Additional Material

Collection of formulas to be included in the exam

As the title says, this collection of formulas will be included in the exam papers.

Note: there have been a few errors in the first version of this file (mainly: the sign in the Jacobi iteration and an integration variable in the Fourier transform), and still at least one error in the second version (in the last formula for Fourier transforms - shame on me). Thanks for Arne Magnus Ulland, Pavel Palchykov, and Anders Holvik for pointing them out. The above link points you to the third version from November 21.

Crank-Nicolson method for the heat equation

These are the slides from November 17 on the Crank Nicolson method. The equation is the same as below (that is, ut = uxx for 0 < x < 1 and t > 0, with initial conditions u(x,0)=0 and boundary conditions u(0,t)=t and u(1,t)=0). The space discretization uses again h=1/4, and the time discretization uses k=1/10. In contrast to the Euler method (see below), the results appear to be perfectly fine. Indeed one can show that, independent of the step sizes, the Crank-Nicolson method is stable in the sense that no self-reinforcing oscillations occur. It can happen that non-physical solutions occur during the iterations (meaning: the temperature may drop below zero), but these will vanish (or almost vanish) in very short time.

Explicit method for the heat equation

These are the slides from November 14 on solving the heat equation using the Euler method, that is, explicit forward time steps. The equation is ut = uxx for 0 < x < 1 and t > 0, with initial conditions u(x,0)=0 and boundary conditions u(0,t)=t and u(1,t)=0. The step size in space is h=1/4 and the step size in time is k=1/10. One sees clearly that the method becomes unstable and yields unphysical results as soon as in the third time step, when at one point the approximated temperature drops below zero. (Heat equation with Euler method)

Gauss-Seidel iteration for the Poisson equation

Here are the slides presented on November 14 on using the Gauss-Seidel iteration for the numerical solution of the linear system that one obtains when applying the finite difference method to the Poisson equation. The equation in this case was Δu = 1, the region is the unit square, and the grid size h=1/3. (Gauss-Seidel for a Poisson equation)

Gauss-Seidel iteration

Here are the slides with an example of the Gauss Seidel iteration, which I presented in the lecture on October 31.

Gaussian elimination

Here are the slides I presented on the lecture on Gaussian elimination and LU factorization on October 28:

Fixed Point iteration

The following images show what happens if one applies fixed point iteration to the function x = exp(x)/3 (cf. the lecture on 10th October). The figures have been created with this Python script. A very simple Python implementation of fixed point iteration can be found here.

Starting Value No. of Iterations
0 3
1 10
1.5 20
1.55 10

Wave Equation

With the Python script in this link you can produce images of certain solutions of the wave equation (I could not show the videos in class because of some technical problems). The script includes at least some documentation, so it might be possible to follow if you are somehow familiar with Python or similar programming languages. I cannot guarantee that saving the animations will work on all operating systems. You find much more information about Python here.

There is also an amazing youtube video showing a bowed violin string in extreme slow motion - the fundamental frequency of the string in that video must be around 200Hz. In particular, you can see there that almost triangular waves actually appear in nature.

2014-11-21, Markus Grasmair