Computational Physics: Problem Solving with Computers Book

1 Introduction.

1.1 Computational Physics and Computational Science.

1.2 How to Use this Book.

2 Computing Software Basics.

2.1 Making Computers Obey.

2.2 Computer Languages.

2.3 Programming Warmup.

2.4 Shells, Editors, and Programs.

2.5 Limited Range and Precision of Numbers.

2.6 Number Representation.

2.7 IEEE Floating Point Numbers.

2.8 Over/Underflows Exercise.

2.9 Machine Precision.

2.10 Determine Your Machine Precision.

2.11 Structured Program Design.

2.12 Summing Series.

2.13 Numeric Summation.

2.14 Good and Bad Pseudocode.

2.15 Assessment.

3 Errors and Uncertainties in Computations.

3.1 Living with Errors.

3.2 Types of Errors.

3.3 Model for Disaster: Subtractive Cancellation.

3.4 Subtractive Cancellation Exercises.

3.5 Model for Roundoff Error Accumulation.

3.6 Errors in Spherical Bessel Functions (Problem).

3.7 Numeric Recursion Relations (Method).

3.8 Implementation and Assessment: Recursion Relations.

3.9 Experimental Error Determination.

3.10 Errors in Algorithms.

3.11 Minimizing the Error.

3.12 Error Assessment.

4 Object-Oriented Programming: Kinematics.

4.1 Problem: Superposition of Motions.

4.2 Theory: Object-Oriented Programming.

4.3 Theory: Newton’s Laws, Equation of Motion.

4.4 OOP Method: Class Structure.

4.5 Implementation: Uniform 1D Motion, unim1d.cpp.

4.6 Assessment: Exploration, shms.cpp.

5 Integration.

5.1 Problem: Integrating a Spectrum.

5.2 Quadrature as Box Counting (Math).

5.3 Algorithm: Trapezoid Rule.

5.4 Algorithm: Simpson’s Rule.

5.5 Integration Error.

5.6 Algorithm: Gaussian Quadrature.

5.7 Empirical Error Estimate (Assessment).

5.8 Experimentation.

5.9 Higher Order Rules.

6 Differentiation.

6.1 Problem 1: Numerical Limits.

6.2 Method: Numeric.

6.3 Forward Difference.

6.4 Central Difference.

6.5 Extrapolated Difference.

6.6 Error Analysis.

6.7 Error Analysis (Implementation and Assessment).

6.8 Second Derivatives.

7 Trial and Error Searching.

7.1 Quantum States in SquareWell.

7.2 Trial-and-Error Root Finding via Bisection Algorithm.

7.2.1 Bisection Algorithm Implementation.

7.3 Newton–Raphson Algorithm.

8 Matrix Computing and N-D Newton Raphson.

8.1 Two Masses on a String.

8.2 Classes of Matrix Problems.

9 Data Fitting.

9.1 Fitting Experimental Spectrum.

9.2 Fitting Exponential Decay.

9.3 Theory: Probability and Statistics.

9.4 Least-Squares Fitting.

9.5 Appendix: Calling LAPACK from C.

10 Deterministic Randomness.

10.1 Random Sequences.

11 Monte Carlo Applications.

11.1 A RandomWalk.

11.2 Radioactive Decay.

11.3 Implementation and Visualization.

11.4 Integration by Stone Throwing.

11.5 Integration by Rejection.

11.6 High-Dimensional Integration.

11.7 Integrating Rapidly Varying Functions.

12 Thermodynamic Simulations: Ising Model.

12.1 Statistical Mechanics.

12.2 An Ising Chain (Model).

12.3 The Metropolis Algorithm.

13 Computer Hardware Basics: Memory and CPU.

13.1 High-Performance Computers.

13.2 The Central Processing Unit.

14 High-Performance Computing: Profiling and Tuning.

14.1 Rules for Optimization.

14.2 Programming for Data Cache.

15 Differential Equation Applications.

15.1 UNIT I. Free Nonlinear Oscillations.

15.2 Nonlinear Oscillator.

15.3 Math: Types of Differential Equations.

15.4 Dynamical Form for ODEs.

15.5 ODE Algorithms.

15.6 Solution for Nonlinear Oscillations.

15.7 Extensions: Nonlinear Resonances, Beats and Friction.

15.8 Implementation: Inclusion of Time-Dependent Force.

15.9 UNIT II. Balls, not Planets, Fall Out of the Sky.

