Numerical Methods for Theoretical Physicists II - NTMF058

Basic information:

in the year 2022/2023, the lecturer is Karel Houfek
lectures and practical courses take place from February 16 to May 19 from 2:50 pm in the lecture room of the Institute of Theoretical Physics

Materials to download

Notes and Mathematica notebooks used during the lectures and practical courses:

Iterative methods of numerical linear algebra
- stationary iterative methods (notes)
  - 1D Poisson equation (notebook, pdf)
  - example from Trefethen's textbook (notebook, pdf)
- multigrid method (notes)
  - 1D Poisson equation (notebook, pdf)
- gradient iterative methods (notes)
  - 1D Poisson equation (notebook, pdf)
  - example from Trefethen's textbook (notebook, pdf)
- eigenvalue problem (notes)
  - basic methods - power, inverse, Rayleigh quotient, QR (notebook pdf)
  - Jacobi diagonalization - LHO using a trigonometric basis (notebook, pdf)
  - Jacobi diagonalization - LHO using a trigonometric basis (notebook, pdf)
Numerical solution of PDEs
- finite-difference method (notes)
  - solution of u_t = u_x using explicit methods (notebook, pdf)
  - solution of u_t = u_xx + u_yy using explicit methods (notebook, pdf)
- finite-element method - model problem and FEM-DVR (notes)
  - 1D Poisson equation solved with FEM-DVR (notebook, pdf)
  - 1D time-dependent Schrödinger equation for free particle solved with FEM-DVR and Crank-Nicolson (notebook, pdf)
  - 1D time-dependent Schrödinger equation for free particle solved with FEM-DVR and generalized Crank-Nicolson (notebook, pdf)
  - 1D time-dependent Schrödinger equation for free particle with CAP solved with FEM-DVR and generalized Crank-Nicolson (notebook, pdf)
- finite-element method - more general formulation and FreeFEM++ (notes)
  - 2D Poisson equation solved with FreeFEM++ (zip)
  - 2D thermal Conduction solved with FreeFEM++ (zip)
  - 2D time-dependent Schrödinger equation for a free particle solved with FreeFEM++ (zip)
  - 2D time-dependent Schrödinger equation for harmonic oscillator solved with FreeFEM++ (zip)
  - 2D time-dependent Schrödinger equation for scattering problem solved with FreeFEM++ (zip)
- Fourier transform and split-operator method for time evolution of quantum systems (notes)
  - Fourier transform via Fast Fourier transfrom (notebook, pdf)
  - Time evolution of quantum systems using the split-operator method (notebook, pdf)

Problems for credits

Finally, I decided to use as a homework problem one of the problems from the Numerical methods I (winter semester) which more or less belongs to the summer semester. Even though the solution of time evolution of 1D quantum system was illustrated during lectures using Mathematica, you should write and debug your own code. Details of the problem in English are here (the Czech version can be found as problem 4 of the winter semester).