Computational Physics II

FSU Jena, SS 2021

Lecture period: 12.04.2021 - 16.07.2021

Last updated: 13.03.2021

Lectures will be online using Zoom

Topics

• Programming

  • Unix & Linux
  • Bash & Console navigation
  • Programming languages
  • Introduction to Python
  • Floating point representation

• Solving equations and Linear systems

  • Root finding: Bisection, Newton-Raphson, Secant
  • Direct methods: Gauss elimination, LU decomposition
  • Iterative methods: Jacobi, Gauss-Seidel, Successive-Over-Relaxation

• Function representation, integrals, derivatives

  • Polynomial approximation of function, interpolant
  • Runge phenomenon
  • Integrals: Riemann sums, Trapezoidal rule, Gaussian quadratures
  • Spectral methods: Orthogonal polynomial, Fourier basis
  • Interpolation: Lagrangian, Splines
  • Derivatives: finite differencing, spectral methods

• Ordinary Differential Equations

  • Initial Value Problems: well-posedness, stability and convergence
  • Euler scheme: stable/unstable vs explicit/implicit
  • Stability analaysis of ODE solvers
  • Runge-Kutta methods
  • Simplecting method: symplectic Euler, Störmer&Verlet
  • Boundary Value Problems: shooting method, solution with methods for linear systems

• Fourier Transform

  • Discrete Fourier Transform
  • Fast Fourier Transform
  • Solution of BVPs via FFT
  • Signal analysis

• Monte Carlo methods

  • Random variables and pseudorandom numbers
  • Multidimensional integration with MC
  • Random walks and Metropolis algorithm

Literature

Books & Lecture notes

• Numerical Recipes in C (2Nd Ed.): The Art of Scientific Computing. William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Cambridge University Press, New York, NY, USA, 1992.
• Geometric Numerical Integration, Hairer, Lubich & Wanne, Springer, 2006.
• Approximation Theory and Approximation Practice, Trefethen, SIAM, 2013
• Exploring ODEs Nick Trefethen
• Nick Trefethen Books
• Chebyshev and Fourier Spectral Methods. John, P. Boyd, Dover Publisher, 2000.
• Morten Hjorth-Jensen lectures notes
• Demmel’s lecture on floating point
• An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
• Richard Fitzpatrick’s lecture notes
• Lecture notes (available on moodle)
• Lecture notes of the previous course (CPI) (German)

Top-10 algorithms

From Nick Higham’s The Top 10 Algorithms in Applied Mathematics (see e.g. this page for PDF links). See also Dongarra & Sullivan’s The Top 10 Algorithms

• Newton and quasi-Newton methods (Broyden (1965))
• Matrix factorizations (LU, Cholesky, QR) (Householder (1958))
• Singular value decomposition, QR and QZ algorithms (Francis (1959)
• Monte-Carlo methods (Metropolis et al (1953))
• Fast Fourier transform (Cooley & Tukey (1965))
• Krylov subspace methods (conjugate gradients, Lanczos, GMRES, minres) (Hestenes & Stiefel (1952))
• JPEG (Wallace(1992))
• PageRank
• Simplex algorithm
• Kalman filter

Web resources and Software

• Python Python is a programming language that lets you work quickly and integrate systems more effectively.
• Jupyter
• NIST Digital Library of Mathematical Functions
• GNU Scientific Library
• Mathematics Source Library
• Python tutorial (One of the many)
• The computational saloon, M. Vallisneri,
• MMA tutorial on iterative methods for linear systems
• WikiBook on numerical methods
• Scholarpedia tutorial on PDE
• Introduction to neural networks by Christos Stergiou and Dimitrios Siganos
• Tensor Flow

Miscellenea

• Arianna Rosenbluth and the Metropolis Monte Carlo Algorithm
• The Fermi-Pasta-Ulam “numerical experiment”: history and pedagogical perspectives Original report

Tutorials

• Gymnastic with bash and Python
• Roots of complex functions and fractals
• Tridiagonal system solver: direct vs iterative methods
• Lagrangian Interpolation
• Finite differencing
• Kepler problem
• Ising 2D model