Tuesdays and Thursdays 9:30-10:45 (Meyer 639)
Instructor: Roman Scoccimarro
This is a *physics* course, in which we will use computational techniques to solve problems in physics.
Analytic approximation techniques will also be an important part of the course.
There will be no exams, grades will be based on homework.
You will need to (or learn how to) program (e.g. C, Fortran, Mathematica, MATLAB), use LaTeX (to write up each homework) and plotting software (e.g. gnuplot, pgplot, if you use C/Fortran).
Please return a brief summary of your work (including figures) as a printout generated by
LaTeX (see below for a template).
You should also hand in analytic calculations in
standard (handwritten) form, and send me the code you use to generate results by email.
There is no formal textbook that I will follow, although Numerical Recipes
can be very useful.
Other general books that you may want to check are,
A.L. Garcia, Numerical Methods for Physics, 2000, Prentice Hall
M.T. Heath, Scientific Computing: An Introductory Survey, 2002, McGraw Hill
A. Iserles, A First Course in the Numerical Analysis of Differential Equations,
1996, Cambridge University Press (more on the mathematical side)
R.H. Landau & M.J.Paez, Computational Physics, 1997, Wiley
T. Pang, An Introduction to Computational Physics, 1997, Cambridge
For analytic methods, see e.g.
B. Kusse & E. Westwig, Mathematical Physics, 1998, John Wiley & Sons
G. Arfken, Mathematical Methods for Physicists, 1985, Academic Press
C.M. Bender & S.A. Orzag, Advanced Mathematical Methods for
Scientists and Engineers, 1978, McGraw Hill
You can find a sample latex file to present your homework here.
There are many tutorials on LaTeX on the web, see e.g.
C, Fortran, Mathematica, MATLAB Help
There is a lot of useful material on the web, see e.g.
You can use Mathematica or MATLAB (see above) or the freely available GNUPLOT, see
Or if you want a graphics subroutine callable from C or Fortran, see e.g.
Basics of Numerics (2 weeks)
- Numerical Math: Roundoff error, representation of numbers, etc
- Interpolations and Approximations
- Computing Derivatives and Integrals
- Random Number Generators
Ordinary Differential Equations (2 weeks)
- Basic Methods: Euler, Runge-Kutta
- Implicit Methods
- Stiff ODE's, Stability.
- Applications: Resonances in the Solar System, Planetary motion in GR, Structure of Quantum Degenerate Stars
Spectral Methods (3 weeks)
- Random Gaussian Fields, Power Spectrum, Correlation Functions, Cumulants
- Fast Fourier Transform
- Windowed Fourier Transforms, Wavelets
- Applications: Spatial and Temporal Distributions: Analysis and Generation, Quantum evolution of wave packets in anharmonic potentials
Partial Differential Equations (3 weeks)
- Finite Differences
- Grid Methods: FFT, Relaxation, Multigrid
- Lax, Lax-Wendroff, Staggered Leapfrog Methods
- Methods of Characteristics
- Applications: Water Waves and Tsunamis, KdV Solitons, Traffic Problems
Renormalization Group and Monte Carlo (3 weeks)
- Basic Ideas of RG: Real Space and Momentum Shell
- Random Walks, Monte Carlo, Markov Chains
- Applications: Ising Model, Phase Transitions