Lab 19: Tuesday June 3, 2014

• We will go through the notebook $UWHPSC/homeworks/project/Heat_Equation.ipynb, also visible at http://nbviewer.ipython.org/url/faculty.washington.edu/rjl/notebooks/Heat_Equation.ipynb.

  This notebook gives a brief introduction to the heat equation and two numerical methods for its solution, an explicit method and the more stable implicit Crank-Nicolson method.

Some things to try

• You might want to make a copy of this notebook before you start playing with it.
• Experiment with different initial conditions for the heat equation.
• Create an animation (in the notebook) of the numerical solution to the heat equation along with the true solution.
• The Crank-Nicolson method is second order accurate: the error should go to zero like \({\cal O}(\Delta t^2 + \Delta x^2)\) as the grid is refined. So increasing both n (the number of spatial points) and nsteps (the number of time steps) by a factor of 2 should reduce the error by a factor of 4. Test this out.
• Compute or look up the Fourier sine series for some interesting function and try this as initial conditions for the heat equation. Compare the true solution with the numerical solution (where the “true solution” might be estimated by adding up a large but finite number of terms in the Fourier series).
• Try using Sympy to compute the coefficients in the Fourier sine series.

There is quiz for Lab 19