CSE/MATH 551: Numerical Solutions of Ordinary Differential Equations Fall 2006

MWF 01:25PM - 02:15PM 116 EE WEST

Instructor: Xiantao Li

• Office: 219C McAllister
• Phone: 3-9081
• Email: xli AT math DOT psu DOT edu
• Webpage: http://www.math.psu.edu/~xli
• Office hours: Tue 1:00-2:00 PM, Wed 11:00-12:00 AM.

Course description:

• General methods for solution of ODEs (27 lectures),
  1. stability of system of ODEs,
  2. one step methods: Runge-Kutta and extrapolation methods,
  3. error estimate and stability analysis,
  4. multi-step methods: consistency, stability and weak stability,
  5. stiff ODEs: system with multiple time scales,
  6. boundary value problems,
• Special topics: molecular dynamics (15 lectures)
  1. introduction to molecular dynamics
  2. molecular dynamics and statistical mechanics
  3. atomistic representation of macroscale quantities e.g. temperature, pressure, stress, viscosity etc.
  4. force calculation; Verlet list; periodic boundary condition; time integration; time/ensemble average; correction for long-range interactions; probe material structure with radial distribution functions,
  5. canonical/isothermal ensemble; static properties: equation of state; dynamic properties: correlation in time,
  6. constant pressure/stress (isobaric) simulations.

Textbook: P. Deuflhard and F. Bornemann, Scientific Computing with Ordinary Differential Equations (Required). Springer-Verlag, New York, 2002.

Other references (on reserve):

• J.D. Lambert, Numerical Methods for Ordinary Differential Systems, Wiley, Chichester,UK, 1991.
• Frenkel and Smit, Understanding Molecular Simulation, Academic Press.
• Allen and Tildesley, Computer simulation of liquids, Clarendon Press.

1. A molecular dynamics primer
2. Embedded Atom potential
3. Case studied in the book by Frenkel and Smit
4. Lecture notes from Prof. Tuckman

Homework: Homework will be assigned and collected every two weeks on Mondays. Some homework includes computer projects.

Final exam: There will be a take-home project for the final.

Homework Assignment

• Homework I
• Homework II
• Homework III
• Homework IV
• Homework V
• Final Project

Sample Fortran Programs

Fortran program: I

Micro-canonical simulation in Lennard-Jones material

Verlet's integrator

Verlet's neighbor list

Sampling the pressure and temperature

Velocity autocorrelation function (VACF)

Radial distribution function (RDF)

Fortran program: II

Canonical simulation in Lennard-Jones material

F.C.C. lattice as initial configuration

Andersen's thermostats

Fortran program: III

Canonical simulation in Lennard-Jones material

Nose-Hoover thermostat

Xiantao Li 2006-08-31