multiplication Differential Equation Applications UNIT I. Free Nonlinear Oscillations Nonlinear Oscillator Math: Types of Differential Equations A Dynamical Form for ODEs ODE Algorithms Euler'sRule Runge-Kutta Algorithm Assessment: rk2 v. rk4 v. rk Solution for Nonlinear Oscillations Precision Assessment: Energy Conservation Extensions: Nonlinear Resonances, Beats and Friction Friction: Model and Implementation Resonances and Beats: Model and Implementation Implementation: Inclusion of Time-Dependent Force UNIT II. Balls, not Planets, Fall Out of the Sky Theory: Projectile Motion with Drag Simultaneous Second Order ODEs Assessment Exploration: Planetary Motion Implementation: Planetary Motion 232

7 XIII Contents 16 Quantum Eigenvalues via ODE Matching Theory: The Quantum Eigenvalue Problem Model: Nucleon in a Box Algorithm: Eigenvalues via ODE Solver + Search Implementation: ODE Eigenvalues Solver Explorations Fourier Analysis of Linear and Nonlinear Signals Harmonics of Nonlinear Oscillations Fourier Analysis Example 1: Sawtooth Function Example 2: Half-Wave Function Summation of Fourier Series(Exercise) Fourier Transforms Discrete Fourier Transform Algorithm (DFT) Aliasing and Antialiasing DFT for Fourier Series 259 i 17.8 Assessments DFT of Nonperiodic Functions (Exploration) Model Independent Data Analysis Assessment Unusual Dynamics of Nonlinear Systems The Logistic Map Properties of Nonlinear Maps Fixed Points Period Doubling, Attractors Explicit Mapping Implementation Bifurcation Diagram Implementation Visualization Algorithm: Binning Random Numbers via Logistic Map Feigenbaum Constants Other Maps Differential Chaos in Phase Space Problem: A Pendulum Becomes Chaotic (Differential Chaos) Equation of Chaotic Pendulum Oscillations of a Free Pendulum Pendulum's "Solution" as Elliptic Integrals Implementation and Test: Free Pendulum Visualization: Phase-Space Orbits Chaos in Phase Space 285

8 Contents XIII Assessment in Phase Space A Assessment: Fourier Analysis of Chaos Exploration: Bifurcations in Chaotic Pendulum Exploration: Another Type of Phase-Space Plot Further Explorations Fractals Fractional Dimension The Sierpiriski Gasket Implementation Assessing Fractal Dimension Beautiful Plants Self-Affine Connection Barnsley's Fern (fern.c) Self-Affinity in Trees (tree.c) Ballistic Deposition Random Deposition Algorithm (film.c) Length of British Coastline Coastline as Fractal Box Counting Algorithm Coastline Implementation Problem 5: Correlated Growth, Forests, and Films Correlated Ballistic Deposition Algorithm (column.c) Globular Cluster Diffusion-Limited Aggregation Algorithm (dla.c) Fractal Analysis of DLA.Graph Problem 7: Fractal* in Bifurcation Graph Parallel Computing Parallel Semantics' Granularity Distributed Memory Programming Parallel Performance Communication Overhead Parallel Computing with MPI Running on a Beowulf An Alternative: BCCD = Your Cluster on a CD Running MPI MPI under a Queuing System Your First MPI Program MPIheUo.c Explained 330

9 XIVI Contents Send/Receive Messages Receive More Messages Broadcast Messages: MPIpi.c Exercise Parallel Tuning: TuneMPI.c A String Vibrating in Parallel A.I MPIstring.c Exercise Deadlock Nonblocking Communication Collective Communication Supplementary Exercises List of MPI Commands Electrostatics Potentials via Finite Differences (PDEs) PDE Generalities Electrostatic Potentials Laplace's Elliptic PDE 353, 23.3 Fourier Series Solution of PDE Shortcomings of Polynomial Expansions Solution: Finite Difference Method A.I Relaxation and Over-Relaxation Lattice PDE Implementation Assessment via Surface Plot Three Alternate Capacitor Problems Implementation and Assessment Other Geometries and Boundary Conditions Heat Flow The Parabolic Heat Equation Solution: Analytic Expansion Solution: Finite Time Stepping (Leap Frog) von Neumann Stability Assessment Implementation Assessment and Visualization PDE Waves on Strings and Membranes The Hyperbolic Wave Equation Solution via Normal Mode Expansion Algorithm: Time Stepping (Leapfrog) Implementation Assessment and Exploration Including Friction (Extension) 388

10 Contents XV Variable Tension and Density Realistic ID Wave Exercises Vibrating Membrane (2D Waves) Analytical Solution Numerical Solution for 2D Waves Solitons; KdeV and Sine-Gordon Chain of Coupled Pendulums (Theory) Wave Dispersion Continuum Limit, the SGE Analytic SGE Solution A Numeric Solution: 2D SGE Solitons D Soliton Implementation Visualization Shallow Water (KdeV) Solitons Theory: The Korteweg-de Vries Equation Analytic Solution: KdeV Solitons Algorithm: KdeV Soliton Solution Implementation: KdeV Solitons Exploration: Two KdeV Solitons Crossing Phase-Space Behavior Quantum Wave Packets Time-Dependent Schrodinger Equation (Theory) Finite Difference Solution Implementation Visualization and Animation Wave Packets Confined to Other Wells (Exploration) Algorithm for 2D Schrodinger Equation Quantum Paths for Functional Integration Feynman's Space-Time Propagation Bound-State Wave Function Lattice Path Integration (Algorithm) Implementation Assessment and Exploration Quantum Bound States via Integral Equations Momentum-Space Schrodinger Equation Integral to Linear Equations Delta-Shell Potential (Model) Implementation 448

11 XVII Contents 29.1 A Wave Function Quantum Scattering via Integral Equations Lippmann-Schwinger Equation Singular Integrals Numerical Principal Values Reducing Integral to Matrix Equations Solution via Inversion, Elimination Solving ie Integral Equations Delta-Shell Potential Implementation Scattering Wave Function 458 A PtPlot: 2D Graphs within Java 461 B Glossary 467 C Fortran 95 Codes 479 i D Fortran 77 Codes 513 E C Language Codes 547 References 583 Index 587