15.10 Theory: Projectile Motion with Drag.

15.11 Exploration: Planetary Motion.

16 Quantum Eigenvalues via ODE Matching.

16.1 Theory: The Quantum Eigenvalue Problem.

17 Fourier Analysis of Linear and Nonlinear Signals.

17.1 Harmonics of Nonlinear Oscillations.

17.2 Fourier Analysis.

17.3 Summation of Fourier Series(Exercise).

17.4 Fourier Transforms.

17.5 Discrete Fourier Transform Algorithm (DFT).

17.6 Aliasing and Antialiasing.

17.7 DFT for Fourier Series.

17.8 Assessments.

17.9 DFT of Nonperiodic Functions (Exploration).

17.10 Model Independent Data Analysis.

17.11 Assessment.

18 Unusual Dynamics of Nonlinear Systems.

18.1 The Logistic Map.

18.2 Properties of Nonlinear Maps.

18.2.1 Fixed Points.

18.2.2 Period Doubling, Attractors.

18.3 Explicit Mapping Implementation.

18.4 Bifurcation Diagram.

18.4.1 Implementation.

18.4.2 Visualization Algorithm: Binning.

18.5 Random Numbers via Logistic Map.

18.6 Feigenbaum Constants.

18.7 Other Maps.

19 Differential Chaos in Phase Space.

19.1 Problem: A Pendulum Becomes Chaotic (Differential Chaos).

19.2 Equation of Chaotic Pendulum.

19.3 Visualization: Phase-Space Orbits.

19.4 Assessment: Fourier Analysis of Chaos.

19.5 Exploration: Bifurcations in Chaotic Pendulum.

19.6 Exploration: Another Type of Phase-Space Plot.

19.7 Further Explorations.

20 Fractals.

20.1 Fractional Dimension.

20.2 The Sierpi ´ nski Gasket.

20.3 Beautiful Plants.

20.4 Ballistic Deposition.

20.5 Length of British Coastline.

20.6 Problem 5: Correlated Growth, Forests, and Films.

20.7 Problem 7: Fractals in Bifurcation Graph.

21 Parallel Computing.

21.1 Parallel Semantics.

21.2 Distributed Memory Programming.

21.3 Parallel Performance.

22 Parallel Computing with MPI.

22.1 Running on a Beowulf.

22.2 Running MPI.

22.3 Parallel Tuning: TuneMPI.c.

22.4 A String Vibrating in Parallel.

22.5 Deadlock.

22.6 Supplementary Exercises.

22.7 List of MPI Commands.

23 Electrostatics Potentials via Finite Differences (PDEs).

23.1 PDE Generalities.

23.2 Electrostatic Potentials.

23.3 Fourier Series Solution of PDE.

23.4 Solution: Finite DifferenceMethod.

23.5 Assessment via Surface Plot.

23.6 Three Alternate Capacitor Problems.

23.7 Implementation and Assessment.

23.8 Other Geometries and Boundary Conditions.

24 Heat Flow.

24.1 The Parabolic Heat Equation.

24.2 Solution: Analytic Expansion.

24.3 Solution: Finite Time Stepping (Leap Frog).

24.4 von Neumann Stability Assessment.

24.5 Assessment and Visualization.

25 PDE Waves on Strings and Membranes.

25.1 The Hyperbolic Wave Equation.

25.2 Realistic 1DWave Exercises.

25.3 Vibrating Membrane (2DWaves).

25.4 Analytical Solution.,

25.5 Numerical Solution for 2DWaves.

26 Solitons; KdeV and Sine-Gordon.

26.1 Chain of Coupled Pendulums (Theory).

26.2 Wave Dispersion.

26.3 Analytic SGE Solution.

26.4 Numeric Solution: 2D SGE Solitons.

26.5 2D Soliton Implementation.

26.6 Visualization.

26.7 Shallow Water (KdeV) Solitons.

26.8 Theory: The Korteweg–de Vries Equation.

27 Quantum Wave Packets.

27.1 Time-Dependent Schrödinger Equation (Theory).

27.2 Wave Packets Confined to OtherWells (Exploration).

28 Quantum Paths for Functional Integration.

28.1 Feynman’s Space–Time Propagation.

29 Quantum Bound States via Integral Equations.

29.1 Momentum–Space Schrödinger Equation.

30 Quantum Scattering via Integral Equations.

30.1 Lippmann–Schwinger Equation.

A PtPlot: 2D Graphs within Java.

B Glossary.

C Fortran 95 Codes.

D Fortran 77 Codes.

E C Language Codes.

References.

Index.