PROJECT II

Modify and test--checking errors and convergence as before--either the C or Matlab code to solve ONE of the following problems, using a semi-implicit method:

1. Solve the advection equation for a Gaussian pulse; what instabilities do you observe?

   $$\frac{\partial }{\partial t} A(x,t) = -v \frac{\partial }{\partial x} A.$$ (5.3)

   
   
   What results do you get for the nonlinear viscid Burger's equation? Note: this is a numerically challenging problem! Try parameters of $\kappa = 0.1$, $v = 1$, with periodic boundary conditions and an initial `square profile' of traffic bunched together, with a uniform density of $A_0 = 1$. You will need to iterate the nonlinear differential operators to obtain the central derivatives. Can you explain the phenomena that you see?
2. Solve the Schrödinger equation for a Gaussian pulse scattering off a square potential well, with parameters chosen so that there is a large reflection. It is best to start the pulse near a window boundary, moving inwards to a potential well at the center of the window:

   $$i \frac{\partial }{\partial t} \psi(x,t) = -\kappa \frac{\partial ^2}{\partial x^2} \psi(x,t) + V(x) \psi(x,t).$$ (5.4)

   
   
   Explain the results in terms of resonant behaviour and interference for the waves created inside the potential well.
3. Solve the nonlinear Schrödinger equation which describes solitons in optical fiber pulse propagation, using an initial condition of form $\mrm{sech}(x)$, where:

   $$i \frac{\partial }{\partial t} \psi(x,t) = -\kappa \frac{\partial ^2}{\partial x^2} \psi(x,t) - \gamma \psi(x,t) \vert\psi(x,t)\vert^2.$$ (5.5)

   
   
   In this example, there is an exact solution of form $\mrm{sech}(x) \exp (i \alpha t)$, for testing purposes. It is best to start with the exact analytic form in order to test the code. You will have to choose the appropriate $\kappa$ and $\gamma$ value so the given input is a soliton, which has an invariant shape. You should check that colliding solitons are unchanged in a collision; moving solitons require a phase modulation of form $\mrm{sech}(x) \exp(\pm i v x)$.

It will be necessary to either solve a tri-diagonal matrix equation to obtain the required results, or to use the split-step FFT method, which involves periodic boundary conditions. This is especially useful in higher dimensions. Note that the semi-implicit method is much more useful in practise than the first project, especially for nonlinear equations, where you can try larger initial amplitudes (for example, twice or three times the soliton amplitude), as a more stringent test. This is a nontrivial exercise, as there is a nonlinear term to be treated implicitly. Ask for help from a demonstrator, if you have problems with this.

In each case, your report should include:

• A general description of the problem, and the physics background.
• A code that you wrote and tested, with test output and error-plot using an analytically soluble case.
• Table of maximum errors with various step-sizes.
• Plots of physically relevant outputs, with a conclusion about the physics and the numerical method.
• The source code should also be attached to an email to cochrane@physics.uq.edu.au, along with the relevant input parameters used to generate the plots.