METHODS OF COMPUTATIONAL PHYSICS

• Monte Carlo (MC) simulation of spins--Ising model
  • Numerical vs. MC integration: Simpson's rule, Gaussian quadrature (orthogonal functions--recursive function evaluation, generating function)
  • Probability: Importance sampling, Markov chain, Metropolis algorithm
  • Random number generation (RNG)
  • Statistics: Variance, standard deviation, standard deviation of the MC mean
  • Cluster analysis: Graphs, search, stack
• MC simulation of stock price--geometric Brownian motion
  • Random walk: Einstein's law, central-limit theorem
  • Random variable: Black-Scholes analysis
  • Coordinate transformation: Jacobian, Box-Muller algorithm for RNG of normal distribution
  • Interpolation: Least square fit of data
  • Quantum MC and kinetic MC simulations
• Molecular dynamics (MD) simulation of particles--Newton's second law of motion
  • Numerical differentiation
  • Ordinary differential equation (ODE): Symplectic integrators
  • Minimization of functions: Conjugate gradient method
  • Hybrid MD/MC simulation
• Quantum dynamics simulation of an electron--time-dependent Schrodinger equation
  • Partial differential equation (PDE)
  • Fourier analysis: Spectral analysis, fast Fourier transform (FFT)
• Electronic structures of molecules--quantum mechanical eigenvalue problem
  • Linear algebra: Matrix, orthogonal transformation, rank, singular value decomposition, Krylov subspace
  • Matrix eigensystems: Housholder transformation, QL decomposition
  • Root finding: Newton-Raphson method