Crank-Nicolson scheme

Time-stepping scheme for the diffusion equation $\partial n/\partial t=D{\partial }^{2}n/\partial z^2$, the average of the explicit and implicit schemes:

$$\frac{n_i^{k+1}-n_i^k}{\Delta t}=\frac{D}{2}\left[\frac{n_{i+1}^{k+1}-2n_i^{k+1}+n_{i-1}^{k+1}}{\Delta z^2}+\frac{n_{i+1}^k-2n_i^k+n_{i-1}^k}{\Delta z^2}\right]$$

where:

• $n_i^k$: value at grid point $i$ and timestep $k$,
• $\Delta z$: grid spacing,
• $\Delta t$: timestep,
• $D$: diffusion coefficient.

Stable for any timestep and second-order accurate in time and space, but long timesteps leave slowly decaying oscillations near sharp gradients. Each timestep is a tridiagonal system, solved with the Thomas algorithm. The explicit scheme is only stable for $\Delta t\le \Delta z^2/2D$.