Homework 7#
Note
You are free to discuss these questions with your classmates and on our class slack, but you must write your own solutions, including your own source code.
All code should be uploaded to Brightspace along with any analytic derivations, notes, etc.
Let’s redo the implicit diffusion solve using a second-order accurate discretization in time: the Crank-Nicolson method. The discretized diffusion equation with this method appears as:
\[\frac{\phi_i^{n+1} - \phi_i^n}{\Delta t} = \frac{k}{2} \left ( \frac{\phi_{i+1}^n - 2 \phi_i^n + \phi_{i-1}^n}{\Delta x^2} + \frac{\phi_{i+1}^{n+1} - 2 \phi_i^{n+1} + \phi_{i-1}^{n+1}}{\Delta x^2} \right )\]The main difference here with what we worked out in class is that the righthand side is now centered in time. If we define:
\[\alpha \equiv \frac{k\Delta t}{\Delta x^2}\]and put all of the “\(n+1\)” terms on one side, we have:
\[-\frac{\alpha}{2} \phi_{i+1}^{n+1} + (1 + \alpha) \phi_i^{n+1} - \frac{\alpha}{2} \phi_{i-1}^{n+1} = \phi_i^n + \frac{\alpha}{2} (\phi_{i+1}^n - 2 \phi_i^n + \phi_{i-1}^n)\]We’ll write the righthand side as \(f_i\):
\[f_i = \phi_i^n + \frac{\alpha}{2} (\phi_{i+1}^n - 2 \phi_i^n + \phi_{i-1}^n)\]Now, just like in class, we need to see what happens at the boundary. For the left most point at the boundary, \(\mathrm{lo}\), with Neumann boundary conditions, we have \(\phi_\mathrm{lo-1} = \phi_\mathrm{lo}\) and the expression becomes:
\[\left ( 1 + \frac{\alpha}{2} \right ) \phi_\mathrm{lo}^{n+1} - \frac{\alpha}{2} \phi_\mathrm{lo+1}^{n+1} = f_\mathrm{lo}\]and similarly at the right boundary. With this, our linear system takes the form:
\[\begin{split}\left ( \begin{array}{ccccccc} 1+\tfrac{\alpha}{2} & -\tfrac{\alpha}{2} & & & & & \\ -\tfrac{\alpha}{2} & 1+\alpha & -\tfrac{\alpha}{2} & & & & \\ & -\tfrac{\alpha}{2} & 1+\alpha & -\tfrac{\alpha}{2}& & & \\ & & \ddots & \ddots & \ddots & & \\ & & & \ddots & \ddots & \ddots & \\ & & & & -\tfrac{\alpha}{2} & 1+\alpha &-\tfrac{\alpha}{2} \\ & & & & & -\tfrac{\alpha}{2} &1+\tfrac{\alpha}{2} \\ \end{array}\right ) \left ( \begin{array}{c} \phi_\mathrm{lo}^{n+1} \\ \phi_\mathrm{lo+1}^{n+1} \\ \phi_\mathrm{lo+2}^{n+1} \\ \vdots \\ \vdots \\ \phi_\mathrm{hi-1}^{n+1} \\ \phi_\mathrm{hi}^{n+1} \\ \end{array} \right ) = \left (\begin{array}{c} f_\mathrm{lo}^{n} \\ f_\mathrm{lo+1}^{n} \\ f_\mathrm{lo+2}^{n} \\ \vdots \\ \vdots \\ f_\mathrm{hi-1}^{n} \\ f_\mathrm{hi}^{n} \\ \end{array} \right ) \end{split}\]Your task: Solve this system by modifying the direct solve example from class (or implementing your own version). Alternately, you can choose to do it via relaxation, again following what we did in class as a guide.
Vary the resolution, and demonstrate that this method converges better than the first-order accurate (in time) method that we implemented in class.