Going Further#
Crank-Nicolson#
The Crank-Nicolson method uses a time-centered discretization of the diffusion term:
\[\frac{\phi_i^{n+1} - \phi_i^n}{\Delta x} = \frac{k}{2}
\left (\frac{\phi_{i+1}^n - 2\phi_i^n + \phi_{i-1}^n}{\Delta x^2} +
\frac{\phi_{i+1}^{n+1} - 2\phi_i^{n+1} + \phi_{i-1}^{n+1}}{\Delta x^2}\right )\]
This is still an implicit update, but the coefficients and righthand side in the linear system will be different
State-dependent transport coefficients#
If we have a non-constant conductivity that depends on \(\phi\) itself, then our discretization is more complicated:
\[\frac{\phi_i^{n+1} - \phi_i^n}{\Delta t} =
\frac{ \{ k \nabla \phi \}_{i+1/2} -
\{ k \nabla \phi \}_{i-1/2}}{\Delta x}\]
Now we need \(k\) at the interfaces. There are several ways to do this, which should be motivated by the physics of what the coefficient \(k\) represents:
\[k_{i+1/2} = \frac{1}{2} (k_i + k_{i+1})\]
or
\[\frac{1}{k_{i+1/2}} = \frac{1}{2} \left ( \frac{1}{k_i} + \frac{1}{k_{i+1}} \right )\]
Then a C-N update appears as:
\[\frac{\phi_i^{n+1} - \phi_i^n}{\Delta t} =
\frac{1}{2} \left \{
\nabla \cdot [ k(\phi^n) \nabla \phi^n ]_i +
\nabla \cdot [ k(\phi^{n+1}) \nabla \phi^{n+1} ]_i
\right \}\]
We can update this using a predictor-corrector scheme:
\[\frac{\phi_i^\star - \phi_i^n}{\Delta t} =
\frac{1}{2} \left \{
\nabla \cdot [ k(\phi^n) \nabla \phi^n ]_i +
\nabla \cdot [ k(\phi^n) \nabla \phi^\star ]_i
\right \}\]
\[\frac{\phi_i^{n+1} - \phi_i^n}{\Delta t} =
\frac{1}{2} \left \{
\nabla \cdot [ k(\phi^n) \nabla \phi^n ]_i +
\nabla \cdot [ k(\phi^\star) \nabla \phi^{n+1} ]_i
\right \}\]
This will be second-order accurate in time.